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.gitignore

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+# Created by http://www.gitignore.io
+
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conclusion.tex

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+%!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.