definitions.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Definitions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\r\n",
    "import matplotlib.pyplot as plt\r\n",
    "from qutip import *\r\n",
    "qutip.settings.has_mkl = False"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Qubits"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Pauli Matricies"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\hat\\sigma_x$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0 & 1.0\\\\1.0 & 0.0\\\\\\end{array}\\right)\\end{equation*}",
      "text/plain": "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\nQobj data =\n[[0. 1.]\n [1. 0.]]"
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sigmax()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\hat\\sigma_y$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0 & -1.0j\\\\1.0j & 0.0\\\\\\end{array}\\right)\\end{equation*}",
      "text/plain": "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\nQobj data =\n[[0.+0.j 0.-1.j]\n [0.+1.j 0.+0.j]]"
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sigmay()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\hat\\sigma_z$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}1.0 & 0.0\\\\0.0 & -1.0\\\\\\end{array}\\right)\\end{equation*}",
      "text/plain": "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\nQobj data =\n[[ 1.  0.]\n [ 0. -1.]]"
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sigmaz()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### States"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "up = basis(2, 0)\r\n",
    "down = basis(2, 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$|\\uparrow\\rangle$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": "Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\\begin{equation*}\\left(\\begin{array}{*{11}c}1.0\\\\0.0\\\\\\end{array}\\right)\\end{equation*}",
      "text/plain": "Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\nQobj data =\n[[1.]\n [0.]]"
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "up"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$|\\downarrow\\rangle$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": "Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0\\\\1.0\\\\\\end{array}\\right)\\end{equation*}",
      "text/plain": "Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\nQobj data =\n[[0.]\n [1.]]"
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "down"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\hat \\sigma_z|\\uparrow\\rangle = +1|\\uparrow\\rangle$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": "Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\\begin{equation*}\\left(\\begin{array}{*{11}c}1.0\\\\0.0\\\\\\end{array}\\right)\\end{equation*}",
      "text/plain": "Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\nQobj data =\n[[1.]\n [0.]]"
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sigmaz() * up"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\hat \\sigma_z|\\downarrow\\rangle = -1|\\downarrow\\rangle$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": "Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0\\\\-1.0\\\\\\end{array}\\right)\\end{equation*}",
      "text/plain": "Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\nQobj data =\n[[ 0.]\n [-1.]]"
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sigmaz() * down"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Spin Matricies\r\n",
    "$\\hat S_i = \\frac\\hbar 2\\hat\\sigma_i,\\quad\\hbar=1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0 & 0.500\\\\0.500 & 0.0\\\\\\end{array}\\right)\\end{equation*}",
      "text/plain": "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\nQobj data =\n[[0.  0.5]\n [0.5 0. ]]"
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "jmat(.5, 'x')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0 & -0.500j\\\\0.500j & 0.0\\\\\\end{array}\\right)\\end{equation*}",
      "text/plain": "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\nQobj data =\n[[0.+0.j  0.-0.5j]\n [0.+0.5j 0.+0.j ]]"
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "jmat(.5, 'y')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}0.500 & 0.0\\\\0.0 & -0.500\\\\\\end{array}\\right)\\end{equation*}",
      "text/plain": "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\nQobj data =\n[[ 0.5  0. ]\n [ 0.  -0.5]]"
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "jmat(.5, 'z')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\hat S_z|\\uparrow\\rangle = \\frac12|\\uparrow\\rangle$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": "Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\\begin{equation*}\\left(\\begin{array}{*{11}c}0.500\\\\0.0\\\\\\end{array}\\right)\\end{equation*}",
      "text/plain": "Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\nQobj data =\n[[0.5]\n [0. ]]"
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "jmat(.5, 'z') * up"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\hat S_z|\\downarrow\\rangle = -\\frac12|\\downarrow\\rangle$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": "Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0\\\\-0.500\\\\\\end{array}\\right)\\end{equation*}",
      "text/plain": "Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\nQobj data =\n[[ 0. ]\n [-0.5]]"
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "jmat(.5, 'z') * down"
   ]
  }
 ],
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