From f32f962d7729e93217bc52885b579019ab8deb6c Mon Sep 17 00:00:00 2001 From: Alessandro Cosentino Date: Mon, 6 Jul 2015 13:49:00 -0400 Subject: [PATCH] Base draft --- .gitignore | 62 +++ README.md | 0 conclusion.tex | 85 +++ discrimination.tex | 616 +++++++++++++++++++++ drawin_dualg.pdf | Bin 0 -> 1347 bytes drawin_dualg.pdf_tex | 58 ++ drawin_dualg.svg | 147 +++++ drawing.pdf | Bin 0 -> 1348 bytes drawing.pdf_tex | 58 ++ drawing.svg | 147 +++++ drawing_super.pdf | Bin 0 -> 1347 bytes drawing_super.pdf_tex | 59 ++ drawing_super.svg | 155 ++++++ drawing_symm.pdf | Bin 0 -> 2029 bytes drawing_symm.pdf_tex | 59 ++ drawing_symm.svg | 166 ++++++ drawing_symm_dual.pdf | Bin 0 -> 2033 bytes drawing_symm_dual.pdf_tex | 61 ++ drawing_symm_dual.svg | 201 +++++++ header.tex | 227 ++++++++ introduction.tex | 199 +++++++ matlab_notebook.tex | 73 +++ mes.tex | 1103 +++++++++++++++++++++++++++++++++++++ preliminaries.tex | 714 ++++++++++++++++++++++++ programs.tex | 1099 ++++++++++++++++++++++++++++++++++++ thesis-blx.bib | 11 + thesis-front.tex | 222 ++++++++ thesis.bib | 1038 ++++++++++++++++++++++++++++++++++ thesis.brf | 106 ++++ thesis.pdf | Bin 0 -> 572357 bytes thesis.run.xml | 89 +++ thesis.tex | 257 +++++++++ ups.tex | 527 ++++++++++++++++++ 33 files changed, 7539 insertions(+) create mode 100644 .gitignore create mode 100644 README.md create mode 100644 conclusion.tex create mode 100644 discrimination.tex create mode 100644 drawin_dualg.pdf create mode 100644 drawin_dualg.pdf_tex create mode 100644 drawin_dualg.svg create mode 100644 drawing.pdf create mode 100644 drawing.pdf_tex create mode 100644 drawing.svg create mode 100644 drawing_super.pdf create mode 100644 drawing_super.pdf_tex create mode 100644 drawing_super.svg create mode 100644 drawing_symm.pdf create mode 100644 drawing_symm.pdf_tex create mode 100644 drawing_symm.svg create mode 100644 drawing_symm_dual.pdf create mode 100644 drawing_symm_dual.pdf_tex create mode 100644 drawing_symm_dual.svg create mode 100644 header.tex create mode 100644 introduction.tex create mode 100644 matlab_notebook.tex create mode 100644 mes.tex create mode 100644 preliminaries.tex create mode 100644 programs.tex create mode 100644 thesis-blx.bib create mode 100644 thesis-front.tex create mode 100644 thesis.bib create mode 100644 thesis.brf create mode 100644 thesis.pdf create mode 100644 thesis.run.xml create mode 100644 thesis.tex create mode 100644 ups.tex diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..749edaf --- /dev/null +++ b/.gitignore @@ -0,0 +1,62 @@ +# Created by http://www.gitignore.io + +### LaTeX ### +*.acn +*.acr +*.alg +*.aux +*.bbl +*.blg +*.dvi +*.fdb_latexmk +*.glg +*.glo +*.gls +*.idx +*.ilg +*.ind +*.ist +*.lof +*.log +*.lot +*.maf +*.mtc +*.mtc* +*.nav +*.nlo +*.out +*.pdfsync +*.ps +*.snm +*.synctex.gz +*.toc +*.vrb +*.xdy +*.tdo + +*.mtc* + +### Jabref ### +*.bib.bak + +### Matlab ### +##--------------------------------------------------- +## Remove autosaves generated by the Matlab editor +## We have git for backups! +##--------------------------------------------------- + +# Windows default autosave extension +*.asv + +# OSX / *nix default autosave extension +*.m~ + +# Compiled MEX binaries (all platforms) +*.mex* + +# Simulink Code Generation +slprj/ + +### SublimeText +*.sublime-project +*.sublime-workspace diff --git a/README.md b/README.md new file mode 100644 index 0000000..e69de29 diff --git a/conclusion.tex b/conclusion.tex new file mode 100644 index 0000000..6cea929 --- /dev/null +++ b/conclusion.tex @@ -0,0 +1,85 @@ +%!TEX root = thesis.tex +%---------------------------------------------------------------------- +\chapter{Conclusions and open problems} +\label{chap:conclusions} +%---------------------------------------------------------------------- + +In this thesis we have used techniques from convex optimization to study +the limitations of LOCC, separable, and PPT measurements for the task of +distinguishing sets of bipartite states. Compared to previous approaches, +our techniques turned out to be effective in providing precise bounds on +the maximum probability of locally distinguishing some interesting sets +of maximally entangled states and unextendable product sets. + +Several specific questions regarding the local distinguishability of sets of +bipartite states remain unsolved. + +In Chapter \ref{chap:mes} we proved a tight bound on the entanglement +cost of discriminating sets of Bell states by means of LOCC protocols. +One could ask the following more general question. +\begin{question} +How much entanglement does it cost to distinguish +maximally entangled states in $\complex^{n}\otimes\complex^{n}$? +\end{question} + +Ghosh et al.~\cite{Ghosh04} have shown that orthogonal maximally +entangled states, which are in canonical form, can always be discriminated, +by means of LOCC protocols, if two copies of each of the states are provided. +One could ask if two copies are always sufficient. In fact, this question is open +even for separable and PPT measurement. +\begin{question} +Are two copies sufficient to discriminate any set of orthogonal pure states +by PPT measurements? +\end{question} + +The techniques presented in the paper are not intrinsically limited +to the setting of bipartite pure states, and applications of these techniques to +the \emph{multipartite} setting are topics for possible future work. + +Global distinguishability of random states was studied by A. Montanaro \cite{Montanaro07}. +More precisely, he put a lower bound on the probability of distinguishing +(by global measurements) an ensembles of $n$ random quantum states in $\complex^{d}$, +in the asymptotic regime where $n/d$ approaches a constant. A similar question +on the distinguishability of random states by PPT and separable measurements +could be investigated by using the convex optimization approach developed in the thesis. + +Apart from quantum states, one could also study the LOCC distinguishability of +quantum operations \cite{Matthews10}. It is a topic that has not been studied as throughly +as in the case of states, but once again, one could approach it through +the lens of convex optimization. + +A more speculative project, yet very exciting, is to give a somewhat useful +characterization of the set dual to the set of LOCC measurement and its corresponding +set of linear mappings (both labeled by a question mark in the diagrams of Figure \ref{fig:measurements-dual}). +As we do not have a nice characterization of the LOCC set itself, we suppose +this is a difficult project. One direction to approach this problem +can be to consider smaller sets that are contained in the LOCC set, such as +one-way LOCC, where the communication is only in one direction, say from +Alice to Bob, or LOCC-$r$ , in which the communication is limited to $r$ rounds. +Let us fantasize we had a characterization of the set dual to the set of LOCC measurement +and let us denote it as $\Meas_{\LOCC}^{\ast}(N, \X:\Y)$. Then we could plug it in the +following cone program and proceed as we did for all the cone programs analyzed +on this thesis. +\begin{center} +\underline{Dual (LOCC measurements)} + \begin{equation} + \label{eq:locc-dual-problem} + \begin{split} + \text{minimize:} \quad & \tr(H)\\ + \text{subject to:} + \quad & + \begin{pmatrix} + H - p_{1}\rho_{1} & & \\ + & \ddots & \\ + & & H - p_{N}\rho_{N} + \end{pmatrix}\in \Meas_{\LOCC}^{\ast}(N, \X:\Y),\\ + \quad & H \in \Herm(\X\otimes\Y). + \end{split} + \end{equation} +\end{center} + +Finally, apart from \cite{Gharibian13}, I am not aware of any results where +cone programming is explicitly used in quantum computing for any cones different than the +cone of semidefinite operators. +I hope this work helps toward the rise of more applications of cone programming +in quantum information theory. diff --git a/discrimination.tex b/discrimination.tex new file mode 100644 index 0000000..0d28be2 --- /dev/null +++ b/discrimination.tex @@ -0,0 +1,616 @@ +%!TEX root = thesis.tex +%------------------------------------------------------------------------------- +\chapter{Bipartite state discrimination} +\label{chap:bipartite-state-discrimination} +%------------------------------------------------------------------------------- + +This chapter introduces the problem of discriminating quantum states +from a known set. +A scenario describing the problem is presented, first for the case where the +unknown state is given to a single individual, and then generalized to a +different scenario where the unknown state is distributed to two parties. +In this thesis we will focus on the bipartite case, leaving for future work +an extension of the results to the multipartite case. + +After the problem description, this chapter reviews relevant background work, +including prior results on the local distinguishability of +two classes of pure states that will be the object of study in the following +chapters: maximally entangled states and product states. + +\minitoc + +%------------------------------------------------------------------------------- +\section{Problem description} +%------------------------------------------------------------------------------- + +\subsection*{Global state discrimination} +An instance of the state discrimination problem is defined by a complex +Euclidean space $\X$, a positive integer $N$, and by an ensemble $\E$ of $N$ +states, that is, +\begin{equation} +\label{eq:global-ensemble} + \E = \big\{ (p_{1}, \rho_{1}), \ldots, (p_{N}, \rho_{N}) \big\}, +\end{equation} +where $(p_{1}, \ldots, p_{N})$ is a probability vector +and $\rho_{1}, \ldots, \rho_{N} \in \Density(\X)$ are density operators representing +quantum states. +We denote the set of all ensemble of this kind by $\Ens(N, \X)$. + +The problem is formally described by the following +scenario, which involves two individuals, Alice and Charlie (the reader who misses +Bob can be reassured that he will join us soon). +Charlie picks an index +\[ + k \in \{1, \ldots, N \}, +\] +according to the probability distribution $(p_{1}, \ldots, p_{N})$, +prepares a quantum register $\reg{X}$ with the state $\rho_{k} \in \Pos(\X)$, +and sends it to Alice. +Alice's task it to identify the index $k$ by performing a measurement on the +register $\reg{X}$. + +For Alice performing a measurement +\begin{equation} +\label{eq:global-measurement} +\mu : \{1, \ldots, N\} \rightarrow \Pos(\X), +\end{equation} + the probability that she correctly distinguishes $\E$ is given by the expression +\begin{equation} +\label{eq:opt-value-fixed-measurement} + \opt(\E, \mu) = \sum_{k=1}^{N}p_{k}\ip{\mu(k)}{\rho_{k}}. +\end{equation} +Since $\mu$ is a measurement and $\rho_{1}, \ldots, \rho_{N}$ are density +operators, it is clear that +\begin{equation} +\label{eq:ip-bounds} + 0 \leq \ip{\mu(k)}{\rho_{k}} \leq 1, +\end{equation} +for each $k \in \{1, \ldots, N \}$. +Moreover, since $p$ is a probability vector, we have that +\begin{equation} + 0 \leq \opt(\E, \mu) \leq 1. +\end{equation} +We denote by $\opt(\E)$ the maximum probability of distinguishing $\E$ for any +possible measurement, that is, +\begin{equation} +\label{eq:opt-value} + \opt(\E) = \max_{\mu\in\Meas(N, \X)}\opt(\E, \mu). +\end{equation} +We say that $\E$ is \emph{distinguishable with probability at least $p$} if we +have $\opt(\E) \geq p$. Whenever $\E$ is distinguishable with probability $1$, we say +that $\E$ is \emph{perfectly distinguishable} or that Alice can distinguish +$\E$ \emph{with certainty}. + +The probability distribution with which the states are selected is not important +if we are only interested in perfect distinguishability. +In fact, from the bounds in Eq.~\eqref{eq:ip-bounds} and a standard convexity argument, +for any probability vector $p = (p_{1}, \ldots, p_{N})$, it holds that +\begin{equation} + \sum_{k=1}^{N}p_{k}\ip{\mu(k)}{\rho_{k}} = 1 +\end{equation} +if and only if +\begin{equation} + \ip{\mu(k)}{\rho_{k}} = 1, +\end{equation} +for each $k \in \supp(p)$. +For this reason, whenever we are only interested in a qualitative result +(whether perfect distinguishability holds or not), we will take $p$ to be the uniform +distribution, that is, $p = (1/N, \ldots, 1/N)$. In such cases, we will simply +denote the ensemble by the list of its states, that is, +\begin{equation} + \E = \{\rho_{1}, \ldots, \rho_{N}\}. +\end{equation} + +We will often be interested in the distinguishability of pure-state ensembles, +where each density operator is a rank-one projector, that is, for each +$k \in \{1, \ldots, N\}$, $\rho_{k} = u_{k}u_{k}^{\ast}$, +for a list of unit vectors $\{u_{1}, \ldots, u_{N}\} \in \X$. +In such case, by a further abuse of notation, we simply denote the ensemble by +a list of its vectors: +\begin{equation} + \E = \{u_{1}, \ldots, u_{N}\}. +\end{equation} + +If the states are mutually orthogonal, that is, +\begin{equation} +\ip{\rho_{i}}{\rho_{j}} = 0, +\end{equation} +for all $i,j \in \{1, \ldots, N\}$ with $i \neq j$, there is a measurement +that perfectly distinguish them. +Consider the spectral decomposition of each state: +\begin{equation} + \rho_{k} = \sum_{i=1}^{r_{k}}\lambda_{i}x_{k,i}x_{k,i}^{\ast}, +\end{equation} +with $\lambda_{1}, \ldots, \lambda_{r_{k}}$ being positive real numbers +and $\{x_{k,1}, \ldots, x_{k,r_{k}}\} \subset \X$ being an orthonormal set. +Alice can then construct the quantum measurement that perfectly distinguish +the states, by defining the following measurement operators: +\begin{equation} + \mu(k) = \sum_{i=1}^{r_{k}}x_{k,i}x_{k,i}^{\ast}, +\end{equation} +for each $k = 1,\ldots,N-1$, and +\begin{equation} + \mu(N) = \I - \sum_{k=1}^{N-1}\mu(k). +\end{equation} +It is clear that $\mu$ is a valid measurement and that +\begin{equation} + \ip{\mu(k)}{\rho_{k}} = 1, +\end{equation} +for all $k \in \{1, \ldots, N\}$. + +The distinguishability of non-orthogonal states is a prolific topic on its own and +a treatment of it is outside the scope of this thesis. +For an exposition of results regarding global state discrimination, +the reader is referred to numerous surveys on the topic \cite{Chefles2000, Bergou2004}. + +\subsection*{Bipartite state discrimination} +This dissertation focuses on a modification of the above scenario in which we +have three individuals involved: Alice, Bob, and Charlie. +In this new scenario, the states to be distinguished lie on the tensor product of +two complex Euclidean spaces, which we label by $\X$ and $\Y$ and which are +held respectively by Alice and Bob. +In other words, the ensemble consists of $N$ bipartite states represented +by the density matrices $\rho_{1}, \ldots, \rho_{N} \in \Density(\X\otimes\Y)$. + +Charlie picks an index $k \in \{1, \ldots, N\}$ and prepares the corresponding +state $\rho_{k} \in \Density(\X\otimes\Y)$ on a pair of quantum registers +$(\reg{X},\reg{Y})$ that belong to Alice and Bob, in the sense that the +underlying space is $\X\otimes\Y$. +Their task is to identify the index $k$ chosen by Charlie, by means of an LOCC +measurement on $(\reg{X},\reg{Y})$. + +We will denote by $\opt_{\LOCC}(\E)$ the maximum success probability for +Alice and Bob to distinguish an ensemble $\E\in\Ens(N, \X\otimes\Y)$ by means of +any LOCC measurement, that is, +\begin{equation} +\label{eq:opt-LOCC} + \opt_{\LOCC}(\E) = \max_{\mu \in \Meas_{\LOCC}(N, \X:\Y)}\opt(\E, \mu). +\end{equation} + +As it was discussed in Section \ref{sec:quantum-measurements}, the set of +LOCC measurements has a complex mathematical structure. +For this reason, the state discrimination problem has been analyzed for more +tractable classes of measurements that approximate, in some sense, +the LOCC measurements. +Among these, the classes of separable and PPT measurements are the most studied, +because of their nice mathematical and computational properties. + +We denote by $\opt_{\Sep}(\E)$ and $\opt_{\PPT}(\E)$ the optimal probability of +distinguishing an ensemble +$ + \E \in \Ens(N, \X\otimes\Y) +$ +by separable and PPT measurements, respectively: +\begin{equation} +\label{eq:opt-Sep} + \opt_{\Sep}(\E) = \max_{\mu \in \Meas_{\Sep}(N, \X:\Y)}\opt(\E, \mu), +\end{equation} +and +\begin{equation} +\label{eq:opt-PPT} + \opt_{\PPT}(\E) = \max_{\mu \in \Meas_{\PPT}(N, \X:\Y)}\opt(\E, \mu). +\end{equation} + + +In lights of the containments pictured by the diagram in Figure ..., +we have the following chain of inequalities: +\begin{equation} +\label{eq:inequality-chain} + \opt_{\LOCC}(\E) \leq \opt_{\Sep}(\E) \leq \opt_{\PPT}(\E) \leq \opt(\E), +\end{equation} +for any ensemble $\E$. + +Interestingly, for each of these inequalities, there exists a set of states +that makes the inequality strict. +In the rest of this chapter we will see some examples for which the separation +is achieved. + +In most of our examples and in most prior works on bipartite state discrimination, +the states are taken to be pure and orthogonal, so that a global measurement can +trivially discriminate them with certainty, that is, $\opt(\E) = 1$. +In such cases, a separation between $\opt(\E)$ and, say, $\opt_{\LOCC}(\E)$ is obtained +by showing that the set of states is not perfectly distinguishable by LOCC measurements. + + +\section{Discriminating between pairs of states} +A case of particular interest is when there are two states to be distinguished, +chosen with equal probability. +This is equivalent to the quantum data hiding challenge in which a secret bit +$b \in \{0,1\}$ is required to be hidden into a bipartite state +$\sigma_{b}\in\Density(\X\otimes\Y)$. In the language of the previous section, +we say that quantum data hiding is possible if there exists an ensemble +\begin{equation} +\E = \left\{\left(\frac{1}{2},\sigma_{0}\right), + \left(\frac{1}{2},\sigma_{1}\right)\right\} +\end{equation} +such that two conditions are simultaneously satisfied: +\begin{itemize} +\item[(a)] $\opt(\E) = 1$, and +\item[(b)] $\opt_{\LOCC}(\E) \leq 1/2 + \varepsilon$, +\end{itemize} +for some ``small'' values of $\varepsilon$. The exact bounds on +$\varepsilon$ define the strength of the hiding scheme and, +of course, is depending on the dimensions of Alice's and Bob's spaces. + +The condition (a) above is equivalent to requiring the two states to be +orthogonal\footnote{We could be more general here and define another parameter +$\delta \approx 0$ so to weaken Condition (a) to be $\opt(\E) \geq 1 - \delta$. +This would not affect the discussion that follows, except for making the +presentation less clean.}. +A consequence of this is that at least one of them must be a mixed, +since a result by Walgate, et al. \cite{Walgate00} shows that any two orthogonal +bipartite \emph{pure} states can be perfectly distinguished by LOCC. + +The problem of discriminating between two quantum states is also +interesting for its connection with operator norms. +In particular, a connection between the trace norm and the +optimal probability of distinguishing two states by means of global measurements +is estabilished by the following theorem. +\begin{theorem}[Holevo-Helstrom] +Given a complex Euclidean space $\X$ and two density operators +$\sigma_{0}, \sigma_{1} \in \Density(\X)$, it holds that + \begin{equation} + \opt(\{\sigma_{0},\sigma_{1}\}) = + \frac{1}{2} + \frac{1}{4}\norm{\sigma_{0}-\sigma_{1}}_{1}. + \end{equation} +\end{theorem} +By reversing the logic direction of this theorem, one can define operator norms +starting from different set of measurements. +This approach was taken in \cite{Matthews09}, where the so-called $\LOCC$-norm +was defined so that the following holds: +\begin{equation} + \opt_{\LOCC}(\{\sigma_{0},\sigma_{1}\}) = + \frac{1}{2} + \frac{1}{4}\norm{\sigma_{0}-\sigma_{1}}_{\LOCC}. +\end{equation} + +Similarly, one may define norms $\norm{\cdot}_{\PPT}$ and $\norm{\cdot}_{\Sep}$ +that correspond to distinguishability by PPT and separable measurements, +respectively. + +\begin{example}[Werner hiding pairs] +\label{example:werner-hiding-pairs} + +One typical quantum data hiding scheme \cite{Terhal01a,DiVincenzo2002} encodes +the hidden classical bit in a \emph{Werner hiding pair}. +For any positive integer $n \geq 2$, let +$W_{n} \in \Unitary(\complex^{n}\otimes\complex^{n})$ be the swap operator +defined in Eq.~\eqref{eq:swap-operator}. +A Werner hiding pair in $\complex^{n}\otimes\complex^{n}$ is defined by two states +\begin{equation} + \label{eq:werner-states} + \sigma_{0}^{(n)} = \frac{\I\otimes\I + W_{n}}{n(n+1)} + \hspace*{1cm}\mbox{and}\hspace*{1cm} + \sigma_{1}^{(n)} = \frac{\I\otimes\I - W_{n}}{n(n-1)}. +\end{equation} +% and let $\E^{(n)} = \{\sigma_{0}^{(n)},\sigma_{1}^{(n)}\}$. +Notice that $\sigma_{0}^{(n)}$ and $\sigma_{1}^{(n)}$ are also the normalized +projections on the symmetric and antisymmetric subspace, respectively. +From the orthogonality of the two states, we have +\begin{equation} + \opt(\E^{(n)}) = 1, +\end{equation} +for any $n$, or equivalently +\begin{equation} + \norm{\sigma_{0}^{(n)} - \sigma_{1}^{(n)}}_{1} = 2. +\end{equation} +In the next chapter we show that +\begin{equation} + \opt_{\PPT}(\E^{(n)}) \leq \frac{1}{2} + \frac{1}{n+1}, +\end{equation} +and therefore this is an example of a set of states that makes the rightmost +inequality in Eq. +Since there is an LOCC measurement that achieves the bound +\cite{DiVincenzo2002}, we also have +\begin{equation} + \opt_{\LOCC}(\E^{(n)}) = \opt_{\Sep}(\E^{(n)}) = \opt_{\PPT}(\E^{(n)}) = + \frac{1}{2} + \frac{1}{n+1}, +\end{equation} +or equivalently +\begin{equation} + \norm{\sigma_{0}^{(n)} - \sigma_{1}^{(n)}}_{\LOCC} = + \norm{\sigma_{0}^{(n)} - \sigma_{1}^{(n)}}_{\Sep} = + \norm{\sigma_{0}^{(n)} - \sigma_{1}^{(n)}}_{\PPT} = \frac{4}{n+1}. +\end{equation} +\end{example} + + +% \subsection{A maximally entangled state and its orthogonal complement} + +% \begin{equation} +% u = \frac{1}{\sqrt{n}}\sum_{j=1}^{n}e_{j}\otimes e_{j} +% \end{equation} + +% \begin{equation} +% \sigma_{0} = uu^{\ast} +% \hspace*{2cm} +% \sigma_{1} = \frac{1}{n^{2}-1}(\I\otimes\I - uu^{\ast}) +% \end{equation} + + +\section{Discrimination of maximally entangled states} +\label{sec:mes-intro} + +When investigating any kind of problem, a typical computer science approach is +to bring the operating parameters of the problem to one extreme. +In order to get a better understanding of the role played by entanglement in +bipartite state distinguishability problems, one can restrict their attention to +the case in which the sets to be distinguished consist of orthogonal +maximally entangled pure states. +Considering states that are maximally entangled, as opposed to partially +entangled, is useful to reduce the number of variables that need to be taken into account, and +it helps to have neater problem statements. It makes the problem easier to handle mathematically +(recall that maximally entangled states are in a one-to-one correspondence with +unitary operators) and, at the same time, it constitutes an edge case that is +interesting from the physical point of view. The reason to consider maximally entangled states +can be summarized into one question: why bother with +more complicated cases when we do not even know how to deal with that? + +In this section, some known results on the distinguishability of +maximally entangled states by LOCC, separable, and PPT measurements are reviewed, +whereas new results are presented in Chapter \ref{chap:mes}. + +The simplest example of a set of LOCC-indistinguishable maximally entangled +states is the standard $2$-qubit Bell basis (Eq.~\eqref{eq:Bell-states}). +It turns out that the maximum probability of distinguishing these $4$ states, +for any LOCC measurement, is $1/2$ \cite{Ghosh01}. In fact, a similar bound +holds more in general: if we are given an equally probable ensemble of $N$ +orthogonal maximally entangled states in $\complex^{n}\otimes\complex^{n}$, +the maximum probability of distinguishing them by LOCC is $n/N$ \cite{Ghosh04}. +This bound holds even for the wider class of separable \cite{Duan09} and PPT +measurements \cite{Yu12}. In Chapter \ref{chap:mes} this result is re-proved +using our cone programming framework. + +The assumption on the sets consisting entirely of maximally entangled states is +particularly significant when we inquire the question of how the size of LOCC-indistinguishable +sets relates to the local dimension of each of Alice's and Bob's subsystems. +In fact, if we allow states that are not maximally entangled to be in the set, +we can construct indistinguishable sets with a fixed size in any dimension we like. +Indeed, whenever we find a set of indistinguishable maximally entangled states +for certain local dimensions, those states remain indistinguishable when embedded +in any larger local dimensions. Nonetheless they are no longer maximally +entangled with respect to the new larger local dimensions. + +Whereas this shows that any set of $N > n$ orthogonal maximally entangled states +can never be locally distinguished with certainty, it leaves open the question +whether there exist sets of $N \leq n$ indistinguishable orthogonal maximally +entangled states in $\complex^{n}\otimes\complex^{n}$. +An answer to this question for the particular case of $n = 3 = N$ was given by +Nathanson \cite{Nathanson05}, who showed that any three orthogonal maximally +entangled states in $\complex^{3}\otimes\complex^{3}$ can be perfectly +distinguished by LOCC. In a followup work \cite{Nathanson13}, +Nathanson proved that $3$ maximally entangled states +in $\complex^{n}\otimes\complex^{n}$, for any $n \leq 3$, are always perfectly +distinguishable by PPT. + +Several results aimed at filling the landscape for the case $3 < N \leq n$. +For the weaker model of \emph{one-way} LOCC protocols, +Bandyopadhyay et al. \cite{Bandyopadhyay11a} showed some explicit examples of +indistinguishable sets of states with the size of the sets being equal to the +dimension of the subsystems, i.e., $N = n$. The states they use for those +examples lie on systems whose local dimension is $n = 4, 5, 6$. +Later in Chapter \ref{chap:mes}, we go back to these examples and give +numerical evidence that the same sets of states cannot be distinguished with +certainty even if we allow the parties to perform PPT measurements. + +Recently, Yu et al. \cite{Yu12} presented the first example of $N$ maximally +entangled states in $\complex^{N}\otimes\complex^{N}$ that cannot be +perfectly distinguished by any PPT measurement, and therefore by any general LOCC protocols. +\begin{equation} + \label{eq:ydy_states} + \begin{aligned} + \ket{\phi_1} & = + \frac{1}{2} \ket{0}\ket{0} + + \frac{1}{2} \ket{1}\ket{1} + + \frac{1}{2} \ket{2}\ket{2} + + \frac{1}{2} \ket{3}\ket{3} \\[2mm] + \ket{\phi_2} & = + \frac{1}{2} \ket{0}\ket{3} + + \frac{1}{2} \ket{1}\ket{2} + + \frac{1}{2} \ket{2}\ket{1} + + \frac{1}{2} \ket{3}\ket{0} \\[2mm] + \ket{\phi_3} & = + \frac{1}{2} \ket{0}\ket{3} + + \frac{1}{2} \ket{1}\ket{2} - + \frac{1}{2} \ket{2}\ket{1} - + \frac{1}{2} \ket{3}\ket{0} \\[2mm] + \ket{\phi_4} & = + \frac{1}{2} \ket{0}\ket{1} + + \frac{1}{2} \ket{1}\ket{0} - + \frac{1}{2} \ket{2}\ket{3} - + \frac{1}{2} \ket{3}\ket{2} + \end{aligned} +\end{equation} +Later in Chapter \ref{chap:mes} we turn their result into a quantitative one, +by showing that PPT measurements can only succeed with probability at most $7/8$ +and that $3/4$ is a tight bound on the probability of distinguishing these states +by separable (and LOCC) measurements. + +Yet another tile in the landscape of maximally entangled states distinguishability +is a result by Fan \cite{Fan04}, for which any $N$ generalized +Bell states in $\complex^{n}\otimes\complex^{n}$ can be perfectly distinguished +by LOCC whenever +\[ + \binom{N}{2}\leq n. +\] + +Table \ref{table:max-ent-states} summarizes known results about the +distinguishability of maximally entangled states by LOCC and PPT +measurements, and compares them with the results obtained in this thesis +(highlighted in gray). + +\begin{table}[!ht] +\centering +\def\arraystretch{1.5} +\begin{tabular}{|c|c|c|c|} + \hline + & PPT & LOCC & References\\ + \hline \hline + $N = 2$ & --- & \emph{all} dist. & \cite{Walgate00}\\ + \hline + $N = 3 = n$ & --- & \emph{all} dist. & \cite{Nathanson05}\\ + \hline + $N = 4 = n$ & \emph{some} indist. & --- & \cite{Yu12}\\ + \hline + \rowcolor{Gray} + $N = n > 4$ & \emph{some} indist. & --- & \cite{Cosentino13}\\ + \hline + \rowcolor{Gray} + $4 < N < n$ & \emph{some} indist. & --- & \cite{Cosentino14,Bandyopadhyay15}\\ + \hline + $N > n$ & \emph{all} indist. & --- & \cite{Yu12,Duan09,Ghosh04} \\ + \hline +\end{tabular} +\caption{Distinguishability of maximally entangled states} +\label{table:max-ent-states} +\end{table} + +\section{Discrimination of product sets} + +Indistinguishability by LOCC is not a prerogative of entangled states. +The famous \emph{domino state} set of \cite{Bennett99}, for example, is a +collection of orthogonal \emph{product} states that cannot be perfectly +discriminated by LOCC protocols. +In this example, the local dimension of the states is $3$, and one takes +$N = 9$, $p_1 = \cdots = p_9 = 1/9$, and +{\setlength{\arraycolsep}{2.5mm}% +\begin{equation} \label{eq:domino} +\begin{array}{ll} + \multicolumn{2}{c}{\ket{\phi_1} = \ket{1}\ket{1},}\\[4mm] + \ket{\phi_2} = \ket{0}\left(\frac{\ket{0} + \ket{1}}{\sqrt{2}}\right), + & \ket{\phi_3} = \ket{2}\left(\frac{\ket{1} + \ket{2}}{\sqrt{2}}\right), + \\[4mm] + \ket{\phi_4} = \left(\frac{\ket{1} + \ket{2}}{\sqrt{2}}\right)\ket{0}, + & \ket{\phi_5} = \left(\frac{\ket{0} + \ket{1}}{\sqrt{2}}\right)\ket{2}, + \\[4mm] + \ket{\phi_6} = \ket{0}\left(\frac{\ket{0} - \ket{1}}{\sqrt{2}}\right), + & \ket{\phi_7} = \ket{2}\left(\frac{\ket{1} - \ket{2}}{\sqrt{2}}\right), + \\[4mm] + \ket{\phi_8} = \left(\frac{\ket{1} - \ket{2}}{\sqrt{2}}\right)\ket{0}, + & \ket{\phi_9} = \left(\frac{\ket{0} - \ket{1}}{\sqrt{2}}\right)\ket{2}. +\end{array} +\end{equation} +}% +A rather complicated argument demonstrates that this collection cannot be +discriminated by LOCC with probability greater than $1 - \varepsilon$ for some choice +of a positive real number $\varepsilon$. +(A simplified proof appears in \cite{Childs13}, where this fact +is proved for $\varepsilon = 1.9 \times 10^{-8}$.) + + + +\section{Entanglement cost of state discrimination} + +As explained in the introduction, three Bell states given with uniform +probabilities can be discriminated by separable measurements with success +probability at most 2/3, while four Bell states can be discriminated with success +probability at most 1/2. +These bounds can be obtained by a fairly trivial selection of LOCC +measurements, and can be shown to hold even for PPT measurements. + +On the other hand, if the parties are given a maximally entangled bit as a resource, +then they can perform a teleportation protocol to send each other their respective parts of +the Bell pair they have been asked to identify. +The set of Bell states constitutes an example of a set that is distinguishable only if we +are willing to consume some entanglement (given as an additional resource) or, in other words, +we say that the entanglement cost of distinguishing the Bell states with certainty is bigger than zero. + +The entanglement cost of quantum operations and measurements, within the +paradigm of LOCC, has been considered previously. +For instance, \cite{Cohen08} studied the entanglement cost of perfectly +discriminating elements of unextendable product sets by LOCC measurements. +Interestingly, his work presents some protocol where entanglement is used more +efficiently than in standard teleportation protocols. +In later work, \cite{Bandyopadhyay09} and \cite{Bandyopadhyay10} considered the +entanglement cost of measurements and established lower bounds on the amount of +entanglement necessary for distinguishing complete orthonormal bases of two +qubits. + +Our work on the entanglement cost of Bell states was inspired by a question left open by Yu, Duan, and Ying, who considered the entanglement cost of state discrimination problems by PPT and separable measurements \cite{Yu14}. + + +%------------------------------------------------------------------------------- +\section{Previous approaches} +%------------------------------------------------------------------------------- + +In the results roundup of the previous sections, we summarized ``positive'' +results, in which it is shown that a certain probability of success can be obtained by +some measurement, as well as ``negative'' results, for which an upper bound +on the probability of success is shown for any measurement in a certain class. + +To prove the first kind of results, one needs to show a protocol (for LOCC), +or a collection of measurement operators (for PPT and separable) that achieves +the given probability. +Some protocols/measurements might be complicated to devise, others are based on +the composition of simple primitives. For instance, when Alice and Bob are +supplied with entangled bits as resource, they can a perform a teleportation +protocol on a part of the states they are asked to distinguish +(an example of such a protocol is shown in Chapter \ref{chap:mes} +for the task of distinguishing three and four Bell states). + +To show that the states are not distinguishable, many techniques have been devised. +One possible approach is the one pursued by Walgate and Hardy \cite{Walgate02}, +which is based on a cases analysis in which all possible measurements are eliminated. + +Another method, considered in \cite{Ghosh01,Ghosh04}, is to reduce the distinguishability +problem to a question on the amount of entanglement that can be distilled from a +certain mixed state. +Say you want to prove that the four Bell states from Eq.~\eqref{eq:Bell-states} +are not perfectly distinguishable by any LOCC protocols. +Suppose that the unknown Bell state is shared among two parties, Alice and Bob, +whose spaces are denoted by $\X_{1}$ and $\Y_{1}$. Let $\X_{2}$ and $\Y_{2}$ two +other spaces of the same dimensions held by two more parties, Charlie and Dan. +Consider the state +\begin{equation} + \rho \in \Density\left((\X_{1}\otimes\Y_{1})\otimes(\X_{2}\otimes\Y_{2})\right) +\end{equation} +defined as +\begin{equation} + \rho = \frac{1}{4}\sum_{i\in\{0,1,2,3\}}\psi_{i}\otimes\psi_{i}. +\end{equation} +By contradiction, assume that $\{\psi_{0}, \ldots, \psi_{3}\} \subset \Density(\X_{1}\otimes\Y_{1})$ +are distinguishable by an LOCC protocol between Alice and Bob. +Then they could communicate the outcome classically to Claire and Danny, +who would use this information to create a shared entangled bit between each other. +This is not possible, since $\rho$ can also be written as +\begin{equation} + \rho = \frac{1}{4}\sum_{i\in\{0,1,2,3\}}\psi_{i}\otimes\psi_{i} + \in D\left((\X_{1}\otimes\X_{2})\otimes(\Y_{1}\otimes\Y_{2})\right), +\end{equation} +and therefore it contradicts the fact that $\rho$ is separable in the cut +between Alice and Claire on one side, and Bob and Danny on the other side, that is, +\begin{equation} + \rho \in \Sep(\X_{1}\otimes\X_{2} : \Y_{1}\otimes\Y_{2}). +\end{equation} +A similar argument shows that no three Bell states, and more in general, +no set of $n+1$ orthogonal maximally entangled states in $\complex^{n}\otimes\complex^{n}$, +can be perfectly distinguished by LOCC\footnote{Later in Section \ref{sec:nathansons-bound}, +we will show a proof of this fact by using the convex programming framework introduced in this thesis.}. +This method is referred in the literature as \emph{GKRSS method}, +by the initials of the authors of \cite{Ghosh01}. + +In \cite{Horodecki03}, a modification of the GKRSS method is presented, +called \emph{HSSH method}. In the GKRSS method, the local distinguishability +problem is reduced to analyzing the entanglement contained in a mixed states +constructed starting from the states that are to be distinguished. +The idea is to compare the entanglement in the mixed state before and after +the distinguishability protocol has run its course. +In HSSH the problem is reduced to comparing the entanglement measures in +\emph{pure states} instead. For some instances of the problem, the HSSH method +turns out to be more powerful, due to the fact that entanglement measures for +pure states are better understood. In fact, through the HSSH method, the problem reduces +to understanding entanglement transformations between pure states, +for which necessary and sufficient conditions were derived by Nielsen \cite{Nielsen99}, and +Jonathan and Plenio \cite{Jonathan99}. +An application of the method by \cite{Horodecki03} is the discovery of the first +set of $n$ indistinguishable states in $\complex^{n}\otimes\complex^{n}$. + +The original proof in \cite{Terhal01a} that the Werner states form a hiding pair +(Example \ref{example:werner-hiding-pairs} above) also exploits the theory of entanglement, +but makes use of an extra observation, that is, the fact that the operators +that constitute an LOCC measurement must be PPT. +The mathematical properties of PPT measurements were also exploited in the recent +proofs by \cite{Yu12,Yu14}, which triggered the work of this thesis. + + + +%Of course diff --git a/drawin_dualg.pdf b/drawin_dualg.pdf new file mode 100644 index 0000000000000000000000000000000000000000..1c098d3d941b9e38e79b5593d66f963f10f194f5 GIT binary patch literal 1347 zcmY!laB(>iH)pQs^E+i#oJY?U7ULOoqxA)%v}l(~wEvYbwR zxV^XEu5Our{9V-_^Nebm(jO$hzP&SW=C1YL?@r9k7M1Wn8RhnhH-vMil#Qdsv8JR6 zz8Bd=Uf)>LRI>f|O7;y#o33rqJ#F7Mw?WFv4A%ue z=Ioq%VOm*6*u)g4=&*J3y3g~7OS<xgr3h_3q1rN{LWB?A+W-2YavVfX9o zjoAW^rNRtz=lti@-rlxI^2+l{*@eAQV*8KywCrnhda9ALIOu zG@aSaLz8!^&Z5TE*W+J#YV~aN=1}+Q5xK7G6SUn|*N-(RrudYb%TxO~fA-zrN-KFU z+1BnN`_N_X#7EmEGvA0UXW!KRh5c0Qw>qxWycB3m0$HFm0E&1J4UBpd6LXL}h-U;( zA>cIOT2bO2TmnoF3N|(%0U*#%GyrlG^qn0Q3>EZ~6M-Z!l>jAxBoIN(F@>9>pdV6M zkgA{`keHqdHZvfxC^fGHs126zd{a|06P@xa6rzDb1`4JYCVHk8mWJj=3T7sjdWJw2 zkQocs>ztog0@Pfr07}V_Wb0m(Us{0VJcw;zQJ9N^fo>>D%quPc*_>Pn6ZTXnDJliK z3FIwjpemQtvdrYvAa^H_5{R$#gHnt0ON)|Iixog&6NKbPkPl$F3E{_LutU8wQ;LE9 z0R=-4-0h_xZzCCChGc*6j.pdf_tex} +%% instead of +%% \includegraphics{.pdf} +%% To scale the image, write +%% \def\svgwidth{} +%% \input{.pdf_tex} +%% instead of +%% \includegraphics[width=]{.pdf} +%% +%% Images with a different path to the parent latex file can +%% be accessed with the `import' package (which may need to be +%% installed) using +%% \usepackage{import} +%% in the preamble, and then including the image with +%% \import{}{.pdf_tex} +%% Alternatively, one can specify +%% \graphicspath{{/}} +%% +%% For more information, please see info/svg-inkscape on CTAN: +%% http://tug.ctan.org/tex-archive/info/svg-inkscape +%% +\begingroup% + \makeatletter% + \providecommand\color[2][]{% + \errmessage{(Inkscape) Color is used for the text in Inkscape, but the package 'color.sty' is not loaded}% + \renewcommand\color[2][]{}% + }% + \providecommand\transparent[1]{% + \errmessage{(Inkscape) Transparency is used (non-zero) for the text in Inkscape, but the package 'transparent.sty' is not loaded}% + \renewcommand\transparent[1]{}% + }% + \providecommand\rotatebox[2]{#2}% + \ifx\svgwidth\undefined% + \setlength{\unitlength}{584.58916016bp}% + \ifx\svgscale\undefined% + \relax% + \else% + \setlength{\unitlength}{\unitlength * \real{\svgscale}}% + \fi% + \else% + \setlength{\unitlength}{\svgwidth}% + \fi% + \global\let\svgwidth\undefined% + \global\let\svgscale\undefined% + \makeatother% + \begin{picture}(1,1.11050938)% + \put(0,0){\includegraphics[width=\unitlength,page=1]{drawin_dualg.pdf}}% + \put(0.4854189,1.07207223){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.20365376\unitlength}\raggedright ?\\ \end{minipage}}}% + \put(0.43374087,0.8560354){\color[rgb]{0,0,0}\makebox(0,0)[lb]{\smash{Sep*}}}% + \put(0.43467636,0.68178262){\color[rgb]{0,0,0}\makebox(0,0)[lb]{\smash{PPT*}}}% + \put(0.45210313,0.50810705){\color[rgb]{0,0,0}\makebox(0,0)[lb]{\smash{Pos}}}% + \end{picture}% +\endgroup% diff --git a/drawin_dualg.svg b/drawin_dualg.svg new file mode 100644 index 0000000..e58864a --- /dev/null +++ b/drawin_dualg.svg @@ -0,0 +1,147 @@ + + + + + + + + + + + + image/svg+xml + + + + + + + + + + + ? 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+\usepackage{mathtools} +\usepackage{dsfont} + +%--------------------------------------------------------------------- +% Theorem-like environments +%--------------------------------------------------------------------- + +\usepackage{amsthm} +\newtheorem{theorem}{Theorem}[chapter] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{prop}[theorem]{Proposition} +\newtheorem{cor}[theorem]{Corollary} +\newtheorem{fact}[theorem]{Fact} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{remark}[theorem]{Remark} +\newtheorem{example}[theorem]{Example} +\newtheorem{property}[theorem]{Property} +\newtheorem{question}[theorem]{Question} + +%--------------------------------------------------------------------- +% Macros: +%--------------------------------------------------------------------- + +\newcommand{\comment}[2]{\begin{quote}\sf + [*** {\color{blue} #1} : #2 ***]\end{quote}} + +\newcommand{\tinyspace}{\mspace{1mu}} +\newcommand{\microspace}{\mspace{0.5mu}} 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+\DeclareMathOperator{\spn}{span} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\opt}{opt} + +\def\PP{\textup{P}} +\def\DP{\textup{D}} + +\usepackage{xcolor,colortbl} +\definecolor{Gray}{gray}{0.90} + + +%--------------------------------------------------------------------- +% Cross referencing +%--------------------------------------------------------------------- +\newcommand{\eqnref}[1]{\hyperref[#1]{{(\ref*{#1})}}} +\newcommand{\thmref}[1]{\hyperref[#1]{{Theorem~\ref*{#1}}}} +\newcommand{\lemref}[1]{\hyperref[#1]{{Lemma~\ref*{#1}}}} +\newcommand{\corref}[1]{\hyperref[#1]{{Corollary~\ref*{#1}}}} +\newcommand{\defref}[1]{\hyperref[#1]{{Definition~\ref*{#1}}}} +\newcommand{\secref}[1]{\hyperref[#1]{{Section~\ref*{#1}}}} +\newcommand{\chapref}[1]{\hyperref[#1]{{Chapter~\ref*{#1}}}} +\newcommand{\figref}[1]{\hyperref[#1]{{Figure~\ref*{#1}}}} +\newcommand{\tabref}[1]{\hyperref[#1]{{Table~\ref*{#1}}}} +\newcommand{\remref}[1]{\hyperref[#1]{{Remark~\ref*{#1}}}} +\newcommand{\appref}[1]{\hyperref[#1]{{Appendix~\ref*{#1}}}} +\newcommand{\claimref}[1]{\hyperref[#1]{{Claim~\ref*{#1}}}} +\newcommand{\propref}[1]{\hyperref[#1]{{Proposition~\ref*{#1}}}} +\newcommand{\exampleref}[1]{\hyperref[#1]{{Example~\ref*{#1}}}} +\newcommand{\conjref}[1]{\hyperref[#1]{{Conjecture~\ref*{#1}}}} + +\usepackage{bibentry} +\usepackage{enumitem} diff --git a/introduction.tex b/introduction.tex new file mode 100644 index 0000000..3aede35 --- /dev/null +++ b/introduction.tex @@ -0,0 +1,199 @@ +%!TEX root = thesis.tex +%------------------------------------------------------------------------------- +\chapter{Introduction} +\label{chap:introduction} +%------------------------------------------------------------------------------- + + +%------------------------------------------------------------------------------- +\section{Motivation} +%------------------------------------------------------------------------------- + +A central subject of study in quantum information theory is +the interplay between entanglement and nonlocality. +An important tool to study this relationship is the paradigm +of local quantum operations and classical communication (LOCC). +This is a subset of all global quantum operations, +with a fairly intuitive physical description. +In a two-party LOCC protocol, +Alice and Bob can perform quantum operations only on their local +subsystems and the communication must be classical. +This restricted paradigm has played a crucial role in the understanding +of the role of entanglement in quantum information. It has also provided a +framework for the description of basic quantum tasks such as quantum key +distribution and entanglement distillation. +Furthermore, LOCC protocols are operationally well-motivated, in the sense +that classical communication is easy to implement. + +The LOCC paradigm does not have a proper classical counterpart. +It is worth noticing that its definition does not impose any restriction on the amount of +classical communication that is allowed between the parties, and therefore it should not be +confused with other setups studied in theoretical computer science where such +constraints are instead imposed, such as communication complexity, or information complexity. + +A fundamental problem that has been studied to understand the limitations of +LOCC protocols is the problem of distinguishing quantum states. +The setup of the problem is pretty simple in the bipartite case. +The two parties are given a single copy of a quantum state chosen +with some probability from a collection of states and their goal is to identify +which state was given, with the assumption that the parties have full +a priori knowledge of the collection. + +When restricted to classical states, this is an easy task, +different strings of bits are always completely distinguishable. +In the quantum case, if the states are orthogonal and global operations +are permitted, then it is always possible to determine the state +with certainty. +When the states are not orthogonal, quantum mechanics does not allow +perfect discrimination \cite{Nielsen11}. +The problem of distinguishing quantum states by global operations +dates back to the '70s \cite{Helstrom1969}, +and since then it has been given different names: \emph{quantum state discrimination}, +\emph{quantum detection}, \emph{quantum hypothesis testing}. + + +Even when the states are orthogonal, things get interesting in the quantum setting +once we impose +restrictions on the measurements that can be performed on the unknown state. +Say the two parties to whom the state is given, Alice and Bob, +have their quantum labs very far apart from each other's and, say, their research budget +pays only for an infrastructure to communicate with each other on a classical network. +In other words, say that only LOCC measurements are allowed on the state. +Then Alice and Bob cannot in general discover the state they have been given, +even if the states are orthogonal. + +The problem of distinguishing among a known set +of orthogonal quantum states by LOCC protocols has been studied by several researchers\footnote{The reader may want to browse through the list of references of this thesis, for a more complete list.} +\cite{Bennett99,Walgate00,Ghosh01,Horodecki03,Fan04,Ghosh04,Nathanson05,Watrous05,Yu11,Yu12}. +It is referred to as the \emph{local state distinguishability problem} (or +\emph{local state discrimination}) and it has some direct applications +to other problems in quantum information theory, +such as secret sharing \cite{Cleve99,Gottesman00}, +data hiding \cite{Terhal01a,DiVincenzo2002}, and the study of quantum channel +capacity (see \cite{Watrous05,Yu11} and references therein). + +Local state distinguishability problems offer insights into how useful entanglement +is in quantum information processing tasks. +The reason why investigating these problems is helpful comes from the fact that the role of entanglement +in such tasks is twofold. On the one hand, many LOCC protocols, such as the ones based on teleportation, +are fueled by entanglement shared by the parties, and therefore entanglement is a helpful resource +for distinguishability. +On the other hand, if the states to be distinguished are themselves entangled, +local measurements on only a part of the states do not always suffice to reveal +all the information hidden in the remaining part. +This observation has led us towards the choice of the sets to analyze in this work, +which ended up being of two antipodal categories: sets consisting only of orthogonal maximally entangled states +and sets consisting only of product states. + +The set of measurements that can be implemented through LOCC has an apparently +complex mathematical structure---no tractable characterization of this set is +known, representing a clear obstacle to a better understanding of the +limitations of LOCC measurements. +For example, given a collection of operators +describing a measurement on a bipartite system, the determination of whether or +not this collection describes an LOCC measurement, or is closely approximated +by an LOCC measurement, is not known to be a computationally decidable +problem. +For this reason, the state discrimination problem described above is sometimes +considered for more tractable classes of measurements that approximate, in some +sense, the set of LOCC measurements, and that are mathematically and computationally +more tractable than the LOCC set. +Among these classes, the set of separable and positive-partial-transpose (PPT) measurements +are the subject of study of this thesis. +Since these classes contain LOCC, any bound on their power is reflected into a +bound on the power of LOCC. + +The key observation of this dissertation is that the set of PPT operators +and the set of separable operators both form closed convex cones and many problems concerning them, +including state distinguishability, can be phrased in terms of cone programming, +which is a convex optimization framework that generalizes semidefinite programming. +In general, we do not have a classical polynomial-time algorithm to solve +cone programs and, in fact, optimizing over separable operators is an NP-hard task. +Nevertheless, cone programming, like semidefinite programming, comes with a rich duality theory, +which can be exploited in order to derive analytical bounds for the problems we are seeking to solve. +From the numerical point of view, we exploit the fact that the particular cone programs +we are interested in can be approximated by efficiently solvable hierarchies of +semidefinite programs \cite{Doherty02}. + +%------------------------------------------------------------------------------- +\section{Summary of the results} +%------------------------------------------------------------------------------- +We prove the following specific results: +\begin{itemize} +\item We obtain an exact formula for the optimal probability of correctly +discriminating any set of either three or four Bell states via separable +measurements, when the parties are given a partially entangled pair of qubits as a resource. +In particular, it is proved that this ancillary pair of qubits must be maximally +entangled in order for three Bell states to be perfectly discriminated +by separable (or LOCC) measurements, which answers an open question of \cite{Yu14}. +\item We build up on a construction by Yu, Duan, and Ying \cite{Yu12}, and show +the first example of a set of $k < n$ orthogonal maximally entangled states +in $\complex^{n}\otimes\complex^{n}$ that is not perfectly distinguishable by LOCC, +One takeaway from this is that the local dimension of the local subsystems does not +play any special role in the nonlocality exhibited by LOCC-indistinguishable sets of +maximally entangled states. +The same example serves to exhibit a gap between the power of separable and PPT measurements +for the task of distinguishing maximally entangled states. +\item We provide an easily checkable characterization of when an unextendable +product set is perfectly discriminated by separable measurements, and we use +this characterization to present an example of an unextendable product set in +$\complex^{4}\otimes\complex^{4}$ that is not perfectly discriminated by +separable measurements. This resolves an open question raised in +\cite{Duan09}. +\item We show that every unextendable product set together +with one extra pure state orthogonal to every member of the unextendable product +set is not perfectly discriminated by separable measurements. +\end{itemize} + +%------------------------------------------------------------------------------- +\section{Overview} +%------------------------------------------------------------------------------- + +We assume that the reader is familiar with basic concepts of quantum computation and +the main target is a researcher in quantum information or a graduate student +who has taken an introductory course to quantum based on Nielsen and Chuang \cite{Nielsen11}. +Familiarity with more advanced concepts of quantum information theory (based +on \cite{Watrous15}, for example) would certain help, but it is not necessary. +The same can be said about notions of convex optimization. The syllabus of an introductory +graduate-level course in convex optimization covers more than it is necessary to grasp +the material of this thesis. +In Chapter \ref{chap:preliminaries} basic notions of quantum information are reviewed. + +Chapter \ref{chap:bipartite-state-discrimination} reviews background material +on bipartite state discrimination, including a comparison between previous approaches +to the problem and ours. + +In Chapter \ref{chap:programs}, we lay out a general cone programming framework +for bipartite state discrimination and we instantiate it for the particular cases +of separable and PPT measurement. + +In Chapters \ref{chap:mes} and \ref{chap:ups}, we apply the framework described in Chapter +\ref{chap:programs} to study the distinguishability of sets of maximally entangled states, +and unextendable product sets, respectively. +These two chapters are independent from each other and they can be read in any order. + +In the last chapter we draw conclusions and ask some open questions that may be +of interest for future work. + +The thesis is based on the following papers: +\begin{itemize} + +\item[$\bullet$] + A. Cosentino. +\textbf{PPT-indistinguishable states via semidefinite programming}. +\textit{Physical Review A}, 2013. +\cite{Cosentino13} + +\item[$\bullet$] +A. Cosentino and V. Russo. +\textbf{Small sets of locally indistinguishable orthogonal maximally entangled states}. +\textit{Quantum Information \& Computation}, 2014. +\cite{Cosentino14} + +\item[$\bullet$] +S. Bandyopadhyay, A. Cosentino, N. Johnston, V. Russo, J. Watrous, and N. Yu. +\textbf{Limitations on separable measurements by convex optimization}. +\textit{IEEE Transactions on Information Theory}, 2015. +\cite{Bandyopadhyay15} + +\end{itemize} \ No newline at end of file diff --git a/matlab_notebook.tex b/matlab_notebook.tex new file mode 100644 index 0000000..97edd91 --- /dev/null +++ b/matlab_notebook.tex @@ -0,0 +1,73 @@ +%!TEX root = thesis.tex +%------------------------------------------------------------------------------- +\chapter{Local distinguishability in Matlab} +\label{chap:AppendixA} +%------------------------------------------------------------------------------- + + +\section*{Setup} +\subsection*{Requirements} + +\begin{itemize} + \item MATLAB\footnote{Unfortunately at the time of writing this, + the package CVX only runs on Matlab. I solemnly promise that I will port + all the code to GNU Octave once the Octave port of CVX will be completed.}; + \item CVX \textgreater= 2.1 \cite{cvx}; + \item QETLAB \textgreater= 0.7 \cite{Johnston2015}; +\end{itemize} + +\subsection*{List of functions} + +\begin{itemize} + \item \texttt{Distinguishability} (by N. Johnston) + \item \texttt{LocalDistinguishability} + \item \texttt{UPBSepDistinguishable} (by N. Johnston) +\end{itemize} + +\sloppy +\definecolor{lightgray}{gray}{0.5} +\setlength{\parindent}{0pt} + +\section*{Examples} + +The code for the following examples can be found in the repository \cite{Cosentino15}. + +\subsection*{Yu--Duan--Ying states} + +\begin{verbatim} +states = 1/2*[vec(kron(Pauli(0),Pauli(0)))'; ... + vec(kron(Pauli(1),Pauli(1)))'; ... + vec(kron(Pauli(2),Pauli(1)))'; ... + vec(kron(Pauli(3),Pauli(1)))']'; + +disp(Distinguishability(states)); +disp(LocalDistinguishability(states, 'copies', 1)); +disp(LocalDistinguishability(states)); +\end{verbatim}\color{lightgray} +\begin{verbatim} + 1 + 0.8750 + 0.7500 +\end{verbatim} + +\color{black} + +\subsection*{Tiles set plus extra orthogonal state (Ex. \ref{ex:tiles-set})} + +\begin{verbatim} +extra_state = 1/2*[1 1 -1 0 0 -1 0 0 0]; +set = [UPB('Tiles') extra_state']; + +disp(LocalDistinguishability(UPB('Tiles'))) +disp(LocalDistinguishability(states, 'copies', 1)); +disp(LocalDistinguishability(states)); +\end{verbatim}\color{lightgray} +\begin{verbatim} + 1.0000 + 1.0000 + 0.9860 +\end{verbatim} + +% \subsection*{Generalized Bell states} + +\color{black} diff --git a/mes.tex b/mes.tex new file mode 100644 index 0000000..7a80dbd --- /dev/null +++ b/mes.tex @@ -0,0 +1,1103 @@ +%!TEX root = thesis.tex +%------------------------------------------------------------------------------- +\chapter{Distinguishability of maximally entangled states} +\label{chap:mes} +%------------------------------------------------------------------------------- + +In this chapter we finally bring the cone programming framework into action, +in order to answer some questions about the distinguishability of maximally entangled states. +As it was discussed in Section \ref{sec:mes-intro}, sets of maximally entangled states +constitute an important testbed for gaging the power of different classes of measurements. + +As a warm-up, we first reprove a bound by \cite{Yu12} on the distinguishability +of any set of maximally entangled states. +Next, we answer an open question regarding the entanglement +cost of Bell states that was raised in \cite{Yu14}. +In the second part we study the set of states \eqref{eq:ydy_states} introduced in \cite{Yu12}. + +\minitoc + +%------------------------------------------------------------------------------% +\section{General bound for maximally entangled states} +\label{sec:nathansons-bound} +%------------------------------------------------------------------------------% + +We show that no PPT measurement can perfectly distinguish more than $n$ +maximally entangled states in $\X\otimes\Y$, where $\X = \Y = \complex^{n}$. +This result appears in \cite{Yu12} and it generalizes +a bound by Nathanson, which is valid against LOCC and separable measurements +\cite{Nathanson05}. The following lemma is central to the proof. +\begin{lemma} + \label{lemma:isometry-ppt-star} + Let $A\in\Unitary(\Y,\X)$ be a unitary operator. + It holds that + \begin{equation} + \label{eq:reduction-operator} + \I_{\X}\otimes\I_{\Y} - \vec(A) \vec(A)^{\ast} \in \PPTStar(\X:\Y). + \end{equation} +\end{lemma} +\begin{proof} +Let $W_{n}\in\Unitary(\X,\Y)$ be the swap operator from Eq.~\eqref{eq:swap-operator}. +Consider the operator +\begin{equation} + U = (\overline{A}\otimes\I_{\Y})W_{n}(A^{\t}\otimes\I_{\Y}). +\end{equation} +Since $A$ and $W_{n}$ are unitary operators, so is $U$. Notice that $U$ is also Hermitian, +and therefore its eigenvalues are either $1$ or $-1$. This implies that +\begin{equation} + \I_{\X}\otimes\I_{\Y} - U \in \Pos(\X\otimes\Y) +\end{equation} +is positive semidefinite. Moreover we have that +\begin{equation} + \begin{aligned} + \pt_{\X}(\I_{\X}\otimes\I_{\Y} - U) &= \I_{\X}\otimes\I_{\Y} - + (A\otimes\I_{\Y})\vec(\I)\vec(\I)^{\ast}(A^{\ast}\otimes\I_{\Y}) \\ + &= \I_{\X}\otimes\I_{\Y} - \vec(A) \vec(A)^{\ast}, + \end{aligned} +\end{equation} +and therefore + \begin{equation} + \I_{\X}\otimes\I_{\Y} - \vec(A) \vec(A)^{\ast} \in \PPTStar(\X:\Y). + \end{equation} +\end{proof} + +Now, suppose that $u_1,\ldots,u_N\in\X\otimes\Y$ are vectors representing +maximally entangled pure states. +An upper-bound on the probability to distinguish these $N$ states, assuming +a uniform selection, is obtained from the dual problem \eqref{eq:ppt-dual-problem} +by considering +\begin{equation} + H = \frac{\I_{\X}\otimes\I_{\Y}}{nN}. +\end{equation} +It holds that $H$ is a feasible solution to the dual problem: +since the states are maximally entangled, for each $k\in\{1,\ldots,N\}$ one may write +\begin{equation} + u_k = \frac{1}{\sqrt{n}}\vec(A_k), +\end{equation} +for some choice of an isometry $A_k\in\Unitary(\Y,\X)$, and therefore +\begin{equation} + H - \frac{1}{N}u_k u_k^{\ast} = \frac{1}{nN}\left( + \I_{\X}\otimes\I_{\Y} - \vec(A_k) \vec(A_k)^{\ast}\right) \in \PPTStar(\X:\Y) +\end{equation} +by Lemma~\ref{lemma:isometry-ppt-star}. +Finally, the value +\begin{equation} +\tr(H) = \frac{n}{N} +\end{equation} +is an upper bound on the probability of distinguishing the states and it is smaller +than $1$ whenever the number of states $N$ is bigger than the dimension $n$ of +each subspace. +\begin{remark} +The statement of Lemma \ref{lemma:isometry-ppt-star} may become very familiar to the reader, +once it is translated in the language of linear mappings via the Choi isomorphism. +It is straightforward to see that the operator in Eq.~\ref{eq:reduction-operator} +is the Choi operator of a mapping $\Phi_{A}:\Lin(\Y)\to\Lin(\X)$ defined as +\begin{equation} + \Phi_{A}(X) = \tr(X)\I - AXA^{\ast}, +\end{equation} +for any $X\in\Lin(\Y)$, which in turn is the composition of a unitary mapping +\begin{equation} + X \to AXA^{\ast} +\end{equation} +and the mapping +\begin{equation} + \Phi(X) = \tr(X)\I - X, +\end{equation} +which is the well-known \emph{reduction map} introduced in \cite{Horodecki99} +(it is the mapping at the basis of the reduction criterion for entanglement detection). +In brief, we related the fact that PPT measurements can distinguish no more than +$n$ maximally entangled states in $\complex^{n}\otimes\complex^{n}$ with the fact +that the reduction map from $\Lin(\complex^{n})\to\Lin(\complex^{n})$ is a decomposable map. +\end{remark} + +%------------------------------------------------------------------------------- +\section{Entanglement cost of distinguishing Bell states} +%------------------------------------------------------------------------------- + +In this section, we study state discrimination problems for sets of three or +four Bell states, by LOCC, separable, and PPT measurements, with the assistance +of an entangled pair of qubits. +In particular, we will assume that Alice and Bob aim to discriminate a set of +Bell states given that they share the additional resource state +\begin{equation} + \label{eq:tau_eps} + \ket{\tau_{\eps}} = \sqrt{\frac{1 + \eps}{2}}\,\ket{0}\ket{0} + + \sqrt{\frac{1 - \eps}{2}}\,\ket{1}\ket{1}, +\end{equation} +for some choice of $\eps \in [0,1]$. +The parameter $\eps$ quantifies the amount of entanglement in the state +$\ket{\tau_{\eps}}$. +Up to local unitaries, this family of states represents every pure state of two +qubits. + +Using the cone programming method discussed in the previous chapter, we obtain +exact expressions for the optimal probability with which any set of three or +four Bell states can be discriminated with the assistance of the state +\eqref{eq:tau_eps} by separable measurements (which match the probabilities +obtained by LOCC measurements in all cases). + +%------------------------------------------------------------------------------% +\subsection{Discriminating three Bell states} +%------------------------------------------------------------------------------% + +Notice that the state $\ket{\tau_{1}} = \ket{0}\ket{0}$ is a product state and +it does not aid the two parties in discriminating any set of Bell states, +so the probability of success for $\eps = 1$ is still at most $2/3$ for a set +of three Bell states. +If $\varepsilon = 0$, then Alice and Bob can use teleportation to perfectly +discriminate all four Bell states perfectly by LOCC measurements, and therefore +the same is true for any three Bell states. +It was proved in \cite{Yu14} that PPT measurements can perfectly +discriminate any set of three Bell states using the resource state +\eqref{eq:tau_eps} if and only if $\eps \leq 1/3$. + +Here we show that a maximally entangled state ($\eps = 0$) is required to +perfectly discriminate any set of three Bell states using separable +measurements, and more generally we obtain an expression for the optimal +probability of a correct discrimination for all values of $\varepsilon$. +Because the permutations of Bell states induced by local unitaries is +transitive, there is no loss of generality in fixing the three Bell states to +be discriminated to be $\ket{\phi_1}$, $\ket{\phi_2}$, and $\ket{\phi_3}$ +(as defined in \eqref{eq:Bell-states}). + +\begin{theorem} + \label{thm:three-bell} + Let $\X_1 = \X_2 = \Y_1 = \Y_2 = \complex^2$, define + $\X = \X_1 \otimes \X_2$ and $\Y = \Y_1 \otimes \Y_2$, and let + $\eps \in [0,1]$ be chosen arbitrarily. + For any separable measurement $\mu\in\Meas_{\Sep}(3,\X:\Y)$, the + success probability of correctly discriminating the states corresponding to + the set + \begin{equation} + \label{eq:set-three-bells} + \bigl\{ \ket{\phi_{1}} \otimes \ket{\tau_{\eps}},\; + \ket{\phi_{2}} \otimes \ket{\tau_{\eps}},\; + \ket{\phi_{3}} \otimes \ket{\tau_{\eps}} \bigr\} + \subset (\X_1\otimes\Y_1)\otimes(\X_2\otimes\Y_2), + \end{equation} + assuming a uniform distribution $p_1 = p_2 = p_3 = 1/3$, is at most + \begin{equation} + \label{eq:probability-three-bell} + \frac{1}{3}\left(2 + \sqrt{1 - \eps^{2}}\right). + \end{equation} +\end{theorem} + +To prove this theorem, we require the following lemma. +The lemma introduces a family of positive maps that, to our knowledge, has not +previously appeared in the literature. + +\begin{lemma} + \label{lemma:3Bell} + Define a linear mapping + $\Xi_{t}: \Lin(\complex^2 \oplus \complex^2)\rightarrow + \Lin(\complex^2 \oplus \complex^2)$ as + \begin{equation} + \Xi_t\begin{pmatrix} + A & B\\ + C & D + \end{pmatrix} + = \begin{pmatrix} + \Psi_t(D) + \Phi(D) & + \Psi_t(B) + \Phi(C)\\[2mm] + \Psi_t(C) + \Phi(B) & + \Psi_t(A) + \Phi(A) + \end{pmatrix} + \end{equation} + for every $t\in(0,\infty)$ and $A,B,C,D\in\Lin(\complex^2)$, where + $\Psi_t:\Lin(\complex^2)\rightarrow\Lin(\complex^2)$ + is defined as + \begin{equation} + \Psi_t + \begin{pmatrix} + \alpha & \beta \\ + \gamma & \delta + \end{pmatrix} + = + \begin{pmatrix} + t \alpha & \beta \\ + \gamma & t^{-1} \delta + \end{pmatrix} + \end{equation} + and $\Phi:\Lin(\complex^2)\rightarrow\Lin(\complex^2)$ is defined as + \begin{equation} + \Phi\begin{pmatrix} + \alpha & \beta \\ + \gamma & \delta + \end{pmatrix} + = \begin{pmatrix} + \delta & -\beta\\ + -\gamma & \alpha + \end{pmatrix}, + \end{equation} + for every $\alpha,\beta,\gamma,\delta\in\complex$. + It holds that $\Xi_t$ is a positive map for all $t\in (0,\infty)$. +\end{lemma} + +\begin{proof} + It will first be proved that $\Xi_1$ is positive. + For every vector + \begin{equation} + u = \begin{pmatrix} + \alpha\\ \beta + \end{pmatrix} + \end{equation} + in $\complex^2$, define a matrix + \begin{equation} + M_u = \begin{pmatrix} + \overline\alpha & \overline\beta\\[1mm] + -\beta & \alpha + \end{pmatrix}. + \end{equation} + Straightforward computations reveal that + \begin{equation} + M_u^{\ast} M_v = u v^{\ast} + \Phi(v u^{\ast}) + \qquad\text{and}\qquad + M_u^{\ast} M_u = \norm{u}^2\tinyspace \I + \end{equation} + for all $u,v\in\complex^2$. + It follows that + \begin{equation} + \Xi_1 \begin{pmatrix} + u u^{\ast} & u v^{\ast}\\ + v u^{\ast} & v v^{\ast} + \end{pmatrix} + = \begin{pmatrix} + v v^{\ast} + \Phi(v v^{\ast}) & + u v^{\ast} + \Phi(v u^{\ast}) \\ + v u^{\ast} + \Phi(u v^{\ast}) & + u u^{\ast} + \Phi(u u^{\ast}) + \end{pmatrix} + = \begin{pmatrix} + \norm{v}^2 \I & M_u^{\ast} M_v\\[1mm] + M_v^{\ast} M_u & \norm{u}^2 \I + \end{pmatrix}, + \end{equation} + which is positive semidefinite by virtue of the fact that + $\norm{M_u^{\ast} M_v}\leq\norm{M_u}\norm{M_v} = \norm{u} \norm{v}$. + As every element of $\Pos(\complex^2\oplus\complex^2)$ can be written as a + positive linear combination of matrices of the form + \begin{equation} + \begin{pmatrix} + u u^{\ast} & u v^{\ast}\\ + v u^{\ast} & v v^{\ast} + \end{pmatrix}, + \end{equation} + ranging over all vectors $u, v \in \complex^2$, it follows that $\Xi_1$ is a + positive map. + + For the general case, observe first that the mapping $\Psi_s$ may be + expressed using the Hadamard (or entry-wise) product as + \begin{equation} + \Psi_s + \begin{pmatrix} + \alpha & \beta \\ + \gamma & \delta + \end{pmatrix} + = + \begin{pmatrix} + s \alpha & \beta \\ + \gamma & s^{-1} \delta + \end{pmatrix} + = + \begin{pmatrix} + s & 1\\ + 1 & s^{-1} + \end{pmatrix} \circ + \begin{pmatrix} + \alpha & \beta \\ + \gamma & \delta + \end{pmatrix} + \end{equation} + for every positive real number $s\in(0,\infty)$. + The matrix + \begin{equation} + \begin{pmatrix} + s & 1\\ + 1 & s^{-1} + \end{pmatrix} + \end{equation} + is positive semidefinite, from which it follows (by the Schur product + theorem) that $\Psi_s$ is a completely positive map. + (See, for instance, Theorem 3.7 of \cite{Paulsen02}.) + Also note that $\Phi = \Psi_s \Phi \Psi_s$ for every $s\in (0,\infty)$, which + implies that + \begin{equation} + \Xi_t = \bigl(\I_{\Lin(\complex^2)} \otimes \Psi_s\bigr) \Xi_1 + \bigl(\I_{\Lin(\complex^2)} \otimes \Psi_s\bigr) + \end{equation} + for $s = \sqrt{t}$. + This shows that $\Xi_t$ is a composition of positive maps for every positive + real number~$t$, and is therefore positive. +\end{proof} + +\begin{proof}[Proof of Theorem \ref{thm:three-bell}] + For the cases that $\eps = 0$ and $\varepsilon = 1$, the theorem is known, + as was discussed previously, so it will be assumed that $\eps \in (0,1)$. + Define a Hermitian operator + \begin{equation} + H_{\eps} = \frac{1}{3}\left[\frac{\I_{\X_1\otimes\Y_1}\otimes + \tau_{\eps}}{2} + \sqrt{1 - \eps^{2}} \, \phi_{4} \otimes + \pt_{\negsmallspace\X_2}(\phi_{4}) \right], + \end{equation} + where $\tau_{\eps} = \ket{\tau_{\eps}}\bra{\tau_{\eps}}$, + $\phi_{4} = \ket{\phi_{4}}\bra{\phi_{4}}$, + and $\pt_{\negsmallspace\X_2}$ denotes partial transposition with respect to + the standard basis of $\X_2$. + It holds that + \begin{equation} + \tr(H_{\eps}) = \frac{1}{3}\left(2 + \sqrt{1 - \eps^{2}}\right), + \end{equation} + so to complete the proof it suffices to prove that $H_{\varepsilon}$ is a + feasible solution to the dual problem \eqref{eq:sep-dual-problem} + for the cone program corresponding to the state discrimination problem being considered. + + In order to be more precise about the task at hand, it is helpful to define a + unitary operator $W$, mapping $\X_1\otimes\X_2\otimes\Y_1\otimes\Y_2$ to + $\X_1\otimes\Y_1\otimes\X_2\otimes\Y_2$, that corresponds to swapping the + second and third subsystems: + \begin{equation} + \label{eq:swap} + W(x_{1}\otimes x_{2}\otimes y_{1}\otimes y_{2}) = + x_{1}\otimes y_{1}\otimes x_{2}\otimes y_{2}, + \end{equation} + for all vectors + $x_{1}\in\X_{1}$, $x_{2}\in\X_{2}$, $y_{1}\in\Y_{1}$, $y_{2}\in\Y_{2}$. + We are concerned with the separability of measurement operators with respect + to the bipartition between $\X_1\otimes\X_2$ and $\Y_1\otimes\Y_2$, so the + dual feasibility of $H_{\varepsilon}$ requires that the operators defined as + \begin{equation} + Q_{k, \eps} = W^{\ast} \left( H_{\eps} - \frac{1}{3}\phi_{k} \otimes + \tau_{\eps} \right) W \in \Herm(\X\otimes\Y) + \end{equation} + be contained in $\BPos(\X:\Y)$ for $k = 1,2,3$. + + Let $\Lambda_{k, \eps}: \Lin(\Y) \rightarrow \Lin(\X)$ be the unique linear + map whose Choi representation satisfies + $J(\Lambda_{k,\varepsilon}) = Q_{k, \eps}$ for each $k = 1,2,3$. + As discussed in Section~\ref{sec:block-positive-operators}, + the block positivity of $Q_{k, \eps}$ + is equivalent to the positivity of $\Lambda_{k, \eps}$. + Consider first the case $k = 1$ and let + \begin{equation} + t = \sqrt{\frac{1+\eps}{1-\eps}}. + \end{equation} + A calculation reveals that + \begin{equation} + \Lambda_{1, \eps}(Y) = \frac{\sqrt{1 - \eps^{2}}}{3} + \left(\sigma_{3} \otimes \I_{\X_2}\right) + \Xi_{t}(Y) + \left(\sigma_{3} \otimes \I_{\X_2}\right), + \end{equation} + where $\Xi_{t}:\Lin(\Y)\rightarrow\Lin(\X)$ is the map defined in + Lemma~\ref{lemma:3Bell} and $\sigma_{3}$ denotes one of the Pauli operators + (see Eq.~\eqref{eq:Pauli-operators} for an explicit definition). + % (in general) + % \begin{equation} \label{eq:Pauli-operators} + % \begin{array}{llll} + % \sigma_{0} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, & + % \sigma_{1} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, & + % \sigma_{2} = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, & + % \sigma_{3} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} + % \end{array} + % \end{equation} + % denote the Pauli operators. + As $\eps \in (0,1)$, it holds that $t\in (0,\infty)$, and therefore + Lemma~\ref{lemma:3Bell} implies that $\Xi_{t}(Y) \in \Pos(\X)$ for every + $Y \in \Pos(Y)$. + As we are simply conjugating $\Xi_{t}(Y)$ by a unitary and scaling it + by a positive real factor, we also have that + $\Lambda_{1, \eps}(Y) \in \Pos(\X)$, for any $Y \in \Pos(Y)$, which in turn + implies that $Q_{1, \eps} \in \BPos(\X:\Y)$. + + For the case of $k=2$ and $k = 3$, first define + $U, V \in \Unitary(\complex^{2})$ as follows: + \begin{equation} + U = \begin{pmatrix} + 1 & 0\\ + 0 & i + \end{pmatrix} + \quad \mbox{and} \quad + V = \frac{1}{\sqrt{2}}\begin{pmatrix} + 1 & i\\ + i & 1 + \end{pmatrix}. + \end{equation} + These operators transform $\phi_{1} = \ket{\phi_1}\bra{\phi_1}$ into + $\phi_{2} = \ket{\phi_2}\bra{\phi_2}$ and $\phi_{3} = + \ket{\phi_3}\bra{\phi_3}$, respectively, and leave $\phi_{4}$ unchanged, in + the following sense: + \begin{equation} + \begin{aligned} + (U^{\ast}\otimes U^{\ast}) \phi_{1} (U\otimes U) = \phi_{2},\\ + (V^{\ast}\otimes V^{\ast}) \phi_{1} (V\otimes V) = \phi_{3},\\ + (U^{\ast}\otimes U^{\ast}) \phi_{4} (U\otimes U) = \phi_{4},\\ + (V^{\ast}\otimes V^{\ast}) \phi_{4} (V\otimes V) = \phi_{4}. + \end{aligned} + \end{equation} + Therefore the following equations hold: + \begin{equation} + \begin{aligned} + Q_{2,\eps} &= \left(U^{\ast}\otimes\I \otimes U^{\ast}\otimes\I\right) + Q_{1,\eps} \left(U\otimes\I \otimes U\otimes\I\right), \\ + Q_{3,\eps} &= \left(V^{\ast}\otimes\I \otimes V^{\ast}\otimes\I\right) + Q_{1,\eps} \left(V\otimes\I \otimes V\otimes\I\right). + \end{aligned} + \end{equation} + It follows that $Q_{2,\eps}\in\BPos(\X:\Y)$ and $Q_{3,\eps}\in\BPos(\X:\Y)$, + which completes the proof. +\end{proof} + +\begin{remark} + The upper bound obtained in Theorem \ref{thm:three-bell} is achievable by an + LOCC measurement, as it is the probability obtained by using the resource + state $\ket{\tau_\varepsilon}$ to teleport the given Bell state from one + player to the other, followed by an optimal local measurement to discriminate + the resulting states. +\end{remark} + +%------------------------------------------------------------------------------% +\subsection{Discriminating four Bell states} +%------------------------------------------------------------------------------% + +It is known that, for the perfect LOCC discrimination of all four Bell states +using an auxiliary entangled state $\ket{\tau_{\varepsilon}}$ as above, +one requires that $\varepsilon = 0$ (i.e., a maximally entangled pair of qubits +is required). +This fact follows from the method of \cite{Horodecki03}, for instance. +Here we prove a more precise bound on the optimal probability of a correct +discrimination, for every choice of $\varepsilon\in[0,1]$, along similar lines +to the bound on three Bell states provided by Theorem~\ref{thm:three-bell}. +In the present case, in which all four Bell states are considered, the result +is somewhat easier: one obtains an upper bound for PPT measurements that +matches a bound that can be obtained by an LOCC measurement, +implying that LOCC, separable, and PPT measurements are equivalent for this +discrimination problem. + +\begin{theorem} + \label{thm:four-bell} + Let $\X_1 = \X_2 = \Y_1 = \Y_2 = \complex^2$, define + $\X = \X_1 \otimes \X_2$ and $\Y = \Y_1 \otimes \Y_2$, and let + $\varepsilon\in [0,1]$. + For any PPT measurement $\{P_1,P_2,P_3,P_4\}\subset\PPT(\X:\Y)$, the success + probability of discriminating the states corresponding to the set + \begin{equation} + \label{eq:set-four-bells} + \left\{ \ket{\phi_{1}} \otimes \ket{\tau_{\eps}},\; + \ket{\phi_{2}} \otimes \ket{\tau_{\eps}},\; + \ket{\phi_{3}} \otimes \ket{\tau_{\eps}},\; + \ket{\phi_{4}} \otimes \ket{\tau_{\eps}}\right\} + \subset (\X_1\otimes\Y_1)\otimes(\X_2\otimes\Y_2), + \end{equation} + assuming a uniform distribution $p_1 = p_2 = p_3 = p_4 = 1/4$, is at most + \begin{equation} + \label{eq:probability-four-bell} + \frac{1}{2}\left(1 + \sqrt{1 - \eps^2}\right). + \end{equation} +\end{theorem} + +\begin{proof} + One may formulate a cone program corresponding to state discrimination by + PPT measurements along similar lines to the cone program for separable + measurements, simply by replacing the cone $\Sep(\X:\Y)$ by the cone + $\PPT(\X:\Y)$ of positive semidefinite operators whose partial transpose is + positive semidefinite. + This cone program is a semidefinite program, as discussed in + \cite{Cosentino13}, by virtue of the fact that partial transpose mapping is + linear. + + Consider the following operator: + \begin{equation} + H_{\varepsilon} = \frac{1}{8} + \Bigl[ + \I_{\X_1 \otimes \Y_1} \otimes \tau_{\varepsilon} + + \sqrt{1 - \varepsilon^2}\, + \I_{\X_1 \otimes \Y_1} \otimes \pt_{\negsmallspace\X_2}(\phi_4) + \Bigr] \in \Herm(\X_1\otimes\Y_1\otimes\X_2\otimes\Y_2). + \end{equation} + It holds that + \begin{equation} + \tr(H_{\eps}) = \frac{1}{2}\left( 1 + \sqrt{1-\eps^2} \right), + \end{equation} + so to complete the proof it suffices to prove that $H_{\varepsilon}$ is + dual feasible for the (semidefinite) cone program representing the + PPT state discrimination problem under consideration. + Dual feasibility will follow from the condition + \begin{equation} + (\pt_{\negsmallspace\X_1} \otimes \pt_{\negsmallspace\X_2}) + \Bigl( + H_{\varepsilon} - \frac{1}{4}\,\phi_k\otimes\tau_{\varepsilon}\Bigr) + \in\Pos(\X_1\otimes\Y_1\otimes\X_2\otimes\Y_2) + \end{equation} + (which is sufficient but not necessary for feasibility) for $k = 1,2,3,4$. + One may observe that + \begin{equation} + \pt_{\negsmallspace\X_2}(\tau_{\varepsilon}) + + \frac{\sqrt{1-\varepsilon^2}}{2}\phi_4 + = \frac{1}{2} + \begin{pmatrix} + 1 + \varepsilon & 0 & 0 & 0\\[1mm] + 0 & \frac{\sqrt{1 - \varepsilon^2}}{2} + & \frac{\sqrt{1 - \varepsilon^2}}{2} & 0\\[2mm] + 0 & \frac{\sqrt{1 - \varepsilon^2}}{2} + & \frac{\sqrt{1 - \varepsilon^2}}{2} & 0\\[2mm] + 0 & 0 & 0 & 1 - \varepsilon + \end{pmatrix} + \end{equation} + is positive semidefinite, from which it follows that + \begin{equation} + (\pt_{\negsmallspace\X_1} \otimes \pt_{\negsmallspace\X_2}) + \Bigl( + H_{\varepsilon} - \frac{1}{4}\,\phi_1\otimes\tau_{\varepsilon}\Bigr) + = \frac{1}{4} \phi_4 \otimes \pt_{\negsmallspace\X_2}(\tau_{\varepsilon}) + + \frac{\sqrt{1 - \varepsilon^2}}{8} \I_{\X_1\otimes\Y_1} \otimes \phi_4 + \end{equation} + is also positive semidefinite. + A similar calculation holds for $k=2,3,4$, which completes the proof. +\end{proof} + +\begin{remark} + Similar to Theorem \ref{thm:three-bell}, one has that the upper bound + obtained by Theorem \ref{thm:four-bell} is optimal for LOCC measurements, + as it is the probability obtained using teleportation. +\end{remark} + + + +%------------------------------------------------------------------------------- +\section{Yu-Duan-Ying states} +%------------------------------------------------------------------------------- + +In this section we prove a tight bound of $3/4$ on the maximum success +probability for any LOCC measurement to discriminate the set of states +\eqref{eq:ydy_states} exhibited by Yu, Duan, and Ying \cite{Yu12}, +assuming a uniform selection of states. +The fact that this bound can be achieved by an LOCC measurement is trivial: +if Alice and Bob measure their parts of the states with respect to the standard +basis, they can easily discriminate $\ket{\phi_1}$, $\ket{\phi_2}$, and +$\ket{\phi_4}$, erring only in the case that they receive $\ket{\phi_3}$. +The fact that this bound is optimal will be proved by exhibiting a +feasible solution $H$ to the dual problem \eqref{eq:sep-dual-problem} +instantiated with the state discrimination problem at hand, such that +\begin{equation} +\tr(H) = \frac{3}{4}. +\end{equation} + +With respect to the vector-operator correspondence, the states \eqref{eq:ydy_states} are given +by tensor products of the Pauli operators \eqref{eq:Pauli-operators} as follows: +\begin{align} +\begin{split} + \ket{\phi_1} = \frac{1}{2}\op{vec}(U_1),\quad + \ket{\phi_2} = \frac{1}{2}\op{vec}(U_2),\\ + \ket{\phi_3} = \frac{1}{2}\op{vec}(U_3),\quad + \ket{\phi_4} = \frac{1}{2}\op{vec}(U_4), +\end{split} +\end{align} +for +\begin{equation} +\label{eq:ydy-operators} +\begin{aligned} + U_1 & = + \sigma_0\otimes\sigma_0 = + \begin{pmatrix} + 1 & 0 & 0 & 0 \\ + 0 & 1 & 0 & 0 \\ + 0 & 0 & 1 & 0 \\ + 0 & 0 & 0 & 1 + \end{pmatrix}, & + U_2 & = + \sigma_1\otimes\sigma_1 = + \begin{pmatrix} + 0 & 0 & 0 & 1 \\ + 0 & 0 & 1 & 0 \\ + 0 & 1 & 0 & 0 \\ + 1 & 0 & 0 & 0 + \end{pmatrix},\\[3mm] + U_3 & = + i \sigma_2 \otimes \sigma_1 = + \begin{pmatrix} + 0 & 0 & 0 & 1 \\ + 0 & 0 & 1 & 0 \\ + 0 & -1 & 0 & 0 \\ + -1 & 0 & 0 & 0 + \end{pmatrix}, \qquad & + U_4 & = + \sigma_3 \otimes \sigma_1 = + \begin{pmatrix} + 0 & 1 & 0 & 0 \\ + 1 & 0 & 0 & 0 \\ + 0 & 0 & 0 & -1 \\ + 0 & 0 & -1 & 0 + \end{pmatrix}. +\end{aligned} +\end{equation} + +\subsubsection*{Bound for separable measurements} +A feasible solution of the dual problem \eqref{eq:sep-dual-problem} is based on +a construction of block positive operators that correspond, via the Choi +isomorphism, to the family of positive maps introduced by Breuer and Hall +\cite{Breuer06,Hall06}. + +\begin{prop}[Breuer--Hall] + \label{prop:breuer-hall} + Let $\X = \Y = \complex^n$ and let $U,V\in\Unitary(\Y,\X)$ be unitary + operators such that $U^{\t}V \in \Unitary(\Y)$ is skew-symmetric: + $(V^{\t}U)^{\t} = -V^{\t}U$. + It holds that + \begin{equation} + \I_{\X}\otimes\I_{\Y} - \vec(U) \vec(U)^{\ast} - + \pt_{\negsmallspace\X}(\vec(V)\vec(V)^{\ast}) + \in \BPos(\X:\Y). + \end{equation} +\end{prop} + +\begin{proof} + For every unit vector $y\in\Y$, one has + \begin{multline} \label{eq:BH} + \qquad + (\I_{\X} \otimes y^{\ast}) + (\I_{\X}\otimes\I_{\Y} - \vec(U) + \vec(U)^{\ast} - \pt_{\negsmallspace\X}(\vec(V)\vec(V)^{\ast})) + (\I_{\X} \otimes y) \\ + = \I_{\X} - U \overline{y} y^{\t} U^{\ast} - \overline{V}y + y^{\ast}V^{\t}. + \qquad + \end{multline} + As it holds that $V^{\t}U$ is skew-symmetric, we have + \begin{equation} + \bigip{\overline{V}y}{U\overline{y}} + = y^{\ast} V^{\t}U \overline{y} + = \bigip{y y^{\t}}{V^{\t}U} + = 0, + \end{equation} + as the last inner product is between a symmetric and a skew-symmetric + operator. + Because $U$ and $V$ are unitary, it follows that + $U\overline{y}y^{\t}U^{\ast} + \overline{V}yy^{\ast}V^{\t}$ is a rank two + orthogonal projection, so the operator represented by \eqref{eq:BH} is also a + projection and is therefore positive semidefinite. +\end{proof} + +\begin{remark} + The assumption of Proposition \ref{prop:breuer-hall} requires $n$ to be even, + as skew-symmetric unitary operators exist only in even dimensions. +\end{remark} + +Now, for the ensemble +\begin{equation} +\label{eq:ydy-ensemble} + \E = \{ \ket{\phi_1}, \ket{\phi_2}, \ket{\phi_3}, \ket{\phi_4} \}, +\end{equation} +one has that the following operator is a feasible solution to the dual problem +\eqref{eq:sep-dual-problem}: +\begin{equation} + \label{eq:H_2} + H = \frac{1}{16}(\I_{\X}\otimes\I_{\Y} - + \pt_{\negsmallspace\X}(\vec(V)\vec(V)^{\ast})) +\end{equation} +for +\begin{equation} + V = i \sigma_2 \otimes \sigma_3 + = \begin{pmatrix} + 0 & 0 & 1 & 0\\ + 0 & 0 & 0 & -1\\ + -1 & 0 & 0 & 0\\ + 0 & 1 & 0 & 0 + \end{pmatrix}. +\end{equation} +Due to Proposition~\ref{prop:breuer-hall}, the feasibility of $H$ follows from +the condition +\begin{equation} + (V^{\t} U_k)^{\t} = - V^{\t} U_k, +\end{equation} +which can be checked by inspecting each of the four cases. +It is easy to calculate that $\tr(H) = 3/4$, and so the required bound has been +obtained. + +\subsubsection*{Bound for PPT measurements} +Interestingly, when it comes to distinguishing the Yu--Duan--Ying states, +PPT measurements can do better than separable (and LOCC) measurement, +yet without achieving perfect distinguishability. As far as we know, this is the +first example of a set consisting only of maximally entangled states for which +such a gap holds. +In this section we exhibit a tight bound of $7/8$ on the probability of distinguishing +the Yu--Duan--Ying ensemble by PPT measurements. + +\begin{theorem} +\label{thm:dual-ppt-ydy} +For $\E$ being the ensemble in Eq.~\eqref{eq:ydy-ensemble}, it holds that $\opt_{\PPT}(\E)\leq 7/8$. +\end{theorem} +\begin{proof} +We show that there exists a solution of the dual program $(\DP_{\PPT})$ which achieves the bound. +It is easy to check that the following operator satisfies the constraints in of the program +and its trace is equal to $7/8$: +\[ +Y = \frac{1}{16}\I\otimes\I - \frac{1}{8}\left(\psi_{2}\otimes\psi_{1}\right). +\] +We will check the constraint $Y \geq \pt_{\A}(\rho_1)$ and the reader can check the remaining constraints with +a similar calculation. By the equations in \eqref{eq:ppt-bell-states}, we have +\begin{align*} + \pt_{\A}(\rho_{1}) &= \pt_{\A}(\psi_{0}\otimes\psi_{0}) = \left(\frac{1}{2}\I - \psi_{2}\right)\otimes\left(\frac{1}{2}\I - \psi_{2}\right) \\ +&= \frac{1}{4}\I\otimes\I - \frac{1}{2}\sum_{i\in\{0,1,3\}}(\psi_{i}\otimes\psi_{2}+\psi_{2}\otimes\psi_{i}), +\end{align*} +and +\begin{align*} + Y - \frac{1}{4}\pt_{\A}(\rho_{1}) &= \frac{1}{8}(\psi_{0}\otimes\psi_{2}+\psi_{1}\otimes\psi_{2}+\psi_{3}\otimes\psi_{2} ++\psi_{2}\otimes\psi_{0}+\psi_{2}\otimes\psi_{3}) \geq 0. +\end{align*} +\end{proof} + +\begin{theorem} +\label{thm:primal-ppt-ydy} +For $\E$ being the ensemble in Eq.~\eqref{eq:ydy-ensemble}, it holds that $\opt_{\PPT}(\E)\geq 7/8$. +\end{theorem} +\begin{proof} +It is enough to show a feasible solution of the primal program ($\PP_{\PPT}$) +that achieves the bound. +Let $Q \in \Pos(\complex^{4}\otimes\complex^{4})$ and $R,S\in\Pos(\complex^{2}\otimes\complex^{2})$ +be the following operators: +\[ + Q = \frac{1}{4}\I\otimes(\psi_{1}+\psi_{2}), \qquad + R = \frac{7}{8}\psi_{0}+\frac{1}{8}\psi_{3}, \qquad + S = \frac{1}{8}\psi_{0}+\frac{7}{8}\psi_{3}. +\] +Then the following operators define a PPT measurement that achieves a success probability of $7/8$: +\begin{align*} + \mu(1) &= Q + (\frac{2}{3}\psi_{0}+\frac{1}{3}\I)\otimes R ,\\ + \mu(2) &= Q + (\frac{1}{3}\psi_{0}+\psi_{1})\otimes S + \frac{1}{3}(\psi_{2}+\psi_{3})\otimes R ,\\ + \mu(3) &= Q + (\frac{1}{3}\psi_{0}+\psi_{2})\otimes S + \frac{1}{3}(\psi_{1}+\psi_{3})\otimes R ,\\ + \mu(4) &= Q + (\frac{1}{3}\psi_{0}+\psi_{3})\otimes S + \frac{1}{3}(\psi_{1}+\psi_{2})\otimes R .\\ +\end{align*} +It is easy to check that these operators define a valid measurement, that is $\sum_{k=1}^{4}\mu(k)=\I$. +Using the equations in \eqref{eq:ppt-bell-states}, we can verify that those operators are also PPT. +Again, we check this for $\mu(1)$. +\[ +\pt_{\A}(\mu(1)) = +(\psi_{1} + \psi_{2} + \psi_{4})\otimes(\frac{1}{3}\psi_{1}+\frac{1}{2}\psi_{2}+\frac{1}{3}\psi_{4}) ++ \frac{1}{4}\psi_{3}\otimes(\psi_{2}+\psi_{3}) \geq 0. +\] +Finally, we have that $\ip{\mu(k)}{\rho_{k}} = 7/8$, for each $k \in \{1,\ldots,4\}$. +\end{proof} + +To recap, for $\E$ being the ensemble of Yu--Duan--Ying states selected uniformly at random, we have +\begin{equation} + \opt_{\LOCC}(\E) = \opt_{\Sep}(\E) = \frac{3}{4} < \frac{7}{8} = \opt_{\PPT}(\E) < \opt(\E) = 1. +\end{equation} + +\subsection{Generalization to higher dimension} + +Here we generalize the Yu--Duan--Ying in order to construct sets of $N$ orthogonal +maximally entangled states in $\complex^{N}\otimes\complex^{N}$ that are distinguishable +by separable operators only with probability $3/4$, for any $N \geq 4$ that is power of $2$. + +Let $t \geq 2$ be a positive integer and, for any choice of $k \in \{ 1, \ldots, 2^{t}\}$, +we recursively define the unitary operator +\begin{equation} +\label{eq:construction} + U_{k}^{(t)} = + \begin{cases} + \sigma_{0}\otimes U_{k}^{(t-1)} &\mbox{if } 1 \leq k \leq 2^{t-1}, \mbox{ or}\\ + \sigma_{1}\otimes U_{k-2^{t-1}}^{(t-1)} &\mbox{if } 2^{t-1} + 1 \leq k \leq 2^{t}.\\ + \end{cases} +\end{equation} +The base of the recursion, $U_{1}^{(2)}, \ldots, U_{4}^{(2)}$, +is given by the operators defined in Eq.~\eqref{eq:ydy_Us}. +In a paper by the author \cite{Cosentino14}, it was shown that the set +\begin{equation} +\label{eq:ydy_higher_dimension} + \left\{ \vec(U_{k}^{(t)})\vec(U_{k}^{(t)})^{\ast} : k = 1, \ldots, 2^{t}\right\} +\end{equation} +can be distinguished with probability at most $7/8$ by PPT measurements, +for any value of $t \geq 2$, when the states are drawn with uniform probability, +$p_{1}, \ldots, p_{2^{t}} = 1/2^{t}$. +In the rest of this section, we show that the maximum success probability for any separable +measurement to distinguish the same set of states under the same assumption is $3/4$ instead. + +As in the theme of this thesis, we will exhibit a feasible solution of +the dual cone program \eqref{eq:sep-dual-problem} for which the value of +the objective function equals $3/4$. +However, we first prove a rather general lemma, which shows how to compose a separable +operator with a block positive operator, in order to construct another block positive +operator in higher dimensions. +\begin{lemma} +\label{lemma:higher-dimension} +Let $\X_{1}, \X_{2}, \Y_{1}$, and $\Y_{1}$ be complex Euclidean spaces. We denote by +$W \in \Unitary(\X_{1}\otimes\X_{2}\otimes\Y_{1}\otimes\Y_{2}, + \X_{1}\otimes\Y_{1}\otimes\X_{2}\otimes\Y_{2})$ +the linear isometry that swaps the second and the third subsystems, which is defined by the following equation: + \begin{equation} + \label{eq:swap-2} + W(x_{1}\otimes x_{2}\otimes y_{1}\otimes y_{2}) = + x_{1}\otimes y_{1}\otimes x_{2}\otimes y_{2}, + \end{equation} +holding for all vectors $x_{1} \in \X_{1}, x_{2} \in \X_{2}, +y_{1} \in \Y_{1}, y_{2} \in \Y_{2}$. +Let $S \in \Sep(\X_{1}:\Y_{1})$ be a separable operator +and $Q \in \BPos(\X_{2}:\Y_{2})$ be +a block positive operator. Then the following holds: + \begin{equation} + W^{\ast}(S \otimes Q)W\in \BPos(\X_{1}\otimes\X_{2}:\Y_{1}\otimes\Y_{2}). + \end{equation} +\end{lemma} +\begin{proof} +By the definition of block-positivity, the claim of the lemma is equivalent +to the following condition: +\begin{equation} + (\I_{\X_{1}\otimes\X_{2}} \otimes y^{\ast})W^{\ast}(S \otimes Q)W + (\I_{\X_{1}\otimes\X_{2}} \otimes y) \in \Pos(\X_{1}\otimes\X_{2}), +\end{equation} +for every $y \in \Y_{1}\otimes\Y_{2}$. + +For an arbitrary $y \in \Y_{1}\otimes\Y_{2}$, consider its Schmidt decomposition, +that is, a positive integer $r$ and orthogonal sets +$\{w_{1}, \ldots, w_{r}\} \subset \Y_{1}$ and $\{z_{1}, \ldots, z_{r}\} \subset \Y_{2}$ +such that + \begin{equation} + y = \sum_{i = 1}^{r}w_{i} \otimes z_{i}. + \end{equation} +It holds that +\begin{multline} +\label{eq:SandQ} + (\I\otimes y^{\ast})W^{\ast}(S \otimes Q)W(\I\otimes y)\\ + \begin{aligned} + &=\left(\sum_{i=1}^{r}\I_{\X_{1}\otimes\X_{2}}\otimes w_{i}^{\ast}\otimes z_{i}^{\ast}\right) + W^{\ast}(S \otimes Q)W + \left(\sum_{i=1}^{r}\I_{\X_{1}\otimes\X_{2}}\otimes w_{i}\otimes z_{i}\right) \\ + &=\left(\sum_{i=1}^{r}\I_{\X_{1}}\otimes w_{i}^{\ast}\otimes\I_{\X_{2}} \otimes z_{i}^{\ast}\right) + (S \otimes Q) + \left(\sum_{i=1}^{r}\I_{\X_{1}}\otimes w_{i}\otimes\I_{\X_{2}} \otimes z_{i}\right) \\ + &= \sum_{i,j=1}^{r}(\I_{\X_{1}}\otimes w_{i}^{\ast})S(\I_{\X_{1}}\otimes w_{j})\otimes(\I_{\X_{2}} \otimes z_{i}^{\ast})Q(\I_{\X_{2}} \otimes z_{j}). + \end{aligned} +\end{multline} +The separable operator $S \in \Sep(\X_{1}:\Y_{1})$ can be expressed as +\begin{equation} + S = \sum_{j = 1}^{m}a_{j}a_{j}^{\ast}\otimes b_{j}b_{j}^{\ast}, +\end{equation} +for vectors $a_{1}, \ldots, a_{m} \in \X_{1}$ and $b_{1}, \ldots, b_{m} \in \Y_{1}$. +By convexity, it is enough to argue about the operator +\begin{equation} + (\I\otimes y^{\ast})W^{\ast}(aa^{\ast}\otimes bb^{\ast} \otimes Q)W(\I\otimes y), +\end{equation} +for any vectors $a \in \X_{1}$ and $b \in \Y_{1}$. +From Eq.~\eqref{eq:SandQ}, we have that +\begin{multline} + (\I\otimes y^{\ast})W^{\ast}(aa^{\ast}\otimes bb^{\ast} \otimes Q)W(\I\otimes y)\\ +\begin{aligned} + &= \sum_{i,j=1}^{r}(aa^{\ast}\otimes w_{i}^{\ast}bb^{\ast}w_{j})\otimes(\I_{\X_{2}} \otimes z_{i}^{\ast})Q(\I_{\X_{2}} \otimes z_{j})\\ + &= aa^{\ast} \otimes \left(\I_{\X_{2}}\otimes\sum_{i=1}^{r} + w_{i}^{\ast} a z_{i}^{\ast}\right)Q + \left(\I_{\X_{2}}\otimes\sum_{i=1}^{r} a^{\ast}w_{i}z_{i}\right) \\ + &= aa^{\ast} \otimes (\I_{\X_{2}}\otimes z^{\ast})Q + (\I_{\X_{2}}\otimes z), +\end{aligned} +\end{multline} +where we have defined the vector $z \in \Y_{2}$ to be such that +$z = \sum_{i=1}^{r} a^{\ast}w_{i}z_{i}$. +From the fact the $Q \in \BPos(\X_{2}:\Y_{2})$, it holds that +\[ + (\I_{\X_{2}}\otimes z^{\ast})Q(\I_{\X_{2}}\otimes z) \in \Pos(\X_{2}), +\] +and therefore we have that +\begin{equation} + (\I_{\X_{1}\otimes\X_{2}} \otimes y^{\ast})W^{\ast} + (S \otimes Q)W + (\I_{\X_{1}\otimes\X_{2}} \otimes y) \in \Sep(\X_{1}\otimes\X_{2}) +\end{equation} +is positive semidefinite. +\end{proof} + +Now we are ready to show a feasible solution of the dual problem \eqref{eq:sep-dual-problem} +for the set of states \eqref{eq:ydy_higher_dimension}. +For any $t \geq 2$, we denote with $\X^{(t)}$ and $\Y^{(t)}$ two isomorphic +copies of the complex Euclidean space $\complex^{2^{t}}$, +so that the states in \eqref{eq:ydy_higher_dimension} lie in $\X^{(t)}\otimes\Y^{(t)}$. + +Consider the operator +\begin{equation} + S = \frac{1}{2}\vec(\sigma_{0} + \sigma_{1})\vec(\sigma_{0} + \sigma_{1})^{\ast} + \in \Pos(\X^{(1)}\otimes\Y^{(1)}). +\end{equation} +Let $W \in \Unitary(\X^{(t)}\otimes\Y^{(t)},\X^{(1)}\otimes\Y^{(1)}\otimes\X^{(t-1)}\otimes\Y^{(t-1)})$ +be the linear isometry defined in the statement of Lemma \ref{lemma:higher-dimension}, +acting on the specified spaces. +The solution of the dual problem may be recursively defined as follows: +\begin{equation} + H^{(t)} = W^{\ast}(S \otimes H^{(t-1)})W, +\end{equation} +for any $t \geq 2$. The base of the +recursion $H^{(2)}$ is the operator defined in Eq. \eqref{eq:H_2}. +In the rest of the section we prove, by induction, that the cone program constraint +\begin{equation} +\label{eq:condiction_Ht} + H^{(t)} - \frac{1}{2^{t}}\vec(U_{k}^{(t)})\vec(U_{k}^{(t)})^{\ast} + \in \BPos(\X^{(t)}:\Y^{(t)}) +\end{equation} +is satisfied for any $k \in \{1, \ldots, 2^{t} \}$. + +Let us consider an arbitrary $1 \leq k \leq 2^{t-1}$ +(the case $2^{t-1} \leq k \leq 2^{t}$ follows from a similar argument). +Eq. \eqref{eq:construction} implies that +\begin{equation} + \begin{split} + \vec(U_{k}^{(t)})\vec(U_{k}^{(t)})^{\ast} + &= W^{\ast}(\vec(\sigma_{0})\vec(\sigma_{0})^{\ast}\otimes + \vec(U_{k}^{(t-1)})\vec(U_{k}^{(t-1)})^{\ast})W \\ + &\leq + W^{\ast}(\vec(\sigma_{0} + \sigma_{1})\vec(\sigma_{0} + \sigma_{1})^{\ast} + \otimes \vec(U_{k}^{(t-1)})\vec(U_{k}^{(t-1)})^{\ast})W, + \end{split} +\end{equation} +and therefore that +\begin{align} + H^{(t)} - \frac{1}{2^{t}}\vec(U_{k}^{(t)})\vec(U_{k}^{(t)})^{\ast} + &= W^{\ast}(S \otimes H^{(t-1)})W - + \frac{1}{2^{t}}\vec(U_{k}^{(t)})\vec(U_{k}^{(t)})^{\ast}\\ + &\geq W^{\ast}(S \otimes (H^{(t-1)} - + \frac{1}{2^{t-1}}\vec(U_{k}^{(t-1)})\vec(U_{k}^{(t-1)})^{\ast})W. +\end{align} +The Peres-Horodecki criterion, along with the fact that +\[ + \pt_{\X}(S) = S \in \Pos(\X_{1}\otimes\Y_{1}), +\] +implies that $S\in\Sep(\X_{1}:\Y_{1})$. +From Lemma \ref{lemma:higher-dimension} and the induction hypothesis, we have that +\begin{equation} + W^{\ast}(S \otimes (H^{(t-1)} - + \frac{1}{2^{t-1}}\vec(U_{k}^{(t-1)})\vec(U_{k}^{(t-1)})^{\ast})W + \in \BPos(\X^{(t)}:\Y^{(t)}), +\end{equation} +and therefore the constraint \eqref{eq:condiction_Ht} is satisfied. +Moreover, we have that $\tr(Q) = 1$ and therefore +\[ + \tr(H^{(t)}) = \tr(H^{(2)}) = \frac{3}{4} +\] + +\subsubsection*{Small sets of locally indistinguishable orthogonal maximally entangled states} + +The main corollary of the proof above is that there exist +LOCC-indistinguishable sets of $k < n$ maximally entangled states in +$\complex^{n}\otimes\complex^{n}$. +Asymptotically, our construction allows for the cardinality $k$ of the indistinguishable +sets to be as small as $Cn$, where $C$ is a constant less than $1$. +In particular, we have that $3/4 \leq C < 1$. +It is possible that this constant can be improved by using +a different construction than the Yu--Duan--Ying states. +A further improvement would be to exhibit +indistinguishable sets of maximally entangled states with cardinality $o(d)$. +One among the smallest indistinguishable sets that come out of the above construction consists +of the states in $\complex^{8}\otimes\complex^{8}$ corresponding the following $7$ unitary operators: +\begin{equation} + \label{eq:ydy_Us} + \begin{aligned} + U_{1} &= \sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0},\\ + U_{2} &= \sigma_{0}\otimes\sigma_{1}\otimes\sigma_{1},\\ + U_{3} &= \sigma_{0}\otimes\sigma_{2}\otimes\sigma_{1},\\ + U_{4} &= \sigma_{0}\otimes\sigma_{3}\otimes\sigma_{1},\\ + U_{5} &= \sigma_{0}\otimes\sigma_{0}\otimes\sigma_{0},\\ + U_{6} &= \sigma_{0}\otimes\sigma_{1}\otimes\sigma_{1},\\ + U_{7} &= \sigma_{0}\otimes\sigma_{2}\otimes\sigma_{1}. + \end{aligned} +\end{equation} + +\begin{remark} +In a very recent result\footnote{A similar result (obtained via a different approach) appears also +in a preprint by Yu and Oh \cite{Yu15}. Notice that this has not been +published yet, neither I have verified it myself yet.}, +Li et al. \cite{Li15} build up on our proof from +\cite{Cosentino14} and show that indistinguishable sets of $N$ orthogonal +maximally entangled states in $\complex^{N}\otimes\complex^{N}$ exists for all +$N$ and not just when $N$ is a power of $2$. +\end{remark} + +\subsubsection*{Entanglement Discrimination Catalysis} +It is worth noting that the ``Entanglement Discrimination Catalysis'' +phenomenon, observed in \cite{Yu12} for the set \eqref{eq:ydy_states}, +also applies to the set +of states in the above example and to any set derived +from our construction. +If Alice and Bob are provided with a maximally entangled state +as a resource, then they are able to distinguish the states +in these sets and, when the protocol ends, +they are still left with an untouched maximally entangled state. +When $t=2$, the catalyst is used to teleport +the first qubit from one party to the other, +say from Alice to Bob. +Bob can then measure the first two qubits in the standard +Bell basis and identify which of the four states was prepared. +Since the third and fourth qubits are not being acted on, +they can be used in a new round of the protocol. +For the case $t > 2$, let us recall the recursive construction +of the states from \eqref{eq:construction}. +Distinguishing between the two cases of the recursion is +equivalent to distinguishing between two Bell states. +And the base case is exactly the case $t=2$ described above, +with only one maximally entangled state involved in the catalysis. + + +\subsubsection*{PPT vs. separable in the perfect discrimination of maximally entangled states} +Michael Nathanson (personal communication) raised the question whether there +always exists a separable measurement that \emph{perfectly} distinguishes maximally entangled +states that are known to be \emph{perfectly} distinguishable by PPT. +The construction that generalize Yu--Duan--Ying states in high dimension provides +a negative answer to Michael's question. It turns out that if we take only $7$ out of the $8$ states +coming from the construction for $t=3$ (for example, the ones corresponding to the operators in +Eq.~\eqref{eq:ydy_Us}), they are distinguishable by separable measurement +with probability at most $6/7$, but they are perfectly distinguishable by PPT. +Unfortunately we do not have a nice-looking closed form for +the PPT measurement operators that achieve perfect distinguishability, +but know, by running a semidefinite programming solver, that such operators exist. + +\subsection{Unambiguous discrimination} + +Interestingly, the optimal probability of unambiguously distinguish the set of Yu--Duan-Ying states +with PPT measurements is $3/4$, which should be compared with the success probability of $7/8$ that can be achieved with a minimum-error strategy +(see Theorem \ref{thm:primal-ppt-ydy}). +Using a semidefinite program solver, we were also able to verify that this bound is actually tight. + +\begin{theorem} +The maximum success probability of \emph{unambiguously} +distinguishing the ensemble in Eq.~\eqref{eq:ydy-ensemble} with PPT measurements is equal to $3/4$. +\end{theorem} +\begin{proof} +We show a feasible solution of the dual problem \eqref{sdp-dual-unambiguous} for which the value of the objective function is $3/4$. Let +\begin{equation} + Y = \frac{1}{16}[(\I-\psi_{1})\otimes(\I - 2\psi_{4}) + \psi_{1}\otimes(-\psi_{1}+3\psi_{2}+3\psi_{3}+\psi_{4})]. +\end{equation} +and +\begin{equation} +\begin{aligned} + Q_{1} &= [(\I - \psi_{3})\otimes\psi_{3} + \psi_{3}\otimes(\psi_{2}+\psi_{3})]/4, \\ + Q_{2} &= [(\psi_{1} + \psi_{2})\otimes\psi_{2} + \psi_{4}\otimes(\I - \psi_{2})]/4, \\ + Q_{3} &= [(\psi_{2} + \psi_{4})\otimes\psi_{2} + \psi_{1}\otimes(\I - \psi_{2})]/4, \\ + Q_{4} &= [(\psi_{1} + \psi_{4})\otimes\psi_{2} + \psi_{2}\otimes(\I - \psi_{2})]/4, \\ + Q_{5} &= (\psi_{3}\otimes\psi_{2})/4. +\end{aligned} +\end{equation} +We can use the equations in \eqref{eq:ppt-bell-states} to verify that +the constraints of the program \eqref{sdp-dual-unambiguous} are satisfied: +\begin{equation} +Y - \frac{1}{4}\rho_{j} + \sum_{\substack{1\leq i \leq k \\ i\neq j}}\rho_{i} = \pt_{\A}(Q_{j}), \quad j=1,\ldots,4 \quad\text{and}\quad Y \geq \pt_{\A}(Q_{5}), +\end{equation} +Finally, we have $\tr(Y) = 3/4$. +\end{proof} + +% move to appendix! +% \section{Generalized Bell states} + +% \begin{example}[Example 1 from \cite{Bandyopadhyay11a}] +% \begin{equation} +% \ket{\phi_{00}}, \ket{\phi_{11}}, \ket{\phi_{32}}, \ket{\phi_{31}} +% \end{equation} +% \end{example} + +% \begin{example}[Example 2 from \cite{Bandyopadhyay11a}] +% \begin{equation} +% \ket{\phi_{00}}, \ket{\phi_{11}}, \ket{\phi_{32}}, \ket{\phi_{31}} +% \end{equation} +% \end{example} + +% \begin{example}[Example 3 from \cite{Bandyopadhyay11a}] +% \begin{equation} +% \ket{\phi_{00}}, \ket{\phi_{11}}, \ket{\phi_{32}}, \ket{\phi_{31}} +% \end{equation} +% \end{example} + diff --git a/preliminaries.tex b/preliminaries.tex new file mode 100644 index 0000000..616fa2b --- /dev/null +++ b/preliminaries.tex @@ -0,0 +1,714 @@ +%!TEX root = thesis.tex +%------------------------------------------------------------------------------- +\chapter{Preliminaries} +\label{chap:preliminaries} +%------------------------------------------------------------------------------- + +In this chapter we summarize basic concepts of quantum information theory +that will be used in the rest of the thesis. +Along the way, we will also pin down the notation that +we use throughout this thesis, although for most part, +we use notation that is standard in quantum information theory. +In particular, we will follow the same terminology and conventions adopted +in \cite{Watrous15}. +This should not serve as in introduction to quantum information. For such +an introduction we refer the reader to a standard textbook \cite{Nielsen11}. + +The last section introduces the basic concepts of convex optimization that are +necessary for analyzing problems in quantum information theory. +For a more extended treatment of semidefinite programming and cone programming, +we refer the reader to \cite{Wolkowicz00,Tuncel12} and the references therein. + +\minitoc + +%------------------------------------------------------------------------------- +\section{Basic notions of quantum information theory} +\label{sec:basic-notions-of-quantum-information-theory} +%------------------------------------------------------------------------------- + +\subsection{Vector spaces, linear operators, and linear mappings} + +All vector spaces considered here are assumed to be complex Euclidean spaces +(or, equivalently, finite-dimensional complex Hilbert spaces) and are denoted +by scripted capital letters from the end of the English alphabet, +such as $\X, \Y, \Z$. +Elements of a complex Euclidean space are denoted by lower-case letters +from the end of the English alphabet, such as $u, v, w, z$. For a complex +Euclidean spaces of dimension $n$, elements of the space can be represented as +vectors in $\complex^{n}$. +The standard basis of such a space is denoted using the Dirac notation as +$\left\{ \ket{0}, \ldots, \ket{n-1} \right\}$. + +The inner product of two vectors $u, v \in \complex^{n}$ is defined as +\begin{equation} + \ip{u}{v} = \sum_{i\in\{1,\ldots,n\}}\overline{u(i)}v(i). +\end{equation} + +We write $\Lin(\X,\Y)$ to denote the space of linear operators from a space $\X$ +to a space $\Y$, and we write $\Lin(\X)$ as shorthand for $\Lin(\X,\X)$. + +For every operator $A\in\Lin(\X,\Y)$, the operator $A^{\ast}\in\Lin(\Y,\X)$ +denotes the adjoint of $A$, that is, the unique operator that satisfies the equation +\begin{equation} + \ip{v}{Au} = \ip{A^{\ast}v}{u}, +\end{equation} +for all $u\in\X$ and $y\in\Y$. +In the matrix representation of linear operators, $A^{\ast}$ is the conjugate transpose of +the matrix corresponding to $A$. + +\renewcommand{\descriptionlabel}[1]{\hspace{\labelsep}\emph{#1}} + +For any space $\X$, we denote some important sets of operators acting on $\X$ as follow: +\begin{description} +\item[Hermitian operators -- $\Herm(\X)$]: operators $X\in\Lin(\X)$ such that $X^{\ast} = X$. +\item[Positive semidefinite operators -- $\Pos(\X)$]: operators $X\in\Lin(\X)$ for which it holds + that $X = Y^{\ast}Y$ for some operator $Y\in\Lin(\X)$. +\item[Density operators -- $\Density(\X)$]: positive semidefinite operators + having trace equal to $1$. To denote density operators, we will use letters from + the Greek alphabet, like $\rho,\sigma,\xi$. +\end{description} +The eigenvalues of an Hermitian operator are all real numbers. +Positive semidefinite operators are Hermitian by definition, and they can be +described as those Hermitian operators that have only nonnegative eigenvalues. +To recap, we have the following chain of containments: +\begin{equation} + \Density(\X) \subset \Pos(\X) \subset \Herm(\X) \subset \Lin(\X). +\end{equation} +The identity operator acting on a given space $\X$ is denoted by $\I_{\X}$, +or just as $\I$ when $\X$ is implicit. +For Hermitian operators $A, B \in \Herm(\X)$ the notations $A\geq B$ +and $B \leq A$ indicate that $A - B$ is positive semidefinite. + +Other important operators are \emph{linear isometries}, which are all operators +$X\in\Lin(\X,\Y)$ such that $X^{\ast}X=\I_{\X}$. The set of linear isometries is +denoted by $\Unitary(\X,\Y)$. +Linear isometries in $\Lin(\X)$ are called \emph{unitary operators} and their +set is denoted by $\Unitary(\X)$. + +We denote the standard Hilbert-Schmidt inner product of operators +$X$ and $Y$ as +\begin{equation} + \ip{X}{Y} = \tr(X^{\ast}Y). +\end{equation} + +The trace norm of an operator $A\in\Lin(\X,\Y)$ is defined as +\begin{equation} + \norm{A}_{1} = \tr(\sqrt{A^{\ast}A}), +\end{equation} +where $\sqrt{X}$ denotes the square root of a positive semidefinite +operator $X$, that is, the unique positive semidefinite operator $Y$ such that $Y^{2} = X$. + +A quantum state is represented by a density operator $\rho\in\Density(\X)$, +for some complex Euclidean space $\X$. A state $\rho\in\Density(\X)$ is said +to be \emph{pure} if and only if it has rank equal to 1, or equivalently, +if there exists a unit vector $u\in\X$ such that +\[ + \rho = uu^{\ast}. +\] + +Along with linear operators, we will consider linear mappings of the form +\[ + \Phi:\Lin(\X) \rightarrow \Lin(\Y), +\] +for complex Euclidean spaces $\X$ and $\Y$. +The \emph{adjoint} of a mapping $\Phi$ is defined to be the unique mapping +\[ + \Phi^{\ast}:\Lin(\Y) \rightarrow \Lin(\X), +\] +which satisfies +\begin{equation} +\label{eq:adjoint-map} + \ip{\Phi(X)}{Y} = \ip{X}{\Phi^{\ast}(Y)}. +\end{equation} +Some important sets of linear mappings that we will consider in this thesis are +the following: +\begin{description} +\item[Hermiticity preserving] -- mappings of the form + $\Phi:\Lin(\X)\to\Lin(\Y)$ such that \[\Phi(X) \in \Herm(\Y),\] for any $X\in\Herm(\X)$. +\item[Positive] -- mappings of the form + $\Phi:\Lin(\X)\to\Lin(\Y)$ such that $\Phi(X) \in \Pos(\Y)$, for any $X\in\Pos(\X)$. +\item[Completely positive] -- mappings of the form + $\Phi:\Lin(\X)\to\Lin(\Y)$, such that + \[\Phi\otimes\I_{\Lin(\Z)}(X)\in\Pos(\Y\otimes\Z),\] + for every complex Euclidean space $\Z$, and any $X\in\Pos(\X\otimes\Z)$. +\item[Trace-preserving] -- mappings of the form + $\Phi:\Lin(\X)\to\Lin(\Y)$ such that + \[ \tr(\Phi(X)) = \tr(X),\] + for all $X\in\Lin(\X)$. +\end{description} +Transformations of a quantum system from one state to another are described by +\emph{quantum channel}, which are completely positive, trace-preserving linear mappings. + +Given the tensor product $\X_{1}\otimes\ldots\otimes\X_{n}$ of $n$ complex Euclidean spaces +$\X_{1},\ldots,\X_{n}$, and a partition +\[ (k_{1},\ldots,k_{i} : k_{i+1},\ldots,k_{n}) \] +of the set $\{1, \ldots, n\}$, we use the notation +\[ + (\X_{k_{1}}\otimes\ldots\otimes\X_{k_{i}} : \X_{k_{i+1}}\otimes\ldots\otimes\X_{k_{n}}) +\] +to denote a bipartition of the entire space. + +It is convenient for the analysis of states in a bipartition $(\X:\Y)$ +to make use of the correspondence between operators and vectors +given by the linear function +\begin{equation} +\label{eq:vec} + \op{vec} : \Lin(\Y,\X) \rightarrow \X\otimes\Y +\end{equation} +defined by the action +\begin{equation} + \op{vec}(\ket{k}\bra{j}) = \ket{k} \ket{j} +\end{equation} +on standard basis vectors (and by linearity to all $\Lin(\Y,\X)$). + +\begin{definition} +\label{def:max-ent-states} +Suppose that $\X$ and $\Y$ are complex Euclidean spaces with $n = \dim(\X)$ and +$m = \dim(\Y)$, and assume $n\geq m$. +A unit vector $u\in\X\otimes\Y$, representing a pure state, is said to be +\emph{maximally entangled} provided that +\begin{equation} + \tr_{\X}(u u^{\ast}) = \frac{\I_{\Y}}{m}. +\end{equation} +This condition is equivalent to +\begin{equation} + u = \frac{1}{\sqrt{m}} \vec(A) +\end{equation} +for $A\in\Unitary(\Y,\X)$ being a linear isometry. +\end{definition} + +\subsection{Pauli operators and Bell states} +One particularly important set of linear operators in $\Lin(\complex^{2})$ is +the set of Pauli operators +\begin{equation} +\label{eq:Pauli-operators} + \begin{array}{llll} + \sigma_{0} = \I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, & + \sigma_{1} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, & + \sigma_{2} = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, & + \sigma_{3} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}. + \end{array} +\end{equation} +The operators $\sigma_{1},\sigma_{2},\sigma_{3}$ are often referred as Pauli $X, Y, Z$, respectively. +The Pauli operators are Hermitian, unitary operators, and moreover, +they are orthogonal under the inner product, thus forming an orthogonal basis for +$\Lin(\complex^{2})$. + +Through the vector-operator correspondence of Eq.~\ref{eq:vec}, +the Pauli operators define an important class of maximally +entangled states, famously known as \emph{Bell states}: +\begin{equation} + \psi_{i} = \frac{1}{2}\vec(\sigma_{i})\vec(\sigma_{i})^{\ast}, +\end{equation} +for $i \in \{0,1,2,3\}$. More explicitly, the Bell states can be written down as follows: +\begin{equation} +\label{eq:Bell-states} + \begin{aligned} + \ket{\psi_0} & = \frac{1}{\sqrt{2}}\ket{0}\ket{0} + + \frac{1}{\sqrt{2}}\ket{1}\ket{1},\\ + \ket{\psi_1} & = \frac{1}{\sqrt{2}}\ket{0}\ket{1} + + \frac{1}{\sqrt{2}}\ket{1}\ket{0},\\ + \ket{\psi_2} & = \frac{1}{\sqrt{2}}\ket{0}\ket{1} + - \frac{1}{\sqrt{2}}\ket{1}\ket{0},\\ + \ket{\psi_3} & = \frac{1}{\sqrt{2}}\ket{0}\ket{0} + - \frac{1}{\sqrt{2}}\ket{1}\ket{1}. + \end{aligned} +\end{equation} +In higher dimensions, one can consider a generalization of the Pauli operators. +For any positive integer $n$, let us the define an $n$-th primitive root of unity as +\begin{equation} + \omega_{n} = \exp(2\pi i/n). +\end{equation} +The generalization of Pauli-$X$ and Pauli-$Z$ in $\Unitary(\complex^{n})$ +are defined as follows: +\begin{equation} + X_{n} = \sum_{j \in \integer_{n}}\ket{j+1}\bra{j}, +\end{equation} +and +\begin{equation} + Z_{n} = \sum_{j \in \integer_{n}}^{n}\omega_{n}^{j}\ket{j}\bra{j}. +\end{equation} +One can define the set of \emph{generalized Pauli operators} in $\Unitary(\complex^{n})$ as the set +\begin{equation} +\label{eq:generalized-Pauli-operators} + \left\{ W_{a,b} = X^{a}Z^{b} : a,b\in\integer_{n}\right\}. +\end{equation} +Starting from these operators we define the \emph{generalized Bell basis} +through the vector-operator bijection: +\begin{equation} +\label{eq:generalized-bell-basis} + \left\{ \frac{1}{\sqrt{n}}\vec(W_{a,b}) : a,b\in\integer_{n} \right\}. +\end{equation} + +%------------------------------------------------------------------------------- +\subsection{The Choi isomorphism} +\label{sec:choi-isomorphism} +%------------------------------------------------------------------------------- + +To a quantum mapping $\Phi : \Lin(\X)\rightarrow\Lin(\Y)$, we associate an +operator $J(\Phi) \in \Lin(\Y\otimes\X)$ defined as follows: +\begin{equation} + J(\Phi) = (\Phi\otimes\I_{\Lin(\X)})(\vec(\I_{\X})\vec(\I_{\X})^{\ast}). +\end{equation} +If we are assuming that $\X$ has dimension $n$, we can alternatively write this as +\begin{equation} + J(\Phi) = \sum_{1\leq i,j \leq n}\Phi(\ket{i}\bra{j})\otimes\ket{i}\bra{j}. +\end{equation} +The operator $J(\Phi)$ is called the \emph{Choi representation} +of $\Phi$. +It is often the case that properties of the Choi representation reveal useful information +on the mapping. For instance, positive semidefinite operators correspond to Choi representations +of completely positive mappings. + +%------------------------------------------------------------------------------- +\section{Quantum measurements} +\label{sec:quantum-measurements} +%------------------------------------------------------------------------------- +When we analyze state distinguishability problems, all the physical operations +performed by the parties involved can be formally phrased in terms of quantum measurements. +A \emph{quantum measurement} is defined as a function +\begin{equation} +\label{eq:def-measurement} + \mu : \{ 1, \ldots, N \} \rightarrow \Pos(\X), +\end{equation} +for some choice of a positive integer $N > 0$ and a complex Euclidean space $\X$, +satisfying the constraint +\begin{equation} +\label{eq:measurement-sum} + \sum_{k=1}^{N} \mu(k) = \I_{\X}. +\end{equation} +The values $\{1, \ldots, N\}$ are the \emph{measurement outcomes} of $\mu$, +and each operator $\mu(k)$ is the \emph{measurement operator} of $\mu$ +associated with the outcome $k$. +The set of all measurements over $\X$ with $N$ outcomes is denoted by +$\Meas(N, \X)$ and it is a subset of all functions of the same kind of $\mu$, that is, +\begin{equation} + \Meas(N, \X) \subset \Pos(\X)^{\{1,\ldots,N\}}. +\end{equation} + +Given a measurement $\mu: \{1, \ldots, N \} \rightarrow \Pos(\X)$, +it is useful to associate a mapping $\Phi_{\mu}:\Lin(\X)\rightarrow\Lin(\complex^{N})$ to it, +defined as follows: +\begin{equation} +\label{eq:measurement-channel} + \Phi_{\mu}(X) = \sum_{k=1}^{N}\ip{\mu(k)}{X}\ket{k}\bra{k}, +\end{equation} +for any $X\in\Lin(\X)$. The mapping $\Phi_{\mu}$ is a quantum channel and, in fact, +it is a \emph{quantum-to-classical} channel. + +In order to capture the limitation of some physical processes, we can define more restricted +classes of measurements, which will be object of study of this thesis. + +\subsection{LOCC measurements} + +We refer the reader to the references \cite{Mancinska13,Watrous15} for the precise +definition of an LOCC channel. +To a measurement $\mu: \{1, \ldots, N \} \rightarrow \Pos(\X\otimes\Y)$ on a bipartite +system, we associate the quantum-to-classical channel +\begin{equation} +\label{eq:measurement-channel-bipartite} + \Phi_{\mu}(X) = \sum_{k=1}^{N}\ip{\mu(k)}{X}\ket{k}\bra{k}\otimes\ket{k}\bra{k}, +\end{equation} +and we say that $\mu$ is an LOCC measurement if the channel $\Phi_{\mu}$ can be +implemented by an LOCC protocol between Alice and Bob. + +Notice that we can define different classes of LOCC, +according to the number (finite or infinite) of rounds that compose the protocols. +For the scope of this thesis, the only important thing to notice is that all the LOCC variants +are contained in the class of separable measurement, which will be defined in the next +section. + +We denote the set of all $N$-outcome LOCC bipartite measurements +on the bipartition $(\X:\Y)$ by $\Meas_{\LOCC}(N, \X:\Y)$. + +\subsection{Separable measurements} + +The class of \emph{separable measurements} represents a commonly studied +approximation of the set of LOCC measurements. +A positive semidefinite operator $P\in\Pos(\X\otimes\Y)$ is said to be +\emph{separable} if it is possible to write +\begin{equation} + P = \sum_{k = 1}^{M} Q_k \otimes R_k, +\end{equation} +for some choice of a positive integer $M$ and positive semidefinite operators +\begin{equation} + Q_1,\ldots,Q_M \in \Pos(\X) + \hspace*{1cm}\mbox{and}\hspace*{1cm} + R_1,\ldots,R_M\in\Pos(\Y). +\end{equation} + +\begin{definition} +Let $\X^{\reg{A}}, \X^{\reg{B}}, \Y^{\reg{A}}$, and $\Y^{\reg{B}}$ be complex +Euclidean spaces. +A completely positive mappings +\[ + \Phi:\Lin(\X^{\reg{A}}\otimes\X^{\reg{B}})\to\Lin(\Y^{\reg{A}}\otimes\Y^{\reg{B}}) +\] +is said to be a +\emph{separable channel} if it is a trace-preserving mappings and it is possible +to write +\begin{equation} + \Phi = \sum_{k = 1}^{M}\Psi_{k}^{\reg{A}}\otimes\Psi_{k}^{\reg{B}}, +\end{equation} +for some choice of a positive integer $M$ and collections of +completely positive mappings +\begin{equation} + \Psi_1^{\reg{A}},\ldots,\Psi_{M}^{\reg{A}} : \Lin(\X^{\reg{A}})\to\Lin(\Y^{\reg{A}}) + \hspace*{1cm}\mbox{and}\hspace*{1cm} + \Psi_1^{\reg{B}},\ldots,\Psi_{M}^{\reg{B}} : \Lin(\X^{\reg{B}})\to\Lin(\Y^{\reg{B}}). +\end{equation} +\end{definition} + +\begin{definition} +Let $\X$ and $\Y$ be complex Euclidean spaces, let $N > 0$ a positive integer, and let +\begin{equation} + \mu: \{1, \ldots, N \} \rightarrow \Pos(\X\otimes\Y) +\end{equation} +be a measurement. Then $\mu$ is said to be a \emph{separable measurement} if the +corresponding quantum-to-classical channel $\Phi_{\mu}$, defined as in +Eq.~\eqref{eq:measurement-channel-bipartite}, is a separable channel. +\end{definition} + +\begin{prop} +\label{prop:separable-each} +Let $\X$ and $\Y$ be complex Euclidean spaces, and let $N > 0$. A measurement +$\mu : \{1,\ldots,N\}\to\Pos(\X\otimes\Y)$ is a separable measurement +if and only if each $\mu(k)$ is a separable operator, that is, +$\mu(k)\in \Sep(\X:\Y)$, for each $k\in\{1, \ldots, N\}$. +\end{prop} + +We refer the reader to \cite{Watrous15} for a proof of Proposition \ref{prop:separable-each}, +as well as for the proof that every LOCC measurement is necessarily a separable measurement. +From this latter statement it follows that any limitation proved to hold for +every separable measurement must also hold for every LOCC measurement. + +\subsection{PPT measurements} +\label{sec:ppt-measurements} + +Another class that represents a relaxation of the set of LOCC measurements is +the class of PPT measurements. +Let $\pt_{\negsmallspace\X}:\Lin(\X\otimes\Y)\rightarrow\Lin(\X\otimes\Y)$ be +the linear mapping representing partial transposition with respect to the +standard basis $\{\ket{0},\ldots,\ket{n-1}\}$ of $\X$. Equivalently, +\begin{equation} + \pt_{\X}(X) = (\pt\otimes\I_{\Lin(\Y)})(X), +\end{equation} +for any operator $X\in\Lin(X\otimes\Y)$, where $\pt:\Lin(\X)\to\Lin(\X)$ +is the transpose mapping. + +We will use the fact that the transpose mapping is its own adjoint and inverse, that is, +\begin{equation} + \ip{\pt(X)}{Y} = \ip{X}{\pt(Y)}, \qquad\mbox{for any }X,Y\in\Lin(\X), +\end{equation} +and +\begin{equation} + \pt(\pt(X)) = X,\qquad\mbox{for any }X\in\Lin(\X). +\end{equation} + +A positive semidefinite operator $P\in \Pos(\X\otimes\Y)$ is a \emph{PPT} +(short for \emph{positive partial transpose}) operator if it holds that +\begin{equation} + \pt_{\negsmallspace\X}(P) \in \Pos(\X\otimes\Y). +\end{equation} +We denote the set of all PPT operators in $\Pos(\X\otimes\Y)$ as +\begin{equation} +\PPT(\X:\Y) = \{ P \in \Pos(\X\otimes\Y) \,:\, + \pt_{\X}(P) \in \Pos(\X\otimes\Y) \}. +\end{equation} +Notice that transpose and partial transpose are basis dependent, but the notion of PPT +is not. Also, in the definition of $\PPT(\X:\Y)$, it is irrelevant which of the two subspaces the partial transpose acts on. In fact, since the transpose is a positive mapping, we have that +\begin{equation} + \pt_{\negsmallspace\Y}(P) \in \Pos(\X\otimes\Y)\Rightarrow + \pt(\pt_{\negsmallspace\Y}(P)) = \pt_{\negsmallspace\X}(P) + \in \Pos(\X\otimes\Y). +\end{equation} + +By definition, a measurement is PPT if all its operators are PPT. +\begin{definition} +\label{def:ppt-measurements} +A measurement $\mu : \{1, \ldots, N\}\rightarrow\Pos(\X\otimes\Y)$ is called +\emph{PPT} if it is represented by a collection +of PPT measurement operators, that is, +\begin{equation} +\mu(k) \in \PPT(\X:\Y), +\end{equation} +for all $k \in \{1,\ldots,N\}$. +\end{definition} + +Every separable operator is a PPT operator, so every separable measurement +(and therefore every LOCC measurement) is a PPT measurement as well. + +\begin{theorem} +Any separable operator $P \in \Sep(\X:\Y)$ is also a PPT operator over the same +bipartition, that is, $P \in \PPT(\X:\Y)$. +\end{theorem} +\begin{proof} +Suppose that $P \in \Sep(\X:\Y)$. Then it holds that +\begin{equation} + P = \sum_{k = 1}^{M} Q_k \otimes S_k, +\end{equation} +for some choice of a positive integer $M > 0$ and collections of operators +\begin{equation} + Q_1,\ldots,Q_M \in \Pos(\X) + \hspace*{1cm}\mbox{and}\hspace*{1cm} + R_1,\ldots,R_M\in\Pos(\Y). +\end{equation} +As the transpose mapping is positive, we have +\begin{equation} + (\pt \otimes \I_{\Lin(\Y)})(S) = \sum_{k=1}^{M}\pt(P_{k})\otimes Q_{k} + \in \Pos(\X\otimes\Y), +\end{equation} +and therefore $S \in \PPT(\X:\Y)$. +\end{proof} + +The PPT criterion is therefore a necessary condition for separability. It is +also sufficient in $\complex^{2}\otimes\complex^{3}$ and $\complex^{2}\otimes\complex^{3}$ +\cite{Peres1996,Horodecki1996}, but it is not in higher dimension, +where there are entangled PPT operators. + +The Choi operator of the transpose mapping $\pt : \Lin(\complex^{n})\to\Lin(\complex^{n})$ +is the \emph{swap operator} +$W_{n} \in \Unitary(\complex^{n}\otimes\complex^{n})$, defined on the standard basis as +\begin{equation} +\label{eq:swap-operator} + W_{n} = \sum_{i,j=0}^{n-1}\ket{i}\bra{j}\otimes\ket{j}\bra{i}. +\end{equation} +The swap operator is not positive semidefinite and therefore the transpose +mapping is not completely positive. + +The primary appeal of the set of PPT measurements is its mathematical simplicity. +In particular, the PPT condition is represented by linear and positive +semidefinite constraints, which allows for an optimization over the collection +of PPT measurements to be represented by a semidefinite program. + +The reader may find useful the Venn diagram of Figure~\ref{fig:classes-measurements}, +which pictures inclusion relationships between the main classes of mesaurements +considered in the thesis. Notice that all inclusion in the diagram are known +to be strict. + +\begin{figure}[!ht] + \centering + \def\svgwidth{200pt} + \scalebox{.75}{ + \input{drawing.pdf_tex}} + \caption{Inclusion relationships between classes of measurements.} + \label{fig:classes-measurements} +\end{figure} + +%------------------------------------------------------------------------------- +\section{Convex optimization} +\label{sec:convex-optimization} +%------------------------------------------------------------------------------- + +All the results of this thesis are based on a mathematical framework +called cone programming, which generalizes semidefinite programming. +There has been an extensive range of applications of semidefinite programming +to quantum information theory, but this is not the case for +the more general cone programming framework. +The success of semidefinite programming in quantum information comes from the fact +that many quantum primitives (states, channels, measurements) +can be represented within the cone of positive semidefinite operators with linear +constraints. +In this section we review basic definition in convex analysis and convex optimization. + +Let $\V$ be an arbitrary vector space over the real or complex number. +A subset $\C$ of $\V$ is a \emph{cone} if $u\in\C$ implies that $\lambda u \in \C$, +for all $\lambda \geq 0$. A cone $\C$ is convex if $u,v\in\C$ implies that +$u + v \in \K$. +A cone program (also known as a \emph{conic program}) expresses the +maximization of a linear function over the intersection of an affine subspace +and a closed convex cone in a finite-dimensional real inner product +space \cite{Boyd04}. + +When describing a cone program, it is sometimes convenient to compose small closed convex +cones in a bigger one, and in order to do that, one can make use of the following fact. +\begin{fact} +\label{fact:direct-sum-closed} +The direct sum $\K \oplus \K'$ of two closed convex cones $\K$ and $K'$ is a +closed convex cone. +\end{fact} + +Linear programming (LP) and semidefinite programming (SDP) are special cases of cone +programming: in linear programming, the closed convex cone over which the +optimization occurs is the positive orthant in $\real^n$, while in semidefinite +programming the optimization is over the cone $\Pos(\complex^n)$ of positive +semidefinite operators on $\complex^n$. +In the case of semidefinite programming, the finite-dimensional real inner +product space is the real vector space $\Herm(\complex^n)$ of Hermitian +operators on $\complex^n$, equipped with the Hilbert-Schmidt inner product. + +\subsection*{Linear programming} +\label{sec:linear-programming} +Let $n,m$ be positive integers, $c\in\real^{n}$ and $b\in\real^{m}$ be vectors +of real numbers, and $A \in\real^{n\times m}$ be a matrix. Then a \emph{linear program} +is defined by the triple $(c,b,A)$ and by the following pair of optimization problems. +\begin{center} + \begin{minipage}{2in} + \centerline{\underline{Primal linear program}}\vspace{-7mm} + \begin{align*} + \text{maximize:}\quad & \ip{c}{x}\\ + \text{subject to:}\quad & A x = b,\\ + & x \geq 0. + \end{align*} + \end{minipage} + \hspace*{1.5cm} + \begin{minipage}{2.4in} + \centerline{\underline{Dual linear program}}\vspace{-7mm} + \begin{align*} + \text{minimize:}\quad & \ip{b}{y}\\ + \text{subject to:}\quad & A^{\t}y \geq c,\\ + & y\in\real^{n}. + \end{align*} + \end{minipage} +\end{center} + + +\subsection*{Semidefinite programming} +\label{sec:semidefinite-programming} + +Let $\X$ and $\Y$ be complex Euclidean spaces, $A\in\Herm(\X)$ and $B\in\Herm(\Y)$ +be Hermitian operators, and $\Phi:\Lin(\X)\to\Lin(\Y)$ be a Hermiticity preserving mapping. +Then a \emph{semidefinite program} is defined by the triple $(A,B,\Phi)$ and by +the following pair of optimization problems. +\begin{center} + \begin{minipage}{2in} + \centerline{\underline{Primal semidefinite program}}\vspace{-7mm} + \begin{align*} + \text{maximize:}\quad & \ip{A}{X}\\ + \text{subject to:}\quad & \Phi(X) = B,\\ + & X \in \Pos(\X). + \end{align*} + \end{minipage} + \hspace*{1.5cm} + \begin{minipage}{2.4in} + \centerline{\underline{Dual semidefinite program}}\vspace{-7mm} + \begin{align*} + \text{minimize:}\quad & \ip{B}{Y}\\ + \text{subject to:}\quad & \Phi^{\ast}(Y) \geq A,\\ + & Y\in\Herm(\Y). + \end{align*} + \end{minipage} +\end{center} + +\subsection*{Cone programming} +\label{sec:cone-programming} +For the purposes of the present thesis, it is sufficient to consider only cone +programs defined over spaces of Hermitian operators (with the Hilbert-Schmidt +inner product). +In particular, let $\Z$ and $\W$ be complex Euclidean spaces and let +$\K\subseteq\Herm(\Z)$ be a closed, convex cone. +For any choice of a linear map +$\Phi:\Herm(\Z)\rightarrow\Herm(\W)$ and +Hermitian operators $A\in\Herm(\Z)$ and $B\in\Herm(\W)$, one has a \emph{cone +program} defined by $(A,B,\Phi)$ and represented by the following pair of optimization problems: +\begin{center} + \begin{minipage}{2in} + \centerline{\underline{Primal cone program}}\vspace{-7mm} + \begin{align*} + \text{maximize:}\quad & \ip{A}{X}\\ + \text{subject to:}\quad & \Phi(X) = B,\\ + & X \in \K. + \end{align*} + \end{minipage} + \hspace*{1.5cm} + \begin{minipage}{2.4in} + \centerline{\underline{Dual cone program}}\vspace{-7mm} + \begin{align*} + \text{minimize:}\quad & \ip{B}{Y}\\ + \text{subject to:}\quad & \Phi^{\ast}(Y) - A \in \K^{\ast},\\ + & Y\in\Herm(\W). + \end{align*} + \end{minipage} +\end{center} + +Here, $\K^{\ast}$ denotes the \emph{dual cone} to $\K$, defined as +\begin{equation} +\label{eq:dual-cone} + \K^{\ast} = \{Y\in\Herm(\Z)\,:\,\ip{X}{Y} \geq 0\;\:\text{for all $X\in\K$}\}, +\end{equation} +and $\Phi^{\ast}:\Herm(\W)\rightarrow\Herm(\Z)$ is the adjoint mapping to +$\Phi$. + +We observe the following elementary fact. +\begin{fact} +\label{fact:pos-self-dual} +The cone of positive semidefinite operators is self-dual, that is, +\begin{equation} +\Pos(\X) = (\Pos(\X))^{\ast}, +\end{equation} +for any complex Euclidean space $\X$. +\end{fact} +In light of this, we have that $\K = \K^{\ast} = \Pos(\X)$ and we can write the +constraint from the cone programming dual problem as +$\Phi^{\ast}(Y) - A \in \Pos(\X)$, that is, $\Phi^{\ast}(Y) \geq A$. + +Most of the definitions we introduce in the rest of the section holds for all linear, semidefinite, and general cone programs. +For a cone program defined by $(A,B,\Phi)$, one defines the +\emph{feasible sets} $\A$ and $\B$ of the primal and dual problems as +\begin{equation} + \A = \bigl\{ X \in \K : \Phi(X) = B\bigr\} + \qquad \text{and} \qquad + \B = \bigl\{ Y \in \Herm(\W) : \Phi^{\ast}(Y) - A \in \K^{\ast} \bigr\}. +\end{equation} +One says that the associated cone program is \emph{primal feasible} if +$\A \neq \emptyset$, and is \emph{dual feasible} if $\B \neq \emptyset$. +The function $X \mapsto \ip{A}{X}$ from $\Herm(\Z)$ to $\real$ is called the +\emph{primal objective function}, and the function $Y \mapsto \ip{B}{Y}$ from +$\Herm(\W)$ to $\real$ is called the \emph{dual objective function}. +The \emph{optimal values} associated with the primal and dual problems are +defined as +\begin{equation} + \label{eq:alpha-and-beta} + \alpha = \sup \bigl\{ \ip{A}{X} : X \in \A \bigr\} + \qquad \text{and} \qquad + \beta = \inf \bigl\{ \ip{B}{Y} : Y \in \B \bigr\}, +\end{equation} +respectively. +(It is conventional to interpret that $\alpha = -\infty$ when $\A = \emptyset$ +and $\beta = \infty$ when $\B = \emptyset$.) +The property of \emph{weak duality}, which holds for all cone programs, is +that the primal optimum can never exceed the dual optimum. + +\begin{prop}[Weak duality for cone programs] +\label{prop:weak-duality-cone} + For any choice of complex Euclidean spaces $\Z$ and $\W$, a closed, convex + cone $\K\subseteq\Herm(\Z)$, Hermitian operators $A\in\Herm(\Z)$ and + $B\in\Herm(\W)$, and a linear map $\Phi:\Herm(\Z)\rightarrow\Herm(\W)$, it + holds that $\alpha \leq \beta$, for $\alpha$ and $\beta$ as defined in + \eqref{eq:alpha-and-beta}. +\end{prop} + +\begin{proof} +The proposition is trivial in case $\A = \emptyset$ (which implies that +$\alpha = -\infty)$ or $\B = \emptyset$ (which implies that $\beta = \infty$), +so we will restrict our attention to the case that both $\A$ and $\B$ are +nonempty. +For any choice of $X \in \A$ and $Y \in \B$, one must have $X \in \K$ and +$\Phi^{\ast}(Y) - A \in \K^{\ast}$, and therefore +$\ip{\Phi^{\ast}(Y) - A}{X} \geq 0$. +It follows that +\begin{equation} + \ip{A}{X} = \ip{\Phi^{\ast}(Y)}{X} - \ip{\Phi^{\ast}(Y) - A}{X} + \leq \ip{Y}{\Phi(X)} = \ip{B}{Y}. +\end{equation} +Taking the supremum over all $X \in \A$ and the infimum over all +$Y \in \B$ establishes that $\alpha \leq \beta$. +\end{proof} + +Weak duality implies that every dual feasible operator $Y \in \B$ provides an +upper bound of $\ip{B}{Y}$ on the value $\ip{A}{X}$ that is achievable over all +choices of a primal feasible $X \in \A$, and likewise every primal feasible +operator $X \in \A$ provides a lower bound of $\ip{A}{X}$ on the value +$\ip{B}{Y}$ +that is achievable over all choices of a dual feasible solution $Y \in \B$. +In other words, it holds that +$\ip{A}{X} \leq \alpha \leq \beta \leq \ip{B}{Y}$, +for every $X \in \A$ and $Y \in \B$. + +Some cone programs also satisfy +the property of \emph{strong duality}, which holds when +the optimal values of the primal program and of the dual program are equal, and +the optimal value of the dual program is attained. +We abstain from a formal treatement of the conditions that guarantee +strong duality. Even though all cone programs described in the following chapters +satisfy strong duality, none of our results depend on that. diff --git a/programs.tex b/programs.tex new file mode 100644 index 0000000..9c1562d --- /dev/null +++ b/programs.tex @@ -0,0 +1,1099 @@ +%!TEX root = thesis.tex +%------------------------------------------------------------------------------- +\chapter{A cone programming framework for local state distinguishability} +\label{chap:programs} +%------------------------------------------------------------------------------- + +Due to the intrinsic complexity of LOCC protocols, it is hard +to come up with techniques for their analysis. +This is true in particular for the analysis of the local state discrimination +problem. All the proof techniques that have been proposed so far for this problem +have their own limitations: they are mathematically cumbersome, or they bound the +power only of limited subclasses of LOCC (one-way LOCC, for instance), or they +can be applied only to very specific set of states. + +In this chapter we provide a more general framework based on convex optimization +to prove bounds on LOCC protocols for the task of bipartite state discrimination. +We build on the idea described in Chapter \ref{chap:preliminaries} that LOCC +measurements can be approximated by more tractable classes of measurements, +in particular the sets of separable and PPT measurements. +It turns out that we can describe them conveniently using convex cones, +and therefore, many problems in which we optimize over them can be cast +into the cone programming paradigm. + +\minitoc + +\section{General cone program} + +The global state discrimination problem was one of the +first applications of semidefinite programming to the theory of quantum +information \cite{Eldar03a}. +Let us recall the parameters that define an instance of the problem. +We are given a complex Euclidean space $\X$, a positive integer $N$, and an +ensemble $\E$ of $N$ states, that is, +\begin{equation} +\label{eq:ensemble} + \E = \big\{ (p_{1}, \rho_{1}), \ldots, (p_{N}, \rho_{N}) \big\}, +\end{equation} +where $\rho_{1}, \ldots, \rho_{N} \in \Density(\X)$ and $(p_{1}, \ldots, p_{N})$ is +a probability vector. +We can construct a family of semidefinite programs parametrized by $\X$ +and $N$, such that one program takes $\E \in \Ens(\X, N)$ as input and +its optimal value corresponds to the maximum probability for any measurement to +distinguish $\E$: +\begin{center} +\underline{Primal (Global measurements)} +\begin{equation} + \label{eq:pos-primal-problem} + \begin{split} + \text{maximize:} \quad & + \sum_{k=1}^{N} p_{k}\ip{\rho_{k}}{\mu(k)},\\ + \text{subject to:} \quad & \sum_{k=1}^{N} \mu(k) = \I_{\X}\\ + & \mu : \{1,\ldots, N\}\rightarrow \Pos(\X) + \end{split} +\end{equation} +\end{center} + +The variables of the program form a collection of operators and the constraints +impose that such collection of operators forms a valid measurements. +In particular, the constraints demand that each operator belongs to the cone of +semidefinite operators and that all the operators sum to identity, as in the +definition of measurement from Section \ref{sec:quantum-measurements}. + +The key observation of this dissertation is that we can generalize the +above semidefinite program to a family of cone programs, where +the set of measurements over which we are optimizing forms a convex cone. +In effect, this generalization turns out to be helpful when the set of +measurements is characterized by the further property that each measurement in +the set can be represented by restricting each of its measurement operators to +belong to a particular convex cone. + +More formally, say we are given a complex Euclidean space $\X$ and consider the +problem of distinguishing the ensemble $\E$ from Eq. \eqref{eq:ensemble} by +any measurement in some class +\begin{equation} + \K \subset \Meas(N,\X). +\end{equation} +Further, suppose that the following characterization of $\K$ holds. +\begin{property} +\label{property:each-measurement-operator} +A measurement $\mu : \{1, \ldots, N\} \rightarrow \Pos(\X)$ belongs to the set $\K$ +if and only if there exists a convex cone $\C \subset \Pos(\X)$ such that each +measurement operator belongs to $\C$, that is, $\mu(k)\in \C$, +for each $k \in \{1,\ldots, N\}$. +\end{property} + +If this property is satisfied, the optimal probability of distinguishing $\E$ +by any measurement in $\K$ is thus given by the optimal solution of the +following cone program: +\begin{center} +\underline{Primal (General cone program)} +\begin{equation} + \label{eq:cone-primal-problem} + \begin{split} + \text{maximize:} \quad & + \sum_{k=1}^{N} p_{k}\ip{\rho_{k}}{\mu(k)},\\ + \text{subject to:} \quad & \sum_{k=1}^{N} \mu(k) = \I_{\X}\\ + & \mu : \{1,\ldots, N\}\rightarrow \C + \end{split} +\end{equation} +\end{center} + +If one is to formally specify this problem according to the general form for +cone programs presented in Section \ref{sec:convex-optimization}, the function +$\mu$ may be represented as a block matrix of the form +\begin{equation} + X = \begin{pmatrix} + \mu(1) & \cdots & \cdot \\ + \vdots & \ddots & \vdots\\ + \cdot & \cdots & \mu(N) + \end{pmatrix} \in \Herm(\X \oplus \cdots \oplus \X) +\end{equation} +with the off-diagonal blocks being left unspecified. +The cone denoted by $\K$ in Section \ref{sec:cone-programming} is taken to be +the cone of operators of this form for which each $\mu(k)$ belongs to the cone +$\C$. + +The mapping $\Phi$ and operators $A$ and $B$ +are chosen in the natural way: +\begin{equation} + A = \begin{pmatrix} + p_1 \rho_1 & \cdots & 0 \\ + \vdots & \ddots & \vdots\\ + 0 & \cdots & p_N\rho_N + \end{pmatrix}, + \qquad + B = \I_{\X}, +\end{equation} +and $\Phi : \Lin(\X\oplus\cdots\oplus\X) \rightarrow \Lin(\X)$ is defined as +\begin{equation} + \Phi\begin{pmatrix} + \mu(1) & \cdots & \cdot \\ + \vdots & \ddots & \vdots\\ + \cdot & \cdots & \mu(N) + \end{pmatrix} + \equiv \mu(1)+\cdots+\mu(N), +\end{equation} +for any $P_{1}, \ldots, P_{N} \in \Lin(\X)$. + +Let $\Y = \complex^{N}$. One can easily verify that the mapping +$\Phi^{\ast}: \Lin(\X)\rightarrow\Lin(\Y\otimes\X)$, +defined as +\begin{equation} + \Phi^{\ast}(H) \equiv \I_{\Y}\otimes H, +\end{equation} +satisfies Equation \eqref{eq:adjoint-map} and therefore is +the adjoint of $\Phi$: +for any $H\in\Lin(\X)$. +Also, let $\C^{\ast} \subset \Herm(\X)$ denote the dual cone of $\C$. +With these definitions in hand, one can write the dual of the Program +\eqref{eq:cone-primal-problem} as follows: +\begin{center} +\underline{Dual (General cone program)} +\begin{equation} + \label{eq:cone-dual-problem} + \begin{split} + \text{minimize:} \quad & \tr(H)\\ + \text{subject to:} \quad & H-p_k\rho_k\in\C^{\ast} + \quad(\text{for each}\;k = 1,\ldots,N)\\ + \quad & H \in \Herm(\X). + \end{split} +\end{equation} +\end{center} + +Throughout the thesis we will use the property of \emph{weak duality} of cone +programs (Proposition \ref{prop:weak-duality-cone}) to upper bound the +optimal solution of the primal program \eqref{eq:cone-primal-problem}. +The cone programs considered in this thesis also possess the property +of \emph{strong duality}. This property depends on the specific cone $\C$, +and we will discuss it whenever we treat a specific $\C$, although +it should be noted that strong duality is not needed for any of our results. + +In the rest of this chapter, we will see different instantiations of the general +program for a variety of measurements classes. +We started this section by presenting the semidefinite program +\eqref{eq:pos-primal-problem} for the problem of +state distinguishability by global measurement. In that case, the cone $\C$ +of the general cone program +corresponded to the cone of semidefinite operators, that is, $\C = \Pos(\X)$ and, +due to Fact \ref{fact:pos-self-dual}, we have that $\C^{\ast} = \C$. +Thus we can write the following dual program of \eqref{eq:pos-primal-problem}: +\begin{center} +\underline{Dual (Global measurements)} +\begin{equation} + \label{eq:global-dual-problem} + \begin{split} + \text{minimize:} \quad & \tr(H)\\ + \text{subject to:} \quad & H-p_k\rho_k \in \Pos(\X) + \quad(\text{for each}\;k = 1,\ldots,N)\\ + \quad & H \in \Herm(\X). + \end{split} +\end{equation} +\end{center} + + +\section{Bipartite measurements} +The above generalization of the optimal measurement cone program turns out to be +particularly helpful for the analysis of the bipartite state discrimination problem, +which is the main focus of this thesis. + +Recall that, as input of the problem, we are given two complex Euclidean +spaces $\X$ and $\Y$, one for each party, a positive integer $N$, and an +ensemble of states that are distributed among the spaces of the two parties, +that is, +\begin{equation} +\label{eq:ensemble-bipartite} + \E = \big\{ (p_{1}, \rho_{1}), \ldots, (p_{N}, \rho_{N}) \big\}, +\end{equation} +with $\rho_{1}, \ldots, \rho_{N} \in \Density(\X\otimes\Y)$. + +Ideally, we would like to solve the following problem: +\begin{center} +\underline{Primal (LOCC measurements)} + \begin{equation} + \label{eq:locc-primal-problem} + \begin{split} + \text{maximize:} \quad & + \sum_{k=1}^{N} p_{k}\ip{\rho_{k}}{\mu(k)},\\ + \text{subject to:} %\quad & \sum_{k=1}^{N} P_{k} = \I_{\X\otimes\Y}\\ + \quad & \mu \in \Meas_{\LOCC}(N, \X:\Y). + \end{split} + \end{equation} +\end{center} + +Phrasing this problem as a cone program is not interesting as such. Even though it is +technically possible, we would not be able to exploit the advantages that come from such formulation, +due to the fact (by now, we have stressed this enough!) that the set of LOCC measurements +is not easy to be handled mathematically. +For the set of LOCC measurement, we do not even have a characterization +on the same lines of Property \ref{property:each-measurement-operator}, so we cannot cast +the problem in the general form of the program in \eqref{eq:cone-primal-problem}. + +As indicated in Chapter \ref{chap:bipartite-state-discrimination}, the LOCC set +can be approximated by other sets of measurements that are easier to be manipulated +mathematically. It turns out that both the sets of PPT and separable measurements are +suitable for the cone programming framework described above. In fact, for both +these sets it is relatively easy to characterize the dual set, and an equivalent +of Property \ref{property:each-measurement-operator} holds, +For example, Proposition \ref{prop:separable-each} characterizes a separable +measurement over the bipartition $(\X:\Y)$ as a collection of operators +belonging to the cone $\Sep(\X:\Y)$. +This property allows us to characterize the maximum probability of +distinguishing the ensemble in Eq.~\eqref{eq:ensemble-bipartite} by any +separable measurement as a cone program of the same form as the one +in~\eqref{eq:cone-primal-problem}, where instead of $\X$, the underlying space +of the operators is $\X\otimes\Y$, and $\C(\X)$ is replaced by the cone of +separable operators $\Sep(\X:\Y)$. + +In the rest of this section, we study the different programs that derive +from the Program~\eqref{eq:cone-primal-problem} when we instantiate $\C$ with +some particular cones corresponding to different classes of bipartite +measurements. In particular, we will mainly look at the programs derived from the cones +of separable operators, PPT operators, and operators with $k$-symmetric extensions. +For each of these programs, we show the dual program and try to make any possible +simplification. Moreover, we point out whenever a program can be +expressed by using only semidefinite constraints, as it was the case for the +Program~\eqref{eq:pos-primal-problem} from above. + +\subsection{PPT measurements} + +We start with the cone of PPT operators and we describe a semidefinite program +that computes $\opt_{\PPT}(\E)$. Using tools of convex optimization to solve +problems concerning the PPT cone is not a novel idea. +Two other applications of convex programming to the realm of PPT operations +are the semidefinite program shown by Rains to compute the maximum fidelity +obtained by a PPT distillation protocol \cite{Rains01} and the hierarchy of +semidefinite programs proposed as separability criteria by Doherty, Parrilo, and +Spedalieri \cite{Doherty02,Doherty04}. + +Recall from Definition \ref{def:ppt-measurements} that a measurement +$\mu : \{1, \ldots, N\} \rightarrow \Pos(\X\otimes\Y)$ is in +$\Meas_{\PPT}(N, \X:\Y)$ if and only if +\begin{equation} + \mu(1), \ldots, \mu(N) \in \PPT(\X:\Y). +\end{equation} +From the definition of $\PPT(\X:\Y)$, we can write the cone program in +the following form: +\begin{center} +\underline{Primal (PPT measurements)} +\begin{equation} + \label{eq:ppt-primal-problem} + \begin{split} + \text{maximize:} \quad & + \sum_{k=1}^{N} p_{k}\ip{\rho_{k}}{\mu(k)},\\ + \text{subject to:} \quad & \sum_{k=1}^{N} \mu(k) = \I_{\X\otimes\Y}\\ + & \mu : \{1,\ldots, N\}\rightarrow \Pos(\X\otimes\Y),\\ + & \pt_{\X}(\mu(k))\in\Pos(\X\otimes\Y) \quad(\text{for each}\;k = 1,\ldots,N). + \end{split} +\end{equation} +\end{center} +An immediate observation is that the cone program above is in fact a semidefinite +program. To see this formally, let us introduce $N$ variables +$Q_{1},\ldots,Q_{N} \in \Herm(\X\otimes\Y)$ and, for each $k \in \{1, \ldots, N\}$, let +\begin{equation} + Q_{k} = \pt_{\X}(\mu(k)). +\end{equation} +One can write the above program as a semidefinite program in the standard form of +Section~\ref{sec:semidefinite-programming}, where +\begin{equation} +\label{eq:ppt-X} +X = \begin{pmatrix} + \mu(1) & \cdots & \cdot\\ + \vdots & \ddots & \vdots\\ + \cdot & \cdots & \mu(N)\\ + \end{pmatrix} + \oplus + \begin{pmatrix} + Q_1 & \cdots & \cdot\\ + \vdots & \ddots & \vdots\\ + \cdot & \cdots & Q_N\\ + \end{pmatrix} +\end{equation} +is the variable over which we optimize, +\begin{equation} + A = \begin{pmatrix} + p_{1}\rho_{1} & \cdots & \cdot\\ + \vdots & \ddots & \vdots\\ + \cdot & \cdots & p_{N}\rho_{N}\\ + \end{pmatrix} + \oplus + \begin{pmatrix} + 0 & \cdots & \cdot\\ + \vdots & \ddots & \vdots\\ + \cdot & \cdots & 0\\ + \end{pmatrix}, +\end{equation} +and +\begin{equation} + B = + \begin{pmatrix} + \I_{\X\otimes\Y} & \cdot & \cdots & \cdot \\ + \cdot & 0 & \cdots & \cdot\\ + \vdots & \vdots & \ddots & \vdots\\ + \cdot & \cdot & \cdots & 0 + \end{pmatrix}. +\end{equation} +are the known inputs of the problem, and the map $\Phi : $ is defined as +\begin{equation} + \Phi(X) \equiv \begin{pmatrix} + \mu(1) + \cdots +\mu(N) & \cdot & \cdots & \cdot \\ + \cdot & \pt_{\X}(\mu(1)) - Q_{1} & \cdots & \cdot\\ + \vdots &\vdots & \ddots & \vdots\\ + \cdot &\cdot & \cdots & \pt_{\X}(\mu(N)) - Q_{N} + \end{pmatrix}, +\end{equation} +for any operator $X$ in the form of Eq.~\eqref{eq:ppt-X}. + +Once the program is in the standard form, one can easily derive its dual. +The variable of the dual program is the Hermitian operator +$Y \in $ +defined as follows: +\begin{equation} +\label{eq:ppt-Y} + Y = + \begin{pmatrix} + H & \cdot & \cdots & \cdot \\ + \cdot & -R_{1} & \cdots & \cdot\\ + \vdots & \vdots & \ddots & \vdots\\ + \cdot & \cdot & \cdots & -R_{N} + \end{pmatrix}, +\end{equation} +for Hermitian operators $Y, R_{1}, \ldots, R_{N} \in \Herm(\X\otimes\Y)$. +The adjoint of $\Phi$ is defined as the mapping +\begin{equation} + \Phi^{\ast}(Y) \equiv + \begin{pmatrix} + H - \pt_{\X}(R_{1}) & \cdots & \cdot\\ + \vdots & \ddots & \vdots\\ + \cdot & \cdots & H - \pt_{\X}(R_{N})\\ + \end{pmatrix} + \oplus + \begin{pmatrix} + R_{1} & \cdots & \cdot\\ + \vdots & \ddots & \vdots\\ + \cdot & \cdots & R_{N}\\ + \end{pmatrix}, +\end{equation} +for any operator $Y$ in the form of Eq.~\eqref{eq:ppt-Y}. +It is easy to verify that the map $\Phi^{\ast}$ satisfies Eq.~\eqref{eq:adjoint-map}. +From the fact the the partial transpose is its own adjoint and inverse, we have +that +\begin{equation} + \ip{A}{B} = \ip{\pt_{X}(A)}{\pt_{X}(B)}, +\end{equation} +for any operators $A, B\in\Lin(\X\otimes\Y)$, which implies +\begin{equation} +\ip{\mu(k)}{\pt_{\X}(R_{k})} = \ip{\pt_{\mu(k)}}{R_{k}}, +\end{equation} +for any $k \in \{1, \ldots, N\}$, and therefore +\begin{equation} + \ip{Y}{\Phi(X)} = \ip{\Phi^{\ast}(Y)}{X}. +\end{equation} +By rearranging everything in a more explicit form, we have the following dual program: +\begin{center} +\underline{Dual (PPT measurements)} +\begin{equation} + \label{eq:ppt-dual-problem} + \begin{split} + \text{minimize:} \quad & \tr(H)\\ + \text{subject to:} \quad & H-p_k\rho_k \geq \pt_{\X}(R_{k}) + \quad(\text{for each}\;k = 1,\ldots,N),\\ + \quad & R_{1}, \ldots, R_{N} \in \Pos(\X\otimes\Y),\\ + \quad & H \in \Herm(\X\otimes\Y). + \end{split} +\end{equation} +\end{center} + +\subsubsection{Decomposable operator} +An equivalent way of deriving the above dual program is by defining the cone +\begin{equation} + \PPT^{\ast}(\X:\Y) = \{ S + \pt_{\X}(R) \,:\, S, R \in \Pos(\X\otimes\Y) \}, +\end{equation} +which satisfies \eqref{eq:dual-cone} and therefore is the dual cone of +$\PPT(\X:\Y)$. +The program above corresponds to an instance of +the generic dual program \eqref{eq:cone-dual-problem}, where $\C^{\ast}$ is +replaced by $\PPT^{\ast}(\X:\Y)$. + +The operators in $\PPT^{\ast}(\X:\Y)$ can also be characterized as +representations of so-called \emph{decomposable maps} from +$\Lin(\Y)$ to $\Lin(\X)$, via the Choi isomorphism +(see Section \ref{sec:choi-isomorphism}) +\begin{definition} +A decomposable map $\Phi:\Lin(\Y)\rightarrow\Lin(\X)$ is a linear map that can +be represented as the sum of a completely positive map and a completely +co-positive map, that is, there exist two completely positive maps +$\Psi,\Xi:\Lin(\Y)\rightarrow\Lin(\X)$, such that, for any $Y\in\Lin(\Y)$, +\begin{equation} + \Phi(Y) = \Psi(Y) + (\pt {\circ}\, \Xi)(Y), +\end{equation} +where $\pt$ denotes the transpose map. +\end{definition} + +\subsubsection{Exploiting symmetries} + +In cases where the ensemble of states we wish to distinguish exhibit some symmetry, +we can simplify the semidefinite program ($\PP_{\PPT}$) to a linear program. +One particular case where this kind of symmetry emerges is when +we consider so-called \emph{lattice states}. Let +\[ + \psi_{i} = \ket{\psi_{i}}\bra{\psi_{i}} \in \Density(\complex^{2}\otimes\complex^{2}), +\] +for $i\in \{0,1,2,3\}$, be the density operators corresponding to the standard Bell +states, as defined in Eq.~\eqref{eq:Bell-states}. +Let $v \in \integer_{4}^{t} $ be a $t$-dimensional vector and let +$\ket{\psi_{v}} \in \complex^{2^{t}}\otimes\complex^{2^{t}}$ +be the maximally entangled state given by the tensor product of Bell states indexed by the vector +$v = (v_{1}, \ldots, v_{t})$, that is, +$$ +\ket{\psi_{v}} = \ket{\psi_{v_{1}}}\otimes\ldots\otimes\ket{\psi_{v_{t}}}. +$$ +In the literature, operators diagonal in the basis +$\{\psi_{v} = \ket{\psi_{v}}\bra{\psi_{v}} : v \in \integer_{4}^{t}\}$ +are called \emph{lattice operators}, or \emph{lattice states} if they are also +density operators \cite{Piani06}. + +The following equations regarding the partial transpose of the Bell states will +be used in the main proof of this section and can be proved by direct inspection. +\begin{equation} + \label{eq:ppt-bell-states} + \begin{aligned} + \pt_{\X}(\psi_{0}) = \frac{1}{2}\I - \psi_{2},\qquad + \pt_{\X}(\psi_{1}) = \frac{1}{2}\I - \psi_{3},\\ + \pt_{\X}(\psi_{2}) = \frac{1}{2}\I - \psi_{0},\qquad + \pt_{\X}(\psi_{3}) = \frac{1}{2}\I - \psi_{1}. + \end{aligned} +\end{equation} +The following proposition is useful for the proof of the main theorem of this +section and, again, it can be easily proved by direct inspection. +\begin{prop} +\label{prop:groupG} +Let $\{\sigma_{0}, \sigma_{1}, \sigma_{2}, \sigma_{3}\} \subset \Herm(\complex^{2})$ +be the set of Pauli operators in defined in Eq.~\eqref{eq:Pauli-operators}. +It holds that the Bell states from Eq.~\eqref{eq:Bell-states} are invariant under the group of local symmetries +\begin{equation} + G = \big\{ \sigma_{i} \otimes \sigma_{i} : i\in\{0,1,2,3\} \big\}, +\end{equation} +that is, $\psi_{i} = U\psi_{i}U^{*}$ for any $U \in G$ and any $i\in\{0,1,2,3\}$. +\end{prop} + +It turns out that in the case when the set to distinguish contains only lattice states, +the semidefinite program ($\PP_{\PPT}$) simplifies remarkably, as it is established +by the following theorem. +\begin{theorem} +If the set to be distinguished consists only of lattice states, then the probability of +successfully distinguishing them by PPT measurements can be expressed as the +optimal value of a linear program. +\end{theorem} +\begin{proof} +We will prove that for any feasible solution of the semidefinite program ($\PP_{\PPT}$), +there is another feasible solution consisting only of lattice operators +for which the objective function takes the same value. + +Let $\Delta : \Lin(\complex^{2}\otimes\complex^{2}) \rightarrow + \Lin(\complex^{2}\otimes\complex^{2})$ be the channel defined as follows: +\begin{equation} + \Delta(X) = \frac{1}{|G|}\sum_{U \in G} UXU^{*}, +\end{equation} +where $G$ is the group of local unitaries defined in Proposition \ref{prop:groupG}. +The channel $\Delta(X)$ acts on $X$ as a completely dephasing channel in the Bell basis. +Let $\X^{(t)} = \Y^{(t)} = \complex^{2^{t}}$ for some positive integer $t > 1$. +Say the states we want to distinguish, +\begin{equation} + \rho_{1}, \ldots, \rho_{N} \in \Density(\X^{(t)}\otimes\Y^{(t)}), +\end{equation} +are lattice states. +Let $\Phi = \Delta^{\otimes t}$ be the $t$-fold tensor product of the map $\Delta$, +that is, +\begin{equation} + \Phi(v_{1}\otimes\cdots\otimes v_{t}) = + \Delta(v_{1})\otimes\cdots\otimes\Delta(v_{t}), +\end{equation} +for any choice of vectors $v_{1}, \ldots, v_{t} \in \X\otimes\Y$. +Assume that a measurement +\[ + \mu : \{1,\ldots,N\}\to\PPT(\X:\Y) +\] +is a feasible solution of the program ($\PP_{\PPT}$) +for the states $\rho_{1}, \ldots, \rho_{N}$. +In the rest of the proof we want to show that a new measurement $\mu'$ constructed by +applying $\Phi$ to each measurement operator of $\mu$ is also a feasible solution +of the program ($\PP_{\PPT}$), for the same set of states. +Since $\Phi$ is a dephasing channel in the lattice basis, this would imply the +statement of the theorem. + +First, let us observe that the value of the objective function for the solution +$\mu'$ is the same as the value for the original solution $\mu$. +The channel $\Phi$ is its own adjoint and therefore we have +\[ + \ip{\mu(k)}{\rho_{k}} = \ip{\mu(k)}{\Phi(\rho_{k})} = + \ip{\Phi(\mu(k))}{\rho_{k}} , +\] +for any $k = 1, \ldots, N$. + +Next, we show that $\mu'$ is a PPT measurement. From the fact that $\Phi$ is unital +(in fact it is a mixed unitary channel), we have +\begin{equation} + \Phi(\mu(1)) + \ldots + \Phi(\mu(N)) = \I, +\end{equation} +and from the fact that $\Phi$ is positive, we have +\begin{equation} + \Phi(\mu(k)) \in\Pos(\X^{(t)}\otimes\Y^{(t)}) +\end{equation} +and +\begin{equation} +\label{eq:pt-mu} + \Phi(\pt_{\X}(\mu(k))) \in\Pos(\X^{(t)}\otimes\Y^{(t)}), +\end{equation} +for any $k \in \{1, \ldots, N \}$. +The first fact implies that $\mu'$ is indeed valid measurement. +The second fact is close to what we want in order to show that $\mu'$ is a PPT measurement. +To complete the proof, we wish to show that the partial transpose mapping commutes +with the channel $\Phi$. +First we observe how the partial transposition modifies the action of local operators. +Given linear operators $A\in \Lin(\X)$, $B \in \Lin(\Y)$ and $X \in \Lin(\X \otimes \Y)$, +we have +\begin{equation} + \pt_{\X} [(A \otimes B)X(A \otimes B)^{*}] = + (\overline{A}\otimes B)\pt_{\X}(X)(\overline{A}\otimes B)^{*}. +\end{equation} +Now, for the Pauli matrices, we have $\overline{\sigma_{j}} = \sigma_{j}$ +for $j \in \{ 0,1,3\}$ and $\overline{\sigma_{2}} = -\sigma_{2}$. +Therefore +\begin{equation} + \Delta(\pt_{\X}(X)) = \pt_{\X}(\Delta(X)), \quad \text{for any $X \in \Lin(\X\otimes\Y)$}. +\end{equation} +This property trivially extends by tensor product to $\Phi$ and therefore +Eq.~\eqref{eq:pt-mu} implies +\begin{equation} + \pt_{\X}(\Phi(\mu(k))) \geq 0, +\end{equation} +for any $k \in \{1, \ldots, N\}$, which concludes the proof. +\end{proof} + +The advantage of the linear programming formulation is in the computational +efficiency of the algorithms that solve the program. +However, for sake of clarity, in the analytic proofs that will follow, +we will always stick to the more general semidefinite programming formulation, +even when we consider distinguishability of lattice states. + +\subsection{Separable measurements} +We have yet to fully exploit the expressive power of the cone programming language. +In this section we do so by showing a connection between convex optimization and +the cone of separable operators, which would not be possible if the only tool at +hand was semidefinite programming. +Surprisingly, there are only few examples in the literature (\cite{Gharibian13} +being one of those) where this connection has been made use of. + +As it is stated by Proposition \ref{prop:separable-each}, a measurement +$\mu : \{1,\ldots,N\} \rightarrow \Pos(\X\otimes\Y)$ is separable when +\begin{equation} + \mu(1), \ldots, \mu(N) \in \Sep(\X:\Y). +\end{equation} +We can write the maximum probability of distinguishing an ensemble $\E$ by +separable measurements, $\opt_{\Sep}(\E)$, as the optimal value of the +following cone program: +\begin{center} +\underline{Primal ($\PP_{\Sep}$)} +\begin{equation} + \label{eq:sep-primal-problem} + \begin{split} + \text{maximize:} \quad & + \sum_{k=1}^{N} p_{k}\ip{\rho_{k}}{\mu(k)},\\ + \text{subject to:} \quad & \sum_{k=1}^{N} \mu(k) = \I_{\X\otimes\Y},\\ + & \mu : \{1,\ldots, N\}\rightarrow \Sep(\X:\Y). + \end{split} +\end{equation} +\end{center} + +An important observation about this program is that, unlike the case of the +PPT program, its constraints cannot be formulated as semidefinite constraints. +% \comment{AC}{``cannot be formulated'' is not a formal statement, make this formal} +Later in Section \ref{sec:computational-aspects} we will see how this has +implications in the computational complexity of the problem. + +We denote the cone dual to $\Sep(\X:\Y)$ by $\BPos(\X:\Y)$, and defer its definition +to the next paragraph, after we write the dual program of \eqref{eq:sep-primal-problem}: +\begin{center} +\underline{Dual ($\DP_{\Sep}$)} +\begin{equation} + \label{eq:sep-dual-problem} + \begin{split} + \text{minimize:} \quad & \tr(H)\\ + \text{subject to:} \quad & H-p_k\rho_k\in\BPos(\X:\Y) + \quad(\text{for each}\;k = 1,\ldots,N),\\ + \quad & H \in \Herm(\X\otimes\Y). + \end{split} +\end{equation} +\end{center} + +\subsubsection{Block-positive operators} +\label{sec:block-positive-operators} + +The cone $\BPos(\X : \Y)$, which is commonly known as the set of +\emph{block-positive operators}, is +\begin{equation} + \BPos(\X:\Y) = \bigl\{ + H\in\Herm(\X\otimes\Y)\,:\, + \ip{P}{H} \geq 0 \;\text{for every $P \in \Sep(\X:\Y)$}\bigr\}. +\end{equation} +There are several equivalent characterizations of this set. +For instance, one has +\begin{multline} + \BPos(\X:\Y) = \bigl\{ + H\in\Herm(\X\otimes\Y)\,:\\ + (\I_{\X} \otimes y^{\ast}) H (\I_{\X} \otimes y)\in\Pos(\X) + \;\text{for every $y\in\Y$}\bigr\}. +\end{multline} +Alternatively, block-positive operators can be characterized as representations +of \emph{positive} linear maps, via the Choi representation. +That is, for any linear map $\Phi:\Lin(\Y)\rightarrow\Lin(\X)$ +mapping arbitrary linear operators on $\Y$ to linear operators on $\X$, the +following two properties are equivalent: +\begin{itemize} +\item[(a)] For every positive semidefinite operator $Y \in \Pos(\Y)$, it +holds that $\Phi(Y) \in \Pos(\X)$. +\item[(b)] The Choi operator +\begin{equation} + J(\Phi) = \sum_{0\leq j,k< m} \Phi\bigl(\ket{j}\bra{k}\bigr) + \otimes \ket{j}\bra{k} +\end{equation} +of $\Phi$ satisfies $J(\Phi) \in \BPos(\X:\Y)$. +\end{itemize} + +Dual to the fact that there are positive semidefinite operators, which are not +separable, is the fact that there are block-positive operators, which are not +positive semidefinite. Such operators are called \emph{entanglement witnesses} +and represent (via the Choi representation) linear maps that are positive, +but not completely positive. + +One example of entanglement witness is the swap operator defined in +\eqref{eq:swap-operator}, which is the Choi representation +of the transpose map. +We will see many other example of entanglement witnesses in later sections; +for a more exhaustive review, see \cite{Chruscinski2014}. + +The Venn diagram in Figure \ref{fig:measurements-dual} depicts the containments +of sets that are dual to the sets of measurements operators we have considered +(a set with one shade of gray is dual to the set with the same shade of gray from +Figure \ref{fig:classes-measurements}). + +\begin{figure}[!ht] + \centering + \begin{minipage}{0.5\textwidth} + \centering + \def\svgwidth{200pt} + \scalebox{.75}{\input{drawin_dualg.pdf_tex}} + \end{minipage}\hfill + \begin{minipage}{0.5\textwidth} + \centering + \def\svgwidth{200pt} + \scalebox{.75}{\input{drawing_super.pdf_tex}} + \end{minipage} + \caption{Sets of operators that are dual to the sets of Figure + \ref{fig:classes-measurements} (on the left) and the + corresponding sets of linear mappings via the Choi isomorphism (on the right).} + \label{fig:measurements-dual} +\end{figure} + +\subsection{Symmetric extensions} +\label{sec:symm-ext} +For any given ensemble of states $\E$, the cone program \eqref{eq:sep-primal-problem} +for separable measurements outputs a better (not always strictly better) approximation of $\opt_{\LOCC}(\E)$, +compared to the output of the semidefinite program \eqref{eq:ppt-primal-problem} for PPT measurements. +The drawback is in the computational complexity: on the one extreme, we have +polynomial time algorithms to solve the PPT semidefinite program, and on the other +extreme, we can prove that optimizing over a set of separable operator is an NP-hard +problem (more details in Section \ref{sec:computational-aspects}). + +Building up on an idea by Doherty, Parrilo, and Spedalieri \cite{Doherty02,Doherty04}, +we are able to interpolate between these two extremes and construct a hierarchy of +semidefinite programs characterized by the following trade-off: whenever we +climb up a level of the hierarchy, the size of the program increases +(and so does the running time of the algorithms that solve it), however the +outputs of the program gives an approximation closer to $\opt_{\Sep}(\E)$. + +In order to formally describe the hierarchy of semidefinite programs that +approximates $\opt_{\Sep}(\E)$, we first need to introduce the concept of +symmetric extension of a positive operator. +First let us define the set of permutation operators. +Let $s$ be a positive integer and let $\X_{1}, \ldots, \X_{s}$ be $s$ isomorphic +copies of the same complex Euclidean space, that is, for some positive integer +$d$ and any $k \in \{1, \ldots, m \}$, $\X_{k} \simeq \complex^{d}$. +Given a permutation $\pi\in\Sym(s)$, we define +a \emph{permutation operator} +\begin{equation} + W_{\pi} \in \Unitary(\X_{1}\otimes\cdots\otimes\X_{s}) +\end{equation} +to be the unitary operator that acts as follows: +\begin{equation} + W_{\pi}(u_{1}\otimes\cdots\otimes u_{s}) = u_{\pi(1)}\otimes\cdots + \otimes u_{\pi(s)}, +\end{equation} +for any choice of vectors $u_{1}, \ldots, u_{s}\in\complex^{d}$. +We are now ready to give the definition of symmetric extension of a positive +semidefinite operator. + +\begin{definition} + Suppose we are given complex Euclidean spaces $\X$ and $\Y$, and an operator + $P\in\Pos(\X\otimes\Y)$. Moreover, let $s$ be a positive integer and let + $\Y_{2},\ldots,\Y_{s}$ be copies of the space $\Y$. An operator + \begin{equation} + X \in \Pos(\X\otimes\Y\otimes\Y_{2}\otimes\cdots\otimes\Y_{s}) + \end{equation} + is called an \emph{$s$-symmetric extension} of $P$ if the following two + properties hold: + \begin{itemize} + \item[(a)] $\tr_{\Y_{2}\otimes\cdots\otimes\Y_{s}}(X) = P$; + \item[(b)] $(\I_{\X}\otimes W_{\pi})X(\I_{\X}\otimes W_{\pi}^{\ast}) = X$, + for every $\pi\in\Sym(s)$. + \end{itemize} +\end{definition} + +Any separable operator possesses $s$-symmetric extensions, for any $s \geq 1$. +This is easy to see from the definition of separable operator: given +$P\in\Sep(\X:\Y)$, there exists a positive integer $M$, positive semidefinite +operators $Q_1,\ldots,Q_M \in \Pos(\X)$ and density matrices $\rho_1,\ldots, \rho_M\in\Density(\Y)$ +such that +\begin{equation} + P = \sum_{k = 1}^M Q_{k} \otimes \rho_{k}. +\end{equation} +Then, for any $s \geq 1$, +\begin{equation} +\label{eq:X-extension} + X = \sum_{k = 1}^M Q_k \otimes \rho_{k}^{\otimes s} +\end{equation} +is an $s$-symmetric extension of $P$. + +Interestingly, the converse is also true, that is, for any entangled operator +\begin{equation} +P \in \Pos(\X:\Y) \setminus \Sep(\X:\Y), +\end{equation} +there must exist a value $s > 1$ for which $P$ does not have a $s$-symmetric +extension. For a proof of this fact, which is more involved, we refer the +reader to the original paper \cite{Doherty02}. + + + +It is worthwhile to consider two additional constraints we can add to the +definition of symmetric extension. +The first one comes from the observation that the symmetric part of the operator +$X$ in Eq.~\eqref{eq:X-extension} is supported by the symmetric subspace, which +is defined as the set of all vectors in $\Y\otimes\Y_{2}\otimes\cdots\otimes\Y_{s}$ +that are invariant under the action of $W_{\pi}$, for every choice of $\pi\in\Sym(s)$. +We denote the symmetric subspace by +\begin{equation} + \Y\ovee\Y_{2}\ovee\cdots\ovee\Y_{s} = \{ y \in \Y\otimes\Y_{2}\otimes\cdots\otimes\Y_{s} + : y = W_{\pi}y \;\mbox{ for every $\pi\in\Sym(s)$} \}, +\end{equation} +and the projection on this subspace by $\Pi_{\Y\ovee\Y_{2}\ovee\cdots\ovee\Y_{s}}$. + +\begin{definition} +\label{def:bosonic} + Suppose we are given complex Euclidean spaces $\X$ and $\Y$, and an operator + $P\in\Pos(\X\otimes\Y)$. Moreover, let $s$ be a positive integer and let + $\Y_{2},\ldots,\Y_{s}$ be copies of the space $\Y$. An operator + \begin{equation} + X \in \Pos(\X\otimes\Y\otimes\Y_{2}\otimes\cdots\otimes\Y_{s}) + \end{equation} + is called an \emph{$s$-symmetric bosonic extension} of $P$ if the following + two properties hold: + \begin{itemize} + \item[(a)] $\tr_{\Y_{2}\otimes\cdots\otimes\Y_{s}}(X) = P$; + \item[(b)] $(\I_{\X}\otimes \Pi_{\Y\ovee\Y_{2}\ovee\cdots\ovee\Y_{s}})X + (\I_{\X}\otimes \Pi_{\Y\ovee\Y_{2}\ovee\cdots\ovee\Y_{s}}) = X$. + \end{itemize} +\end{definition} + +A further observation is that we want the semidefinite program at the first level +of the symmetric-extension hierarchy to be at least as powerful as the program +($\PP_{\PPT}$). In order to do so, we add a third condition to Definition +\ref{def:bosonic}: + \begin{itemize} + \item[(c)] + \begin{description} + \item[] $\pt_{\X}(X) \in \Pos(\X\otimes\Y\otimes\Y_{2}\otimes\cdots\otimes\Y_{s})$, and + \item[] $\pt_{\Y_{j}\otimes\Y_{j+1}\otimes\cdots\otimes\Y_{s}}(X) \in + \Pos(\X\otimes\Y\otimes\Y_{2}\otimes\cdots\otimes\Y_{s})$, + for all $j \in \{2, \ldots, s\}$. + \end{description} + \end{itemize} + +Finally, we can put all the constraint together in a hierarchy of semidefinite programs +in which, at the $s$-th level, we optimize over measurements whose operators possess +$s$-symmetric bosonic PPT extensions. +The following is the semidefinite program corresponding to the level $s = 2$ of +the hierarchy (the programs corresponding to higher levels of the hierarchy can +be easily inferred from this one). + +\begin{center} +\underline{Primal ($\PP_{\Sym}$)} +\begin{equation} + \label{eq:sym-primal-problem} + \begin{split} + \text{maximize:} \quad & + \sum_{k=1}^{N} p_{k}\ip{\rho_{k}}{\mu(k)},\\ + \text{subject to:} \quad & + \sum_{k=1}^{N} \mu(k) = \I_{\X\otimes\Y},\\[2ex] + & \left.\kern-\nulldelimiterspace + \!\!\begin{aligned} + &\tr_{\Y_{2}}(X_{k}) = \mu(k),\\[1ex] + &(\I_{\X}\otimes \Pi_{\Y\ovee\Y_{2}})X_{k} + (\I_{\X}\otimes \Pi_{\Y\ovee\Y_{2}}) = X_{k},\\[1ex] + &\pt_{\X}(X_{k}) \in \Pos(\X\otimes\Y\otimes\Y_{2}),\\[1ex] + &\pt_{\Y_{2}}(X_{k}) \in \Pos(\X\otimes\Y\otimes\Y_{2}), + \end{aligned}\,\right\} \quad(\text{for each}\;k = 1,\ldots,N),\\[2ex] + \quad & X_{1}, \ldots, X_{N} \in \Pos(\X\otimes\Y\otimes\Y_{2}). + \end{split} +\end{equation} +\end{center} +\vspace{10pt} +The dual program can be derived by moving to the standard form and again, +by observing that the partial transpose is its own adjoint. +\vspace{10pt} +\begin{center} +\underline{Dual ($\DP_{\Sym}$)} +\begin{equation} + \label{eq:sym-dual-problem} + \begin{split} + \text{minimize:} \quad & + \tr(H),\\[1ex] + \text{subject to:} + \quad \\[-3.5ex] + & \left.\kern-\nulldelimiterspace + \!\!\begin{aligned} + &H - Q_{k} \geq p_{k}\rho_{k},\\[1ex] + & Q_{k}\otimes\I_{\Y_{2}} + (\I_{\X}\otimes \Pi_{\Y\ovee\Y_{2}})R_{k}(\I_{\X}\otimes \Pi_{\Y\ovee\Y_{2}}) - R_{k} \\ + & \qquad - \pt_{\X}(S_{k}) - \pt_{\Y}(Z_{k}) + \in \Pos(\X\otimes\Y\otimes\Y_{2}), + \end{aligned}\,\right\} \quad(\text{for each}\;k = 1,\ldots,N),\\[2ex] + \quad & H, Q_{1}, \ldots, Q_{N}\in\Herm(\X\otimes\Y),\\ + \quad & R_{1}, \ldots, R_{N}\in\Herm(\X\otimes\Y\otimes\Y_{2})\\ + \quad & S_{1}, \ldots, S_{N},Z_{1},\ldots,Z_{N} + \in\Pos(\X\otimes\Y\otimes\Y_{2}).\\ + \end{split} +\end{equation} +\end{center} + +\begin{figure}[!htbp] + \centering + \def\svgwidth{200pt} + \scalebox{.85}{ + \input{drawing_symm.pdf_tex}} + \caption{Symmetric extension hierarchy.} + \label{fig:symm-extension} +\end{figure} + +\begin{figure}[!htbp] + \centering + \def\svgwidth{200pt} + \scalebox{.85}{ + \input{drawing_symm_dual.pdf_tex}} + \caption{Dual of the symmetric extension hierarchy of Figure \ref{fig:symm-extension}.} + \label{fig:symm-extension-dual} +\end{figure} + +%---------------------------------------------------------------------- +\section{An example: Werner hiding pair} +%---------------------------------------------------------------------- + +In order to demonstrate an analytic method to use the cone programming framework +presented above, we apply it here to a simple example. +We reprove the result from Example \ref{example:werner-hiding-pairs}, +that is, a tight bound on the LOCC-distinguishability of a pair of quantum hiding states. +For any positive integer $n \leq 2$, consider the equiprobable ensemble $\E^{(n)}$ of +the two extremal Werner states +$\sigma_{0}^{(n)}, \sigma_{1}^{(n)} \in \Density(\complex^{n}\otimes\complex^{n})$ +defined in Eq. \eqref{eq:werner-states}. +Here we prove that +\begin{equation} + \opt_{\PPT}(\E^{(n)}) \leq \frac{1}{2} + \frac{1}{n+1}, +\end{equation} +for any $n \geq 2$. +For this particular case, the semidefinite program ($\PP_{\PPT}$) is +sufficient to prove a tight bound on the distinguishability of the states. +This is not always the case, as we will see in the next chapter. + +Let the integer $n$ be fixed and let $\X$ and $\Y$ be copies of $\complex^{n}$. +First, we instantiate Program \eqref{eq:ppt-dual-problem} for the ensemble $\E^{(n)}$: +\begin{equation} + \label{eq:ppt-dual-problem-werner} + \begin{split} + \text{minimize:} \quad & \tr(H)\\ + \text{subject to:} + \quad & H - \frac{1}{2}\sigma_0^{(n)} \in \PPT^{\ast}(\X:\Y),\\ + \quad & H - \frac{1}{2}\sigma_1^{(n)} \in \PPT^{\ast}(\X:\Y),\\ + \quad & H \in \Herm(\X\otimes\Y).\\ + \end{split} +\end{equation} +Next, we define the Hermitian operator $H \in \Herm(\X\otimes\Y)$ as +\begin{equation} +\label{eq:H-werner} + H = \frac{1}{2}\sigma_{0}^{(n)} + \frac{1}{n+1}\sigma_{1}^{(n)}, +\end{equation} +and observe that the trace is equal to the bound we seek to prove, that is, +\begin{equation} + \tr(H) = \frac{1}{2} + \frac{1}{n+1}. +\end{equation} +It is left to be checked that $H$ is a feasible solution of the dual program. +By the definition of the cone $\PPT^{\ast}(\X\otimes\Y)$, +we need to show that there exist positive semidefinite operators +$R_{0}, R_{1} \in \Pos(\X\otimes\Y)$ such that +\begin{equation} +H - \frac{1}{2}\sigma_0^{(n)} \geq \pt_{\X}(R_{0}) +\end{equation} +and +\begin{equation} +H - \frac{1}{2}\sigma_1^{(n)} \geq \pt_{\X}(R_{1}). +\end{equation} +Let +\begin{equation} +u = \frac{1}{\sqrt{n}}\sum_{i=0}^{n-1}\ket{i}\ket{i} +\end{equation} +be the canonical maximally entangled state in $\X\otimes\Y$ and let +\begin{equation} + R_{0} = 0 + \hspace*{1cm}\mbox{and}\hspace*{1cm} + R_{1} = \frac{1}{n+1}uu^{\ast}. +\end{equation} +We have +\begin{equation} + H - \frac{1}{2}\sigma_{0}^{(n)} = \frac{1}{n+1}\sigma_{1}^{(n)} \geq 0 = \pt_{\X}(R_{0}), +\end{equation} +and +\begin{equation} + H - \frac{1}{2}\sigma_{1}^{(n)} = \frac{1}{2}\sigma_{0}^{(n)} - + \frac{n-1}{2(n+1)}\sigma_{1}^{(n)} = \frac{1}{n(n+1)}W_{n} = \pt_{\X}(R_{1}), +\end{equation} +where $W_{n}\in\Unitary(\X\otimes\Y)$ in the last equation is the swap operator +defined in \eqref{eq:swap-operator}. + +%------------------------------------------------------------------------------- +\section{A discussion on computational aspects} +\label{sec:computational-aspects} +%------------------------------------------------------------------------------- + +The proof approach we outlined for the Werner hiding pair (and that we are going to +pursue all over in the following chapters) may leave an uneasy feeling to the reader, +even though it is mathematically legitimate. We started by defining the operator $H$ in +Eq.~\eqref{eq:H-werner} whose trace was exactly equal to the bound we wanted to prove, +and then we proved that $H$ is indeed a feasible solution of the dual problem. +A posteriori our strategy did work out just fine, but how did we know that $H$ was +a good candidate for the solution we were seeking? + +For simple problems such as the one above, one could come up with some +insights after trying a few candidates that look promising and eventually tweak +the solution until things work out. +For more complicated instances this is not always a good strategy, and most often we are not +blessed by magic insights. What comes to rescue in difficult situations is a numerical +approach: we can get a good candidate solution by running an actual computer implementation of +one of the programs described above. +The output of the program often gives insights on a potential solution, which can +be then verified analytically, as we did above for the Werner hiding pair example. +When running a particular instance of a convex optimization problem, +attention should be paid to the time and the memory space that computer solvers need, +which is the topic of discussion of the rest of this section. + +Optimizing over separable operators (and therefore over separable measurements) is \mbox{NP-hard} \cite{Gharibi10}, +which simply means that solving the cone program ($\PP_{\Sep}$) is not feasible. +However, there do exist algorithms that solve semidefinite programs efficiently. +This very fact motivated us (and \cite{Doherty04} before us) to introduce the hierarchy of semidefinite programs +in Section \ref{sec:symm-ext}. +More precisely, let the dimensions of Alice's and Bob's subspaces be $\dim(\X) = n$ and +$\dim(\Y) = m$, respectively. Then the \emph{ellipsoid method} \cite{MartinGroetschel93} +solves the program corresponding to the $s$-th level of the symmetric extension hierarchy +in a running time that is \emph{polynomial} in $N$ (the number of states), $n$, $m^{s}$, +and the maximum bit-length of the entries of the density matrices specifying the +input states\footnote{Actual software implementations of semidefinite programming +solvers may use other methods that are not guaranteed to run as efficiently in theory, +but that behave very well in practice, the \emph{interior point method} being one of those.}. +Notice that the value of $s$ needed in order to reach a good approximation +of the separable problem can grow larger than a constant (and in fact larger than $m^{o(1)}$), +which is why this does not contradict the above-mentioned NP-hardness of the problem (or more precisely, +any complexity theory hypothesis). + +One way to make the algorithm more efficient in terms of memory used is to observe that +ultimately we are really optimizing only over operators in the space +\begin{equation} + \X\otimes\Y_{1}\ovee\cdots\ovee\Y_{s}, +\end{equation} +which has dimension +\begin{equation} + d = n\binom{s+m-1}{m-1}, +\end{equation} +that is, much smaller than $nm^{s}$. + +Let us see how this works for the case $s=2$, the other cases being +a straightforward generalization. +Let $A \in \Unitary(\Y\otimes\Y_{2},\Y\ovee\Y_{2})$ be the linear isometry such +that $AA^{\ast} = \Pi_{\Y\ovee\Y_{2}}$, where +$\Pi_{\Y\ovee\Y_{2}}$ is the projection on the symmetric subspace +$\Y\ovee\Y_{2}$. +Then we can replace the two sets of constraints +\begin{equation} +\begin{aligned} + &\tr_{\Y_{2}}(X_{k}) = \mu(k),\\ + &(\I_{\X}\otimes \Pi_{\Y\ovee\Y_{2}})X_{k} + (\I_{\X}\otimes \Pi_{\Y\ovee\Y_{2}}) = X_{k} +\end{aligned} +\end{equation} +in the Program \eqref{eq:sym-primal-problem}, with a set of constraints of the form +\begin{equation} + \tr_{\Y_{2}}((\I_{\X}\otimes A)X_{k}(\I_{\X}\otimes A^{\ast})) = \mu(k), +\end{equation} +where the only variables of the program are now of the form +\begin{equation} + X_{k}\in\Pos(\complex^{d}). +\end{equation} + +A MATLAB function that checks for separable/PPT distinguishability +has been developed as part of N. Johnston's QETLAB toolbox \cite{Johnston2015}. +See Appendix~\ref{chap:AppendixA} for more details on the implementation, +as well as a tutorial on how to use the function. + +\section{Unambiguous state discrimination} + +\label{sec:unambiguous-program} +In the previous sections, we analyzed the problem of distinguishing quantum states +using measurements that minimize the probability of error. +Here we consider a strategy in which Alice and Bob can give an inconclusive, yet +never incorrect answer. Such measurement strategies are called \emph{unambiguous}. + +If there are $N$ states to be distinguished, an unambiguous measurement +\[ + \mu : \{1, \ldots, N+1\} \rightarrow \Pos(\X\otimes\Y) +\] +consists of $N+1$ operators, where the outcome of the operator $\mu(N+1)$ corresponds +to the inconclusive answer. + +The cone programming approach has already been used to study unambiguous +discrimination by global measurement \cite{Eldar03}, +but never, as far as we know, to study unambiguous local discrimination. +In fact, we believe that unambiguous LOCC discrimination in general has not +been thoroughly investigated yet. + +The optimal value of the following cone program is equal to the success +probability of unambiguously distinguishing an ensemble of states +$\{ \rho_{1}, \ldots, \rho_{k} \}$ using measurements whose operators belong +to a convex cone $\C \subset \Pos(\X\otimes\Y)$. +Again, we assume that the states are drawn with a uniform probability. +\begin{center} + \centerline{\underline{Primal problem}}\vspace{-4mm} + \begin{align} + \text{maximize:}\quad & \sum_{k = 1}^N p_{k}\ip{\mu(k)}{\rho_{k}}\notag\\ + \text{subject to:}\quad & \sum_{k=1}^{N+1} \mu(k)= \I_{\X\otimes\Y}, \label{sdp-primal-unambiguous}\\ + & \mu : \{1, \ldots, N+1\} \rightarrow \C, \notag\\ + & \ip{\mu(i)}{\rho_{j}} = 0, \qquad 1 \leq i,j \leq N, \quad i \neq j. \notag + \end{align} +\end{center} +The dual program can be derived by routine calculation. +\begin{center} + \centerline{\underline{Dual problem}}\vspace{-4mm} + \begin{align} + \text{minimize:}\quad & \tr(H)\notag\\ + \text{subject to:}\quad & H - p_{k}\rho_{k} + \sum_{\substack{1\leq i \leq N \\ i\neq k}} + y_{i,k}\rho_{i} \in \C^{\ast}, \quad k=1,\ldots,N \; ,\label{sdp-dual-unambiguous}\\ + & H \in \Herm(\X\otimes\Y),\notag\\ + & y_{i,j} \in \real, \quad 1 \leq i,j \leq N, \quad i \neq j.\notag + \end{align} +\end{center} diff --git a/thesis-blx.bib b/thesis-blx.bib new file mode 100644 index 0000000..f1ead3e --- /dev/null +++ b/thesis-blx.bib @@ -0,0 +1,11 @@ +@Comment{$ biblatex control file $} +@Comment{$ biblatex version 1.7 $} +Do not modify this file! + +This is an auxiliary file used by the 'biblatex' package. +This file may safely be deleted. It will be recreated as +required. + +@Control{biblatex-control, + options = {1.7:0:0:1:0:0:1:1:0:0:0:0:1:1:3:1:79:+}, +} diff --git a/thesis-front.tex b/thesis-front.tex new file mode 100644 index 0000000..ce4f613 --- /dev/null +++ b/thesis-front.tex @@ -0,0 +1,222 @@ +%!TEX root = thesis.tex +% +% T I T L E P A G E +% ------------------- +% Last updated May 24, 2011, by Stephen Carr, IST-Client Services +% The title page is counted as page `i' but we need to suppress the +% page number. We also don't want any headers or footers. +\pagestyle{empty} +\pagenumbering{roman} + +% The contents of the title page are specified in the "titlepage" +% environment. +\begin{titlepage} + \begin{center} + \vspace*{1.0cm} + + \Huge + {\bf Quantum State Local Distinguishability via Convex Optimization} + + \vspace*{1.0cm} + + \normalsize + by \\ + + \vspace*{1.0cm} + + \Large + Alessandro Cosentino \\ + + \vspace*{3.0cm} + + \normalsize + A thesis \\ + presented to the University of Waterloo \\ + in fulfillment of the \\ + thesis requirement for the degree of \\ + Doctor of Philosophy \\ + in \\ + Computer Science \\ + + \vspace*{2.0cm} + + Waterloo, Ontario, Canada, 2015 \\ + + \vspace*{1.0cm} + + \end{center} +\end{titlepage} + +\noindent\textbf{Copyright notice.} +Chapters~\ref{chap:programs} and~\ref{chap:mes} contain material +from~\cite{Cosentino13}, which is copyrighted by the American Physical Society. +Chapters~\ref{chap:programs},~\ref{chap:mes} and ~\ref{chap:ups} +contain material from~\cite{Cosentino14}, which is copyrighted by Rinton Press, +and from~\cite{Bandyopadhyay15}, copyrighted by IEEE. +\\ \\ +Remaining material is: +\copyright\ Alessandro Cosentino 2015 \\ + + +% The rest of the front pages should contain no headers and be numbered using Roman numerals starting with `ii' +\pagestyle{plain} +\setcounter{page}{2} + +\cleardoublepage % Ends the current page and causes all figures and tables that have so far appeared in the input to be printed. +% In a two-sided printing style, it also makes the next page a right-hand (odd-numbered) page, producing a blank page if necessary. + + + +% D E C L A R A T I O N P A G E +% ------------------------------- + % The following is the sample Delaration Page as provided by the GSO + % December 13th, 2006. It is designed for an electronic thesis. + \noindent +I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. + + \bigskip + + \noindent +I understand that my thesis may be made electronically available to the public. + +\cleardoublepage +%\newpage + +% A B S T R A C T +% --------------- + +\begin{center}\textbf{Abstract}\end{center} + +Entanglement and nonlocality play a fundamental role in quantum computing. +To understand the interplay between these phenomena, researchers have considered the +model of local operations and classical communication, or LOCC for short, +which is a restricted subset of all possible operations +that can be performed on a multipartite quantum system. +The task of distinguishing states from a set that is known a priori to all parties +is one of the most basic problems among those used to test the power of LOCC protocols, +and it has direct applications to quantum data hiding, secret sharing and quantum channel capacity. + +The focus of this thesis is on state distinguishability problems for +classes of quantum operations that are more powerful than LOCC, +yet more restricted than global operations, namely the +classes of separable and positive-partial-transpose (PPT) measurements. +We build a framework based on convex optimization (on cone programming, in particular) +to study such problems. +Compared to previous approaches to the problem, +the method described in this thesis provides precise numerical bounds and quantitative analytic results. +By combining the duality theory of cone programming with the channel-state duality, +we also establish a novel connection between the state distinguishability problem +and the study of positive linear maps, which is a topic of independent interest in quantum information theory. + +We apply our framework to several questions that were left open in previous works +regarding the distinguishability of maximally entangled states and unextendable product sets. +First, we exhibit sets of $k < n$ orthogonal maximally entangled states in $\complex^{n}\otimes\complex^{n}$ +that are not perfectly distinguishable by LOCC. +As a consequence of this, we show a gap between the power of PPT and separable measurements +for the task of distinguishing sets consisting only of maximally entangled states. +Furthermore, we prove tight bounds on the entanglement cost that is necessary to +distinguish any sets of Bell states, thus showing that quantum teleportation is optimal for this task. +Finally, we provide an easily checkable characterization of when an unextendable product set is +perfectly discriminated by separable measurements, along with the first known +example of an unextendable product set that cannot be perfectly discriminated +by separable measurements. + +\cleardoublepage +\newpage + +% A C K N O W L E D G E M E N T S +% ------------------------------- + +\begin{center}\textbf{Acknowledgements}\end{center} + +First and foremost, I would like to thank my advisor John Watrous for his guidance +over my PhD years. +His meticulous style of writing has been and always will be an inspiration for me. +I also thank him for teaching two terrific courses on quantum information theory +and semidefinite programming. From his lectures I learned almost all the mathematical tools +I needed for writing this thesis. + +Next, I am grateful to professors Richard Cleve, Runyao Duan, Debbie Leung, and Ashwin Nayak +for agreeing to be in my defense committee, and to professor Andrew Childs for being in +my comprehensive exam committee. + +Most of the questions addressed in this thesis were explained to me by Som Bandyopadhyay, +Nengkun Yu, and Michael Nathanson. I thank them for a lot of insightful discussions on the problem. +The ``state distinguishability'' research community is small, but has welcomed me warmly. + +I owe some of the ideas presented in this work to my colleagues and co-authors +Vincent Russo and Nathaniel Johnston. I must thank them also for helping me +with coding in MATLAB the optimization problems presented in this thesis. + +From my great nerdy friends Robin and Vinayak, I learned too much: lots of theoretical computer science, +many research skills and, more importantly, rationality. + +I wish to thank grad school fellows Stacey, Adam, Ansis, Jamie, Laura, and Abel, for making +my life at IQC more enjoyable, and for being the coolest bunch of quantum kids in the world. +I hope our paths will cross many times in the years to come. + +Thanks to my roommate Daniele for bringing a piece of Italy into my everyday life. +(And I am not talking \emph{only} about Baricelle olive oil!) + +To all my soccer team-mates, I'm very sad +that I will no longer be playing for Hopeless Experts. +Every game, every defeat, every win with you guys has been fantastic. +I wish the team every success in the future seasons. + +Throughout the past years, my \emph{classical} friend Diego has been reminding me of +the joy of computer hacking and of how damn interesting every branch of computer science can be. +Thanks for that and for keeping me updated on your favorite musical discoveries. + +I have to save my last thanks to my family for their love and support. +I missed you! + + +\cleardoublepage +\newpage + +% D E D I C A T I O N +% ------------------- + +% \begin{center}\textbf{Dedication}\end{center} + +% This is dedicated to the one I love. +% \cleardoublepage +%\newpage + +% T A B L E O F C O N T E N T S +% --------------------------------- +\renewcommand\contentsname{Table of Contents} +\tableofcontents +\cleardoublepage +\phantomsection +%\newpage + +% L I S T O F T A B L E S +% --------------------------- +\addcontentsline{toc}{chapter}{List of Tables} +\listoftables +\cleardoublepage +\phantomsection % allows hyperref to link to the correct page +%\newpage + +% L I S T O F F I G U R E S +% ----------------------------- +\addcontentsline{toc}{chapter}{List of Figures} +\listoffigures +\cleardoublepage +\phantomsection % allows hyperref to link to the correct page +%\newpage + +% L I S T O F S Y M B O L S +% ----------------------------- +% To include a Nomenclature section +% \addcontentsline{toc}{chapter}{\textbf{Nomenclature}} +% \renewcommand{\nomname}{Nomenclature} +% \printglossary +% \cleardoublepage +% \phantomsection % allows hyperref to link to the correct page +% \newpage + +% Change page numbering back to Arabic numerals +\pagenumbering{arabic} + diff --git a/thesis.bib b/thesis.bib new file mode 100644 index 0000000..5ec8d69 --- /dev/null +++ b/thesis.bib @@ -0,0 +1,1038 @@ +% This file was created with JabRef 2.10. +% Encoding: UTF-8 + + +@Article{Bandyopadhyay11, + Title = {More Nonlocality with Less Purity}, + Author = {Bandyopadhyay, Somshubhro}, + Journal = {Phys. 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Comput.}, + Year = {2014}, + + Month = oct, + Number = {13-14}, + Pages = {1098--1106}, + Volume = {14}, + + Acmid = {2685167}, + Address = {Paramus, NJ}, + ISSN = {1533-7146}, + Issue_date = {October 2014}, + Keywords = {LOCC, PPT, semidefinite programming, state distinguishability}, + Numpages = {9}, + Publisher = {Rinton Press, Incorporated}, + Url = {http://dl.acm.org/citation.cfm?id=2685164.2685167} +} + +@Article{DiVincenzo2002, + Title = {Quantum data hiding}, + Author = {DiVincenzo, David P. and Leung, D.W. and Terhal, B.M.}, + Journal = {Information Theory, IEEE Transactions on}, + Year = {2002}, + + Month = {Mar}, + Number = {3}, + Pages = {580-598}, + Volume = {48}, + + Doi = {10.1109/18.985948}, + ISSN = {0018-9448} +} + +@Article{DiVincenzo03, + Title = {Unextendible Product Bases, Uncompletable Product Bases and Bound Entanglement}, + Author = {DiVincenzo, David P. and Mor, Tal and Shor, Peter W. and Smolin, John A. and Terhal, Barbara M.}, + Journal = {Communications in Mathematical Physics}, + Year = {2003}, + Number = {3}, + Pages = {379-410}, + Volume = {238}, + + Doi = {10.1007/s00220-003-0877-6}, + ISSN = {0010-3616}, + Language = {English}, + Owner = {acosenti}, + Publisher = {Springer-Verlag}, + Timestamp = {2015.04.16}, + Url = {http://dx.doi.org/10.1007/s00220-003-0877-6} +} + +@Article{Doherty04, + Title = {Complete family of separability criteria}, + Author = {Doherty, Andrew C. and Parrilo, Pablo A. and Spedalieri, Federico M.}, + Journal = {Phys. 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Rev. A}, + Year = {2011}, + + Month = {Jul}, + Pages = {012304}, + Volume = {84}, + + Doi = {10.1103/PhysRevA.84.012304}, + Eprint = {1010.2664}, + Issue = {1}, + Numpages = {3}, + Publisher = {American Physical Society} +} + +@Article{Yu15, + Title = {Detecting the local indistinguishability of maximally entangled states}, + Author = {Sixia Yu and C.H. Oh}, + Year = {2015}, + + Month = {Feb}, + + Eprint = {1502.01274}, + Owner = {acosenti}, + Timestamp = {2015.06.29} +} + +@Book{Wolkowicz00, + Title = {Handbook of Semidefinite Programming}, + Editor = {Henry Wolkowicz and Romesh Saigal and Lieven Vandenberghe}, + Publisher = {Springer US}, + Year = {2000}, + + Doi = {10.1007/978-1-4615-4381-7} +} + diff --git a/thesis.brf b/thesis.brf new file mode 100644 index 0000000..af86131 --- /dev/null +++ b/thesis.brf @@ -0,0 +1,106 @@ +\backcite {Cosentino13}{{ii}{(document)}{Doc-Start}} +\backcite {Cosentino14}{{ii}{(document)}{Doc-Start}} +\backcite {Bandyopadhyay15}{{ii}{(document)}{Doc-Start}} +\backcite {Nielsen11}{{2}{1.1}{section.1.1}} +\backcite {Helstrom1969}{{2}{1.1}{section.1.1}} +\backcite {Bennett99,Walgate00,Ghosh01,Horodecki03,Fan04,Ghosh04,Nathanson05,Watrous05,Yu11,Yu12}{{2}{1.1}{section.1.1}} +\backcite {Cleve99,Gottesman00}{{2}{1.1}{section.1.1}} +\backcite {Terhal01a,DiVincenzo2002}{{2}{1.1}{section.1.1}} +\backcite 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{Terhal01a,DiVincenzo2002}{{25}{3.2}{theorem.3.2}} +\backcite {DiVincenzo2002}{{26}{3.2}{equation.3.2.26}} +\backcite {Ghosh01}{{27}{3.3}{section.3.3}} +\backcite {Ghosh04}{{27}{3.3}{section.3.3}} +\backcite {Duan09}{{27}{3.3}{section.3.3}} +\backcite {Yu12}{{27}{3.3}{section.3.3}} +\backcite {Nathanson05}{{27}{3.3}{section.3.3}} +\backcite {Nathanson13}{{27}{3.3}{section.3.3}} +\backcite {Bandyopadhyay11a}{{27}{3.3}{section.3.3}} +\backcite {Yu12}{{28}{3.3}{section.3.3}} +\backcite {Fan04}{{28}{3.3}{equation.3.3.29}} +\backcite {Bennett99}{{28}{3.4}{section.3.4}} +\backcite {Walgate00}{{29}{3.3}{equation.3.3.29}} +\backcite {Nathanson05}{{29}{3.3}{equation.3.3.29}} +\backcite {Yu12}{{29}{3.3}{equation.3.3.29}} +\backcite {Cosentino13}{{29}{3.3}{equation.3.3.29}} +\backcite {Cosentino14,Bandyopadhyay15}{{29}{3.3}{equation.3.3.29}} +\backcite {Yu12,Duan09,Ghosh04}{{29}{3.3}{equation.3.3.29}} +\backcite {Childs13}{{29}{3.4}{equation.3.4.30}} +\backcite {Cohen08}{{30}{3.5}{section.3.5}} +\backcite {Bandyopadhyay09}{{30}{3.5}{section.3.5}} +\backcite {Bandyopadhyay10}{{30}{3.5}{section.3.5}} +\backcite {Yu14}{{30}{3.5}{section.3.5}} +\backcite {Walgate02}{{30}{3.6}{section.3.6}} +\backcite {Ghosh01,Ghosh04}{{31}{3.6}{section.3.6}} +\backcite {Ghosh01}{{31}{3.6}{equation.3.6.34}} +\backcite {Horodecki03}{{31}{3.6}{equation.3.6.34}} +\backcite {Nielsen99}{{32}{3.6}{equation.3.6.34}} +\backcite {Jonathan99}{{32}{3.6}{equation.3.6.34}} +\backcite {Horodecki03}{{32}{3.6}{equation.3.6.34}} +\backcite {Terhal01a}{{32}{3.6}{equation.3.6.34}} +\backcite {Yu12,Yu14}{{32}{3.6}{equation.3.6.34}} +\backcite {Eldar03a}{{34}{4.1}{section.4.1}} +\backcite {Rains01}{{38}{4.2.1}{subsection.4.2.1}} +\backcite {Doherty02,Doherty04}{{38}{4.2.1}{subsection.4.2.1}} +\backcite {Piani06}{{41}{4.2.1}{section*.13}} +\backcite {Gharibian13}{{44}{4.2.2}{subsection.4.2.2}} +\backcite {Chruscinski2014}{{45}{4.2.2}{equation.4.2.44}} +\backcite {Doherty02,Doherty04}{{46}{4.2.3}{subsection.4.2.3}} +\backcite {Doherty02}{{47}{4.2.3}{equation.4.2.50}} +\backcite {Gharibi10}{{53}{4.4}{section.4.4}} +\backcite {Doherty04}{{53}{4.4}{section.4.4}} +\backcite {MartinGroetschel93}{{53}{4.4}{section.4.4}} +\backcite {Johnston2015}{{54}{4.4}{equation.4.4.69}} +\backcite {Eldar03}{{54}{4.5}{section.4.5}} +\backcite {Yu12}{{56}{5}{chapter.5}} +\backcite {Yu14}{{56}{5}{chapter.5}} +\backcite {Yu12}{{56}{5}{chapter.5}} +\backcite {Yu12}{{57}{5.1}{section.5.1}} +\backcite {Nathanson05}{{57}{5.1}{section.5.1}} +\backcite {Horodecki99}{{58}{5.2}{equation.5.1.12}} +\backcite {Yu14}{{59}{5.2.1}{subsection.5.2.1}} +\backcite {Paulsen02}{{61}{5.2.1}{equation.5.2.25}} +\backcite {Horodecki03}{{63}{5.2.2}{subsection.5.2.2}} +\backcite {Cosentino13}{{63}{5.2.2}{equation.5.2.37}} +\backcite {Yu12}{{64}{5.3}{section.5.3}} +\backcite {Breuer06,Hall06}{{65}{5.3}{section*.15}} +\backcite {Cosentino14}{{68}{5.3.1}{equation.5.3.54}} +\backcite {Li15}{{72}{5.13}{theorem.5.13}} +\backcite {Cosentino14}{{72}{5.13}{theorem.5.13}} +\backcite {Yu12}{{72}{5.3.1}{section*.18}} +\backcite {Yu15}{{72}{1}{Hfootnote.5}} +\backcite {Bennett99}{{75}{6}{equation.6.0.1}} +\backcite {Bennett99}{{76}{6.1}{section.6.1}} +\backcite {DiVincenzo03}{{76}{6.1}{section.6.1}} +\backcite {Feng06}{{79}{6.3}{theorem.6.3}} +\backcite {Johnston2015}{{80}{6.1}{equation.6.1.11}} +\backcite {Terhal01,Bandyopadhyay05}{{80}{6.2}{section.6.2}} +\backcite {Bandyopadhyay11}{{80}{6.2}{section.6.2}} +\backcite {Terhal01}{{81}{6.2}{equation.6.2.13}} +\backcite {Terhal01}{{82}{6.6}{equation.6.2.24}} +\backcite {Ghosh04}{{84}{7}{theorem.7.1}} +\backcite {Montanaro07}{{85}{7}{theorem.7.2}} +\backcite {Matthews10}{{85}{7}{theorem.7.2}} +\backcite {Gharibian13}{{85}{7}{equation.7.0.1}} +\backcite {cvx}{{87}{A}{section*.22}} +\backcite {Johnston2015}{{87}{A}{section*.22}} +\backcite {Cosentino15}{{87}{A}{section*.24}} diff --git a/thesis.pdf b/thesis.pdf new file mode 100644 index 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provides (file)+> + <!ELEMENT requires (file)+> + <!ELEMENT generic (#PCDATA)> + <!ELEMENT binary (#PCDATA)> + <!ELEMENT option (#PCDATA)> + <!ELEMENT infile (#PCDATA)> + <!ELEMENT outfile (#PCDATA)> + <!ELEMENT file (#PCDATA)> + <!ATTLIST requests + version CDATA #REQUIRED + > + <!ATTLIST internal + package CDATA #REQUIRED + priority (9) #REQUIRED + active (0 | 1) #REQUIRED + > + <!ATTLIST external + package CDATA #REQUIRED + priority (1 | 2 | 3 | 4 | 5 | 6 | 7 | 8) #REQUIRED + active (0 | 1) #REQUIRED + > + <!ATTLIST provides + type (static | dynamic | editable) #REQUIRED + > + <!ATTLIST requires + type (static | dynamic | editable) #REQUIRED + > + <!ATTLIST file + type CDATA #IMPLIED + > +]> +<requests version="1.0"> + <internal package="biblatex" priority="9" active="1"> + <generic>latex</generic> + <provides type="dynamic"> + <file>thesis.aux</file> + <file>thesis-blx.bib</file> + </provides> + <requires type="dynamic"> + <file>thesis.bbl</file> + </requires> + <requires type="static"> + <file>blx-compat.def</file> + <file>biblatex.def</file> + <file>numeric.bbx</file> + <file>standard.bbx</file> + <file>numeric.cbx</file> + <file>biblatex.cfg</file> + <file>english.lbx</file> + </requires> + </internal> + <external package="biblatex" priority="5" active="1"> + <generic>bibtex</generic> + <cmdline> + <binary>bibtex</binary> + <option>-min-crossrefs 2</option> + <infile>thesis</infile> + </cmdline> + <input> + <file>thesis.aux</file> + </input> + <output> + <file>thesis.bbl</file> + </output> + <provides type="dynamic"> + <file>thesis.bbl</file> + </provides> + <requires type="dynamic"> + <file>thesis.aux</file> + <file>thesis-blx.bib</file> + </requires> + <requires type="editable"> + </requires> + <requires type="static"> + <file>biblatex.bst</file> + </requires> + </external> +</requests> diff --git a/thesis.tex b/thesis.tex new file mode 100644 index 0000000..165d7c1 --- /dev/null +++ b/thesis.tex @@ -0,0 +1,257 @@ +% uWaterloo Thesis Template for LaTeX +% Last Updated May 24, 2011 by Stephen Carr, IST Client Services +% FOR ASSISTANCE, please send mail to rt-IST-CSmathsci@ist.uwaterloo.ca + +% Effective October 2006, the University of Waterloo +% requires electronic thesis submission. See the uWaterloo thesis regulations at +% http://www.grad.uwaterloo.ca/Thesis_Regs/thesistofc.asp. + +% DON'T FORGET TO ADD YOUR OWN NAME AND TITLE in the "hyperref" package +% configuration below. THIS INFORMATION GETS EMBEDDED IN THE PDF FINAL PDF DOCUMENT. +% You can view the information if you view Properties of the PDF document. + +% Many faculties/departments also require one or more printed +% copies. This template attempts to satisfy both types of output. +% It is based on the standard "book" document class which provides all necessary +% sectioning structures and allows multi-part theses. + +% DISCLAIMER +% To the best of our knowledge, this template satisfies the current uWaterloo requirements. +% However, it is your responsibility to assure that you have met all +% requirements of the University and your particular department. +% Many thanks to the feedback from many graduates that assisted the development of this template. + +% ----------------------------------------------------------------------- + +% By default, output is produced that is geared toward generating a PDF +% version optimized for viewing on an electronic display, including +% hyperlinks within the PDF. + +% E.g. to process a thesis called "mythesis.tex" based on this template, run: + +% pdflatex mythesis -- first pass of the pdflatex processor +% bibtex mythesis -- generates bibliography from .bib data file(s) +% pdflatex mythesis -- fixes cross-references, bibliographic references, etc +% pdflatex mythesis -- fixes cross-references, bibliographic references, etc + +% If you use the recommended LaTeX editor, Texmaker, you would open the mythesis.tex +% file, then click the pdflatex button. Then run BibTeX (under the Tools menu). +% Then click the pdflatex button two more times. If you have an index as well, +% you'll need to run MakeIndex from the Tools menu as well, before running pdflatex +% the last two times. + +% N.B. The "pdftex" program allows graphics in the following formats to be +% included with the "\includegraphics" command: PNG, PDF, JPEG, TIFF +% Tip 1: Generate your figures and photos in the size you want them to appear +% in your thesis, rather than scaling them with \includegraphics options. +% Tip 2: Any drawings you do should be in scalable vector graphic formats: +% SVG, PNG, WMF, EPS and then converted to PNG or PDF, so they are scalable in +% the final PDF as well. +% Tip 3: Photographs should be cropped and compressed so as not to be too large. + +% To create a PDF output that is optimized for double-sided printing: +% +% 1) comment-out the \documentclass statement in the preamble below, and +% un-comment the second \documentclass line. +% +% 2) change the value assigned below to the boolean variable +% "PrintVersion" from "false" to "true". + +% --------------------- Start of Document Preamble ----------------------- + +% Specify the document class, default style attributes, and page dimensions +% For hyperlinked PDF, suitable for viewing on a computer, use this: +\documentclass[letterpaper,12pt,titlepage,oneside,]{book} +\usepackage[utf8]{inputenc} + +% For PDF, suitable for double-sided printing, change the PrintVersion variable below +% to "true" and use this \documentclass line instead of the one above: +%\documentclass[letterpaper,12pt,titlepage,openright,twoside,final]{book} + +% Some LaTeX commands I define for my own nomenclature. +% If you have to, it's better to change nomenclature once here than in a +% million places throughout your thesis! +\newcommand{\package}[1]{\textbf{#1}} % package names in bold text +\newcommand{\cmmd}[1]{\textbackslash\texttt{#1}} % command name in tt font +\newcommand{\href}[1]{#1} % does nothing, but defines the command so the + % print-optimized version will ignore \href tags (redefined by hyperref pkg). +%\newcommand{\texorpdfstring}[2]{#1} % does nothing, but defines the command +% Anything defined here may be redefined by packages added below... + +% This package allows if-then-else control structures. +\usepackage{ifthen} +\newboolean{PrintVersion} +\setboolean{PrintVersion}{false} +% CHANGE THIS VALUE TO "true" as necessary, to improve printed results for hard copies +% by overriding some options of the hyperref package below. +\usepackage{ifdraft} + +%\usepackage{nomencl} % For a nomenclature (optional; available from ctan.org) +\usepackage{amsmath,amssymb,amstext} % Lots of math symbols and environments +\usepackage[pdftex]{graphicx} % For including graphics N.B. pdftex graphics driver + +% Hyperlinks make it very easy to navigate an electronic document. +% In addition, this is where you should specify the thesis title +% and author as they appear in the properties of the PDF document. +% Use the "hyperref" package +% N.B. HYPERREF MUST BE THE LAST PACKAGE LOADED; ADD ADDITIONAL PKGS ABOVE +\usepackage[pdftex,pagebackref=true]{hyperref} % with basic options + % N.B. pagebackref=true provides links back from the References to the body text. This can cause trouble for printing. +\hypersetup{ + plainpages=false, % needed if Roman numbers in frontpages + % pdfpagelabels=true, % adds page number as label in Acrobat's page count + % bookmarks=true, % show bookmarks bar? + unicode=true, % non-Latin characters in Acrobat’s bookmarks + pdftoolbar=true, % show Acrobat’s toolbar? + pdfmenubar=true, % show Acrobat’s menu? + pdffitwindow=false, % window fit to page when opened + pdfstartview={FitH}, % fits the width of the page to the window +% pdftitle={uWaterloo\ LaTeX\ Thesis\ Template}, % title: CHANGE THIS TEXT! +% pdfauthor={Author}, % author: CHANGE THIS TEXT! and uncomment this line +% pdfsubject={Subject}, % subject: CHANGE THIS TEXT! and uncomment this line +% pdfkeywords={keyword1} {key2} {key3}, % list of keywords, and uncomment this line if desired + pdfnewwindow=true, % links in new window + colorlinks=true, % false: boxed links; true: colored links + linkcolor=blue, % color of internal links + citecolor=green, % color of links to bibliography + filecolor=magenta, % color of file links + urlcolor=cyan % color of external links +} +\ifthenelse{\boolean{PrintVersion}}{ % for improved print quality, change some hyperref options +\hypersetup{ % override some previously defined hyperref options +% colorlinks,% + citecolor=black,% + filecolor=black,% + linkcolor=black,% + urlcolor=black} +}{} % end of ifthenelse (no else) + +\hypersetup{final} +\input{header} + +% Setting up the page margins... +% uWaterloo thesis requirements specify a minimum of 1 inch (72pt) margin at the +% top, bottom, and outside page edges and a 1.125 in. (81pt) gutter +% margin (on binding side). While this is not an issue for electronic +% viewing, a PDF may be printed, and so we have the same page layout for +% both printed and electronic versions, we leave the gutter margin in. +% Set margins to minimum permitted by uWaterloo thesis regulations: +\setlength{\marginparwidth}{0pt} % width of margin notes +% N.B. If margin notes are used, you must adjust \textwidth, \marginparwidth +% and \marginparsep so that the space left between the margin notes and page +% edge is less than 15 mm (0.6 in.) +\setlength{\marginparsep}{0pt} % width of space between body text and margin notes +\setlength{\evensidemargin}{0.125in} % Adds 1/8 in. to binding side of all +% even-numbered pages when the "twoside" printing option is selected +\setlength{\oddsidemargin}{0.125in} % Adds 1/8 in. to the left of all pages +% when "oneside" printing is selected, and to the left of all odd-numbered +% pages when "twoside" printing is selected +\setlength{\textwidth}{6.375in} % assuming US letter paper (8.5 in. x 11 in.) and +% side margins as above +\raggedbottom + +% The following statement specifies the amount of space between +% paragraphs. Other reasonable specifications are \bigskipamount and \smallskipamount. +\setlength{\parskip}{\medskipamount} + +% The following statement controls the line spacing. The default +% spacing corresponds to good typographic conventions and only slight +% changes (e.g., perhaps "1.2"), if any, should be made. +\renewcommand{\baselinestretch}{1} % this is the default line space setting + +% By default, each chapter will start on a recto (right-hand side) +% page. We also force each section of the front pages to start on +% a recto page by inserting \cleardoublepage commands. +% In many cases, this will require that the verso page be +% blank and, while it should be counted, a page number should not be +% printed. The following statements ensure a page number is not +% printed on an otherwise blank verso page. +\let\origdoublepage\cleardoublepage +\newcommand{\clearemptydoublepage}{% + \clearpage{\pagestyle{empty}\origdoublepage}} +\let\cleardoublepage\clearemptydoublepage + +%===== +% \usepackage{hyperref} +% \let\Contentsline\contentsline +% \renewcommand\contentsline[3]{\Contentsline{#1}{#2}{}} +%===== + +%===================================================================== +% L O G I C A L D O C U M E N T -- the content of your thesis +%===================================================================== +\begin{document} + +% For a large document, it is a good idea to divide your thesis +% into several files, each one containing one chapter. +% To illustrate this idea, the "front pages" (i.e., title page, +% declaration, borrowers' page, abstract, acknowledgements, +% dedication, table of contents, list of tables, list of figures, +% nomenclature) are contained within the file "uw-ethesis-frontpgs.tex" which is +% included into the document by the following statement. +%---------------------------------------------------------------------- +% FRONT MATERIAL +%---------------------------------------------------------------------- +\input{thesis-front} + +%---------------------------------------------------------------------- +% MAIN BODY +%---------------------------------------------------------------------- +% Because this is a short document, and to reduce the number of files +% needed for this template, the chapters are not separate +% documents as suggested above, but you get the idea. If they were +% separate documents, they would each start with the \chapter command, i.e, +% do not contain \documentclass or \begin{document} and \end{document} commands. +%====================================================================== + +% For an explanation of the following line, +% see http://tex.stackexchange.com/a/36874/64973 +\setcounter{mtc}{2} + +\input{introduction} +\input{preliminaries} +\input{discrimination} +\input{programs} +\input{mes} +\input{ups} +\input{conclusion} + +\appendix + +% Add a title page before the appendices and a line in the Table of Contents +\chapter*{APPENDICES} +\addcontentsline{toc}{chapter}{APPENDICES} +%====================================================================== +\input{matlab_notebook} + +%---------------------------------------------------------------------- +% END MATERIAL +%---------------------------------------------------------------------- + +% B I B L I O G R A P H Y +% ----------------------- + +% The following statement selects the style to use for references. It controls the sort order of the entries in the bibliography and also the formatting for the in-text labels. +\bibliographystyle{alphaurl} +% This specifies the location of the file containing the bibliographic information. +% It assumes you're using BibTeX (if not, why not?). +\cleardoublepage % This is needed if the book class is used, to place the anchor in the correct page, + % because the bibliography will start on its own page. + % Use \clearpage instead if the document class uses the "oneside" argument +\phantomsection % With hyperref package, enables hyperlinking from the table of contents to bibliography +% The following statement causes the title "References" to be used for the bibliography section: +\renewcommand*{\bibname}{References} + +% Add the References to the Table of Contents +\addcontentsline{toc}{chapter}{\textbf{References}} + +\bibliography{thesis} +% Tip 5: You can create multiple .bib files to organize your references. +% Just list them all in the \bibliogaphy command, separated by commas (no spaces). + +% The following statement causes the specified references to be added to the bibliography +% even if they were not cited in the text. The asterisk is a wildcard that causes all +% entries in the bibliographic database to be included (optional). +\nocite{*} + +\end{document} diff --git a/ups.tex b/ups.tex new file mode 100644 index 0000000..76182c7 --- /dev/null +++ b/ups.tex @@ -0,0 +1,527 @@ +%!TEX root = thesis.tex +%------------------------------------------------------------------------------- +\chapter{Distinguishability of unextendable product sets} +\label{chap:ups} +%------------------------------------------------------------------------------- + +In this chapter, we study the state discrimination problem for collections of +states formed by unextendable product sets. +An orthonormal collection of product vectors +\begin{equation} + \A = \{ u_{k}\otimes v_{k} : k = 1, \ldots, N \} \subset \X \otimes \Y, +\end{equation} +for complex Euclidean spaces $\X=\complex^n$ and $\Y=\complex^m$, is said to be +an \emph{unextendable product set} if it is impossible to find a nonzero +product vector $u \otimes v \in \X \otimes \Y$ that is orthogonal to every +element of $\A$ \cite{Bennett99}. +That is, $\A$ is an unextendable product set if, for every choice of vectors +$u\in\X$ and $v\in\Y$ satisfying either $\ip{u}{u_{k}} = 0$ or +$\ip{v}{v_{k}} = 0$ for each $k\in\{1,\ldots,N\}$, one has that either $u = 0$ +or $v = 0$ (or both). + +The first section establishes a simple criterion for the states formed by +any unextendable product set to be perfectly discriminated by separable +measurements, and the second subsection proves that any set of states formed +by taking the union of an unextendable product set $\A \subset \X \otimes \Y$ +together with any pure state $z \in \X\otimes\Y$ orthogonal to every element of +$\A$ cannot be perfectly discriminated by a separable measurement. +(It is evident that PPT measurements allow a perfect discrimination in both +cases.) + +\minitoc + +%------------------------------------------------------------------------------% +\section{A criterion for perfect separable discrimination of +unextendable product sets} +%------------------------------------------------------------------------------% + +Here we provide a simple criterion for when an unextendable product set can be +perfectly discriminated by separable measurements, and we use this criterion to +show that there is an unextendable product set $\A \subset \X\otimes\Y$ that is +not perfectly discriminated by any separable measurement when +$\X = \Y = \complex^4$. +It is known that no unextendable product set $\A \subset \X\otimes\Y$ +spanning a proper subspace of $\X\otimes\Y$ can be perfectly discriminated by +an LOCC measurement \cite{Bennett99}, while every unextendable product +set can be discriminated perfectly by a PPT measurement. +It is also known that every unextendable product set $\A\subset\X\otimes\Y$ can +be perfectly discriminated by separable measurements in the case +$\X = \Y = \complex^3$ \cite{DiVincenzo03}. + +The following notation will be used throughout this section. +For $\X=\complex^n$, $\Y=\complex^m$, and +$\A = \{ u_{k}\otimes v_{k} : k = 1, \ldots, N \} \subset \X\otimes\Y$ +being an unextendable product set, we will write +\begin{equation} + \A_k = \A \backslash \{u_k \otimes v_k\}, +\end{equation} +and define a set of rank-one +product projections +\begin{equation} +\label{eq:P_k-sets} +\P_k = \bigl\{ x x^{\ast} \otimes y y^{\ast}\,:\, +x\in\X,\,y\in\Y,\,\norm{x} = \norm{y} =1,\;\text{and}\; +x\otimes y\perp \A_k\bigr\} +\end{equation} +for each $k = 1,\ldots,N$. +One may interpret each element $x x^{\ast} \otimes y y^{\ast}$ of +$\P_k$ as corresponding to a product vector $x \otimes y$ that could replace +$u_k \otimes v_k$ in $\A$, yielding a (not necessarily unextendable) +orthonormal product set. + +The following theorem states that the sets $\P_1,\ldots,\P_N$ defined above +determine whether or not an unextendable product set can be perfectly +discriminated by separable measurements. + +\begin{theorem}\label{thm:upb_sep_characterize} + Let $\X=\complex^n$ and $\Y=\complex^m$ be complex Euclidean spaces and let + \begin{equation} + \A = \{ u_{k}\otimes v_{k} : k = 1, \ldots, N \} \subset \X\otimes\Y + \end{equation} + be an unextendable product set. + The following two statements are equivalent: + \begin{enumerate} + \item + There exists a separable measurement $\mu \in \Meas_{\Sep}(N, \X:\Y)$ + that perfectly discriminates the + states represented by $\A$ (for any choice of nonzero probabilities + $p_1,\ldots,p_N$). + \item + For $\P_1,\ldots,\P_N$ as defined in \eqref{eq:P_k-sets}, one has that the + identity operator $\I_{\X}\otimes\I_{\Y}$ can be written as a nonnegative + linear combination of projections in the set $\P_1\cup\cdots\cup \P_N$. + \end{enumerate} +\end{theorem} + +\begin{proof} + Assume first that statement 2 holds, so that one may write + \begin{equation} + \I_{\X}\otimes\I_{\Y} = + \sum_{k = 1}^N \sum_{j = 1}^{M_k} \lambda_{k,j}\, + x_{k,j} x_{k,j}^{\ast}\otimes y_{k,j} y_{k,j}^{\ast} + \end{equation} + for some choice of positive integers $M_1,\ldots,M_N$, nonnegative real + numbers $\{\lambda_{k,j}\}$, and product vectors + $\{x_{k,j} \otimes y_{k,j}\}$ satisfying + \begin{equation} + x_{k,j} x_{k,j}^{\ast}\otimes y_{k,j} y_{k,j}^{\ast} \in \P_k + \end{equation} + for each $k\in\{1,\ldots,N\}$ and $j \in \{1,\ldots,M_k\}$. + Define + \begin{equation} \label{eq:P_k-enumeration} + \mu(k) = \sum_{j = 1}^{M_k} \lambda_{k,j}\, + x_{k,j} x_{k,j}^{\ast}\otimes y_{k,j} y_{k,j}^{\ast} + \end{equation} + for each $k\in\{1,\ldots,N\}$. + It is clear that $\mu$ is a separable measurement, and by the + definition of the sets $\P_1,\ldots,\P_N$ it necessarily holds that + \begin{equation} + \ip{\mu(k)}{u_\ell u_\ell^* \otimes v_\ell v_\ell^*} = 0, + \end{equation} + when $k\not=\ell$. + This implies that $\mu$ perfectly discriminates the states + represented by $\A$, and therefore implies that statement 1 holds. + + Now assume that statement 1 holds: there exists a separable measurement + \[\mu \in \Meas_{\Sep}(N, \X:\Y)\] + that perfectly discriminates the states represented by $\A$. + As each measurement operator $\mu(k)$ is separable, it is possible to write + \begin{equation} + \mu(k) = \sum_{j = 1}^{M_k} \lambda_{k,j}\, + x_{k,j} x_{k,j}^{\ast}\otimes y_{k,j} y_{k,j}^{\ast} + \end{equation} + for some choice of nonnegative integers $\{M_k\}$, positive real numbers + $\{\lambda_{k,j}\}$, and unit vectors + $\{x_{k,j}\,:\,j=1,\ldots,M_k\}\subset\X$ and + $\{y_{k,j}\,:\,j=1,\ldots,M_k\}\subset\Y$. + The assumption that this measurement perfectly discriminates $\A$ implies that + $x_{k,j}\otimes y_{k,j} \perp \A_k$, and therefore + $x_{k,j} x_{k,j}^{\ast} \otimes y_{k,j} y_{k,j}^{\ast} \in \P_k$, for each + $k = 1,\ldots,N$ and $j = 1,\ldots,M_k$. + As we have that $\mu(1)+\cdots+\mu(N) = \I_{\X} \otimes \I_{\Y}$, it follows that statement 2 + holds. +\end{proof} + +It is not immediately clear that Theorem~\ref{thm:upb_sep_characterize} is +useful for determining whether or not any particular unextendable product set +can be discriminated by separable measurements, but indeed it is. +What makes this so is the fact that each set $\P_k$ is necessarily finite, as +the following proposition establishes. + +\begin{prop}\label{lem:upb_finite} + Let $\X$ and $\Y$ be complex Euclidean spaces, let + \[ + \A = \{ u_{k}\otimes v_{k} : k = 1, \ldots, N \} \subset \X\otimes\Y + \] + be an unextendable product set, and let $\P_1,\ldots,\P_N$ be as defined in + \eqref{eq:P_k-sets}. + The sets $\P_1,\ldots,\P_N$ are finite. +\end{prop} + +\begin{proof} + Assume toward contradiction that $\P_k$ is infinite for some choice of + $k\in\{1,\ldots,N\}$. + There are finitely many subsets $S\subseteq \{1,\ldots,k-1,k+1,\ldots,N\}$, + so there must exist at least one such subset $S$ with the property that + there are infinitely many pairwise nonparallel product vectors of the form + $x\otimes y$ such that $x \perp u_j$ for every $j\in S$ and $y\perp v_j$ for + every $j\not\in\S$. + This implies that both the subspace of $\X$ orthogonal to + $\{u_j\,:\,j\in S\}$ and the subspace of $\Y$ orthogonal to + $\{v_j\,:\,j\not\in S\}$ have dimension at least 1, and + at least one of them has dimension at least~2. + It follows that there must exist a unit product vector $x \otimes y$ with + three properties: + (i) $x \perp u_j$ for every $j\in S$, + (ii) $y \perp v_j$ for every $j\not\in\S$, and + (iii) $x\otimes y\perp u_k \otimes v_k$. + This contradicts the fact that $\A$ is unextendable, and therefore completes + the proof. +\end{proof} + +Given Proposition~\ref{lem:upb_finite}, it becomes straightforward to make use +of Theorem~\ref{thm:upb_sep_characterize} computationally. +The sets $\P_1,\ldots,\P_N$ can be computed by iterating over all +$S \subseteq \{1,\ldots,k-1,k+1,\ldots,N\}$ and +finding the (at most one) product state orthogonal to $\{ u_j : j \in S \}$ on +$\X$ and $\{ v_j : j \notin S \}$ on $\Y$. +Then, the second statement in Theorem~\ref{thm:upb_sep_characterize} can be +checked through the use of linear programming (and even by hand in some cases). + +\begin{example}[Feng's unextendable product set] +We now present an example of an unextendable product set in $\X\otimes\Y$, +for $\X = \Y = \complex^4$, that cannot be perfectly discriminated by separable +measurements. +In particular, let $\A$ be the unextendable product set consisting of $8$ +states that were found in \cite{Feng06}: +\begin{equation} + \begin{array}{l} + \ket{\phi_1} = \ket{0}\ket{0}, \\ + \ket{\phi_2} = \ket{1}\left(\ket{0} - \ket{2} + \ket{3}\right)/\sqrt{3},\\ + \ket{\phi_3} = \ket{2}\left(\ket{0} + \ket{1} - \ket{3}\right)/\sqrt{3}, \\ + \ket{\phi_4} = \ket{3}\ket{3}, \\ + \ket{\phi_5} = \left(\ket{1} + \ket{2} + \ket{3}\right)\left(\ket{0} + - \ket{1} + \ket{2}\right)/3, \\ + \ket{\phi_6} = \left(\ket{0} - \ket{2} + \ket{3}\right)\ket{2}/\sqrt{3}, \\ + \ket{\phi_7} = \left(\ket{0} + \ket{1} - \ket{3}\right)\ket{1}/\sqrt{3}, \\ + \ket{\phi_8} = \left(\ket{0} - \ket{1} + \ket{2}\right)\left(\ket{0} + + \ket{1} + \ket{2}\right)/3. + \end{array} +\end{equation} + +For each $k = 1, \ldots, 8$, there are exactly $6$ product states contained in +$\P_k$ for each choice of $k$, which we represent by product vectors +$\ket{\phi_{k,j}}$ for $j = 1, \ldots, 6$. +To be explicit, these states are as follows (where we have omitted +normalization factors for brevity):\vspace{3mm} + +\noindent\hspace{8pt} +\begin{minipage}{0.48\textwidth} + $\ket{\phi_{1,1}} = \ket{0}\ket{0}$,\\ + $\ket{\phi_{1,2}} = \left(\ket{0} + \ket{1} - \ket{3}\right) + \left(\ket{0} + \ket{2}\right)$,\\ + $\ket{\phi_{1,3}} = \left(\ket{0} - \ket{1}\right) + \left(\ket{0} + \ket{1} - \ket{3}\right)$, +\end{minipage}\hfill +\begin{minipage}{0.48\textwidth} + $\ket{\phi_{1,4}} = \left(\ket{0} - \ket{1} + + \ket{2}\right)\left(\ket{0} - \ket{1} + \ket{2}\right)$, \\ + $\ket{\phi_{1,5}} = \left(\ket{0} + \ket{2}\right) + \left(\ket{0} - \ket{2} + \ket{3}\right)$,\\ + $\ket{\phi_{1,6}} = \left(\ket{0} - \ket{2} + \ket{3}\right) + \left(\ket{0} - \ket{1}\right)$, +\end{minipage} + +\vspace{2mm} + +\noindent\hspace{8pt} +\begin{minipage}{0.48\textwidth} + $\ket{\phi_{2,1}} = \ket{1}\left(\ket{0} - \ket{2} + \ket{3}\right)$,\\ + $\ket{\phi_{2,2}} = \left(\ket{0} + \ket{1} - \ket{3}\right)\ket{2}$,\\ + $\ket{\phi_{2,3}} = \left(\ket{0} + \ket{1}\right)\ket{3}$, +\end{minipage}\hfill +\begin{minipage}{0.48\textwidth} + $\ket{\phi_{2,4}} = \left(\ket{0} - \ket{1} + \ket{2}\right) + \left(\ket{1} - 2\ket{2} + \ket{3}\right)$, \\ + $\ket{\phi_{2,5}} = \left(\ket{1} + \ket{2} + \ket{3}\right) + \left(\ket{0} - \ket{1} - 2\ket{2}\right)$, \\ + $\ket{\phi_{2,6}} = \left(\ket{1} - \ket{3}\right)\ket{0}$, +\end{minipage} + +\vspace{2mm} + +\noindent\hspace{8pt} +\begin{minipage}{0.48\textwidth} + $\ket{\phi_{3,1}} = \ket{2}\left(\ket{0} + \ket{1} - \ket{3}\right)$,\\ + $\ket{\phi_{3,2}} = \left(\ket{0} - \ket{2} + \ket{3}\right)\ket{1}$,\\ + $\ket{\phi_{3,3}} = \left(\ket{2} - \ket{3}\right)\ket{0}$, +\end{minipage}\hfill +\begin{minipage}{0.48\textwidth} + $\ket{\phi_{3,4}} = \left(\ket{1} + \ket{2} + \ket{3}\right) + \left(\ket{0} + 2\ket{1} + \ket{2}\right)$, \\ + $\ket{\phi_{3,5}} = \left(\ket{0} - \ket{1} + \ket{2}\right) + \left(2\ket{1} - \ket{2} - \ket{3}\right)$, \\ + $\ket{\phi_{3,6}} = \left(\ket{0} - \ket{2}\right)\ket{3}$, +\end{minipage} + +\vspace{2mm} + +\noindent\hspace{8pt} +\begin{minipage}{0.48\textwidth} + $\ket{\phi_{4,1}} = \ket{3}\ket{3}$,\\ + $\ket{\phi_{4,2}} = \left(\ket{0} + \ket{1} - \ket{2}\right) + \left(\ket{2} + \ket{3}\right)$,\\ + $\ket{\phi_{4,3}} = \left(\ket{1} + \ket{3}\right) + \left(\ket{0} + \ket{1} - \ket{3}\right)$, +\end{minipage}\hfill +\begin{minipage}{0.48\textwidth} + $\ket{\phi_{4,4}} = \left(\ket{2} + \ket{3}\right) + \left(\ket{0} - \ket{2} + \ket{3}\right)$, \\ + $\ket{\phi_{4,5}} = \left(\ket{1} + \ket{2} + \ket{3}\right) + \left(\ket{1} + \ket{2} + \ket{3}\right)$, \\ + $\ket{\phi_{4,6}} = \left(\ket{0} - \ket{2} + \ket{3}\right) + \left(\ket{1} + \ket{3}\right)$, +\end{minipage} + +\vspace{2mm} + +\noindent\hspace{8pt} +\begin{minipage}{0.48\textwidth} + $\ket{\phi_{5,1}} = \left(\ket{1} + \ket{2} + \ket{3}\right) + \left(\ket{0} - \ket{1} + \ket{2}\right)$,\\ + $\ket{\phi_{5,2}} = \ket{1}\left(2\ket{0} + \ket{2} - \ket{3}\right)$,\\ + $\ket{\phi_{5,3}} = \ket{3}\ket{0}$, +\end{minipage}\hfill +\begin{minipage}{0.48\textwidth} + $\ket{\phi_{5,4}} = \left(\ket{0} - \ket{2} - 2\ket{3}\right)\ket{2}$, \\ + $\ket{\phi_{5,5}} = \ket{2}\left(2\ket{0} - \ket{1} + \ket{3}\right)$, \\ + $\ket{\phi_{5,6}} = \left(\ket{0} + \ket{1} + 2\ket{3}\right)\ket{1}$, +\end{minipage} + +\vspace{2mm} + +\noindent\hspace{8pt} +\begin{minipage}{0.48\textwidth} + $\ket{\phi_{6,1}} = \left(\ket{0} - \ket{2} + \ket{3}\right)\ket{2}$,\\ + $\ket{\phi_{6,2}} = \ket{3}\left(\ket{0} - \ket{2}\right)$,\\ + $\ket{\phi_{6,3}} = \ket{0}\left(\ket{2} - \ket{3}\right)$, +\end{minipage}\hfill +\begin{minipage}{0.48\textwidth} + $\ket{\phi_{6,4}} = \left(\ket{0} - \ket{1} - 2\ket{2}\right) + \left(\ket{1} + \ket{2} + \ket{3}\right)$, \\ + $\ket{\phi_{6,5}} = \ket{2}\left(\ket{0} - \ket{2} + \ket{3}\right)$, \\ + $\ket{\phi_{6,6}} = \left(\ket{1} - 2\ket{2} + \ket{3}\right) + \left(\ket{0} - \ket{1} + \ket{2}\right)$, +\end{minipage} + +\vspace{2mm} + +\noindent\hspace{8pt} +\begin{minipage}{0.48\textwidth} + $\ket{\phi_{7,1}} = \left(\ket{0} + \ket{1} - \ket{3}\right)\ket{1}$,\\ + $\ket{\phi_{7,2}} = \ket{0}\left(\ket{1} - \ket{3}\right)$,\\ + $\ket{\phi_{7,3}} = \ket{1}\left(\ket{0} + \ket{1} - \ket{3}\right)$, +\end{minipage}\hfill +\begin{minipage}{0.48\textwidth} + $\ket{\phi_{7,4}} = \left(\ket{0} + 2\ket{1} + \ket{2}\right) + \left(\ket{1} + \ket{2} + \ket{3}\right)$, \\ + $\ket{\phi_{7,5}} = \left(2\ket{1} - \ket{2} - \ket{3}\right) + \left(\ket{0} - \ket{1} + \ket{2}\right)$, \\ + $\ket{\phi_{7,6}} = \ket{3}\left(\ket{0} + \ket{1}\right)$, +\end{minipage} + +\vspace{2mm} + +\noindent\hspace{8pt} +\begin{minipage}{0.48\textwidth} + $\ket{\phi_{8,1}} = \left(\ket{0} - \ket{1} + \ket{2}\right) + \left(\ket{0} + \ket{1} + \ket{2}\right)$,\\ + $\ket{\phi_{8,2}} = \ket{1}\left(\ket{0} - \ket{2} - 2\ket{3}\right)$,\\ + $\ket{\phi_{8,3}} = \left(2\ket{0} - \ket{1} + \ket{3}\right)\ket{1}$, +\end{minipage}\hfill +\begin{minipage}{0.48\textwidth} + $\ket{\phi_{8,4}} = \ket{0}\ket{3}$, \\ + $\ket{\phi_{8,5}} = \left(2\ket{0} + \ket{2} - \ket{3}\right)\ket{2}$, \\ + $\ket{\phi_{8,6}} = \ket{2}\left(\ket{0} + \ket{1} + 2\ket{3}\right)$. +\end{minipage} + +\vspace{3mm} + +\noindent +One may verify by a computer that $\I\otimes\I$ is not contained in the convex +cone generated by +\begin{equation} + \label{eq:Feng-replacement-set} + \bigl\{ \ket{\phi_{k,j}}\bra{\phi_{k,j}}\,:\,k = 1,\ldots,8,\;j=1,\ldots,6 + \bigr\}. +\end{equation} +(In fact, $\I\otimes\I$ is not in the linear span of the set +\eqref{eq:Feng-replacement-set}.) +Theorem~\ref{thm:upb_sep_characterize} therefore implies that this unextendable +product set is not perfectly discriminated by separable measurements. +\end{example} + +The computational procedure described above was implemented in MATLAB as part +of the QETLAB Toolbox (\cite{Johnston2015}, \texttt{UPBSepDistinguishable} function). + + +%------------------------------------------------------------------------------% +\section[Impossibility to distinguish an unextendable product set plus one more +pure state]{Impossibility to distinguish an unextendable\newline +product set plus one more pure state} +%------------------------------------------------------------------------------% + +Next, we prove an upper bound on the probability to correctly discriminate +any unextendable product set, together with one extra pure state orthogonal +to the members of the unextendable product set, by a separable measurement. +Central to the proof of this statement is a family of positive linear maps +previously studied in the literature \cite{Terhal01,Bandyopadhyay05}. + +Before proving this fact, we note that it is fairly straightforward to obtain +a qualitative result along similar lines: +if a separable measurement were able to perfectly discriminate a particular +product set from any state orthogonal to this product set, there would +necessarily be a separable measurement operator orthogonal to the space spanned +by the product set, implying that some nonzero product state must be orthogonal +to the product set (and therefore the product set must be extendable). +Related results based on this sort of argument may be found in \cite{Bandyopadhyay11}. +An advantage of the method described here is that one obtains +precise bounds on the optimal discrimination probability, as opposed to a +statement that a perfect discrimination is not possible. + +The following lemma is required for the proof of the theorem below. + +\begin{lemma}[Terhal] + \label{lemma:lambda} + For given complex Euclidean spaces $\X=\complex^n$ and $\Y=\complex^m$, and + any unextendable product set + \begin{equation} + \A = \{ u_{k}\otimes v_{k} : k = 1, \ldots, N \} \subset \X\otimes\Y, + \end{equation} + there exists a positive real number $\lambda_{\A} > 0$ such that + \begin{equation} + \left(\I_{\X} \otimes y^{\ast}\right) + \left( \sum_{k = 1}^{N}u_{k}u_{k}^{\ast}\otimes v_{k}v_{k}^{\ast} \right) + \left(\I_{\X} \otimes y\right) + - \lambda_{\A}\norm{y}^{2}\I_{\X} \in \Pos(\X), + \end{equation} + for every $y \in \Y$. +\end{lemma} + +\noindent +A proof of the lemma, as well as a constructive procedure +to calculate a bound on $\lambda_{\A}$, can be found in \cite{Terhal01}. + +\begin{theorem} + \label{thm:upb} + Let $\X = \complex^n$ and $\Y=\complex^m$ be complex Euclidean spaces, let + \begin{equation} + \A = \{ u_{k}\otimes v_{k} : k = 1, \ldots, N \} \subset \X\otimes\Y + \end{equation} + be an unextendable product set, and let + \begin{equation} + z \in \X\otimes\Y + \end{equation} + be a unit vector orthogonal to $\A$. + We have that + \begin{equation} + \opt_{\Sep}(\A \cup \{ z \}) \leq 1 - \frac{\lambda_{\A}}{(N + 1)\delta}, + \end{equation} + where $\lambda_{\A}$ is a positive real number satisfying the requirements + of Lemma~\ref{lemma:lambda} and + \[ + \delta = \norm{\tr_{\X}(z z^{\ast})}. + \] +\end{theorem} + +\begin{proof} + Consider the following Hermitian operator: + \begin{equation} + H = \frac{1}{N+1} \left( \sum_{k = 1}^{N}u_{k}u_{k}^{\ast}\otimes + v_{k}v_{k}^{\ast} + \left( 1- \frac{\lambda_{\A}}{\delta} \right) + zz^{\ast} \right). + \end{equation} + We want to show that $H$ is a feasible solution of the dual problem + \eqref{eq:sep-dual-problem} for the state discrimination problem under + consideration. + It is clear that + \begin{equation} + H - \frac{1}{N+1}u_{k}u_{k}^{\ast}\otimes v_{k}v_{k}^{\ast} \in + \Pos(\X\otimes\Y) \subset \BPos(\X:\Y) + \end{equation} + for $k = 1, \ldots, N$. + The remaining constraint left to be checked is the following: + \begin{equation} + \label{eq:upb_constraint} + H - \frac{1}{N+1}zz^{\ast} = + \frac{1}{N+1}\left( + \sum_{k = 1}^{N}u_{k}u_{k}^{\ast}\otimes v_{k}v_{k}^{\ast} - + \frac{\lambda_{\A}}{\delta} zz^{\ast} \right) \in \BPos(\X:\Y). + \end{equation} + Using the fact that + \begin{equation} + \delta\norm{y}^{2}\I_{\X} - + \left(\I_{\X} \otimes y^{\ast}\right)zz^{\ast}\left(\I_{\X} \otimes y\right) + \in \Pos(\X), + \end{equation} + for any $y \in \Y$, together with Lemma \ref{lemma:lambda}, one has that + \begin{equation} + \left( \I \otimes y \right)^{\ast} + \left(\sum_{k = 1}^{N}u_{k}u_{k}^{\ast}\otimes v_{k}v_{k}^{\ast} - + \frac{\lambda_{\A}}{\delta} zz^{\ast}\right) + \left( \I \otimes y \right) \in \Pos(\X) + \end{equation} + and therefore the constraint \eqref{eq:upb_constraint} is satisfied. + Finally, it holds that + \begin{equation} + \tr(H) = 1 - \frac{\lambda_{\A}}{(N + 1)\delta}, + \end{equation} + which completes the proof. +\end{proof} + +%------------------------------------------------------------------------------% +\begin{example}[Tiles set] +\label{ex:tiles-set} +%------------------------------------------------------------------------------% + +Theorem \ref{thm:upb} allow us to find specific bounds for the probability +of correctly discriminating certain sets of states. +For instance, here we consider the following unextendable product set +$\A \subset \X\otimes\Y$ for $\X = \Y = \complex^3$, commonly known as the +\emph{tiles set}: +\begin{equation} + \begin{array}{llll} + \ket{\phi_1} = \ket{0}\left(\frac{\ket{0}-\ket{1}}{\sqrt{2}}\right), + &\ket{\phi_2} = \ket{2}\left(\frac{\ket{1}-\ket{2}}{\sqrt{2}}\right), + &\ket{\phi_3} = \left(\frac{\ket{0}-\ket{1}}{\sqrt{2}}\right)\ket{2}, + &\ket{\phi_4} = \left(\frac{\ket{1}-\ket{2}}{\sqrt{2}}\right)\ket{0},\\[4mm] + \multicolumn{4}{c}{\ket{\phi_5} = + \frac{1}{3}\left(\ket{0}+\ket{1}+\ket{2}\right) + \left(\ket{0}+\ket{1}+\ket{2}\right).} + \end{array} +\end{equation} +For a pure state orthogonal to this set, one may take +\begin{equation} + \ket{\psi} = \frac{1}{2}\left(\ket{0}\ket{0} + \ket{0}\ket{1} - + \ket{0}\ket{2} - \ket{1}\ket{2}\right). +\end{equation} +Using the procedure described in \cite{Terhal01}, one obtains +\begin{equation} + \lambda_{\A} \geq \frac{1}{9}\left( 1 - \sqrt{\frac{5}{6}} \right)^{2}. +\end{equation} +Therefore, if we assume that each state is selected with probability $1/6$, +the maximum probability of correctly discriminating the set +$\left\{ \ket{\phi_1}, \ldots, \ket{\phi_5}, \ket{\psi} \right\}$ by a +separable measurement is at most +\begin{equation} + \opt_{\Sep}(\A) \leq 1 - \frac{1}{54}\frac{\left( 1 - \sqrt{\frac{5}{6}} \right)^{2}} + {\cos\left(\frac{\pi}{8}\right)^{2}} < 1 - 1.647 \times 10^{-4}. +\end{equation} +This bound is not tight, as numerical optimization (see Appendix \ref{chap:AppendixA}) +shows that +\begin{equation} + \opt_{\Sep}(\A) < 0.9861 . +\end{equation} + +\end{example} +