superop-contract.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# QuTiP Example: Superoperators, Pauli Basis and Channel Contraction"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[Christopher Granade](http://www.cgranade.com/) <br>\n",
    "Institute for Quantum Computing\n",
    "$\\newcommand{\\ket}[1]{\\left|#1\\right\\rangle}$\n",
    "$\\newcommand{\\bra}[1]{\\left\\langle#1\\right|}$\n",
    "$\\newcommand{\\cnot}{{\\scriptstyle \\rm CNOT}}$\n",
    "$\\newcommand{\\Tr}{\\operatorname{Tr}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Introduction"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this notebook, we will demonstrate the ``tensor_contract`` function, which contracts one or more pairs of indices of a Qobj. This functionality can be used to find rectangular superoperators that implement the partial trace channel $S(\\rho) = \\Tr_2(\\rho)$, for instance. Using this functionality, we can quickly turn a system-environment representation of an open quantum process into a superoperator representation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Preamble"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Features"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We enable a few features such that this notebook runs in both Python 2 and 3."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from __future__ import division, print_function"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Imports"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import qutip as qt\n",
    "\n",
    "from qutip.ipynbtools import version_table"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plotting Support"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Settings"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "qt.settings.colorblind_safe = True"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Superoperator Representations and Plotting"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We start off by first demonstrating plotting of superoperators, as this will be useful to us in visualizing the results of a contracted channel."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In particular, we will use Hinton diagrams as implemented by [``qutip.visualization.hinton``](http://qutip.org/docs/3.0.1/apidoc/functions.html#qutip.visualization.hinton), which\n",
    "show the real parts of matrix elements as squares whose size and color both correspond to the magnitude of each element. To illustrate, we first plot a few density operators."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
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ZkqQeMZglSeoRg3lMkuuAA1W1sDbV+bnjXQrcVlXb1+N4kqTeuxPY2mH7I2tV\nkbXSOpiTnAvcBPxVM38J8F7glcA5wB9V1W0d9vc+4D8Bvwr8BPgH4H1V1WosUUnS5lNVl290HdZa\nl7GydwL3VNVTzfxpjAbkfhewuIpjXwrcCrwGeANwFPhCkjNXsS9JkuZCl0vZOxn7skZVHQAOACS5\nreuBq+r3x+eTXA38CPgd4LNd9ydJ0jxo1WNOcjpwCaNPYK2VFzX1+WHbAkl2r/Txa0mShqhtj/nN\nwINV9b01rMvNwP2MPp3VSlXtBfauWY0kSVpnbYN5B81l67WQ5EbgYuDiqpr6A5qSJA1V22BeALav\nRQWSfBi4Enh9VT2yFseQJGko2j6VfTuwI8lMByRJcjNwFfCGqvrWLPctSdIQtQ3ag8AzwOuAuwGS\nnAa8vFl/CnBukguBJ6vq+yfbYZJbgKuBK4AfJjmrWfV0VT3dvgmSJM2PVj3mGo1/tp/RK1NLLgK+\n3kxbgA80vz+4tEGSa5JUku3L7PY6Rk9i3w08Nja9t2sjJEmaF10uTd8B7AHeCVBVXwJykjLnAQ8B\nhyZXVNXJykqStOl0Gfnri8AZzeXqtnYA11fV0W7VkiRpc2rdY66qZ4EXd9l5Vb2qc40kSdrEuvSY\nN8ICow9nSJK0KfT6e8zN5yUNZknSptH3HrMkSZuKwSxJUo8YzJIk9YjBLElSjxjMkiT1iMEsSVKP\nGMySJPWIwSxJUo8YzJIk9YjBLElSjxjMkiT1iMEsaVNbXFzc6Cp0NsQ6q71ef8RCktbavn37NroK\n0vPYYx45vNEVkKRV8v9fcyZVtdF1kCRJDXvMkiT1iMEsSVKPGMySJPWIwSxJUo8YzJIk9YjBLElS\njxjMkiT1iMEsSVKPGMySJPWIwSxJUo8YzJIk9YjBLElSjxjMkiT1iMEsSVKPGMySJPWIwSxJUo8Y\nzJIk9YjBLElSjxjMkiT1iMEsSVKP/H+KJcjsONnE0wAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x114f4f9b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "qt.visualization.hinton(qt.identity([2, 3]).unit());"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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g4QkAAzwkAQC2QecJAIN0ngAwwK0qALANOk8AGKTzBIAB3Z1z584tu4wdE54ATGoVOs+r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\nBIBBwhMABv1/y2jnpeFbQdEAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11867e9b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "qt.visualization.hinton(qt.Qobj([\n",
    "    [1, 0.5],\n",
    "    [0.5, 1]\n",
    "]).unit());"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We show superoperators as matrices in the *Pauli basis*, such that any Hermicity-preserving map is represented by a real-valued matrix. This is especially convienent for use with Hinton diagrams, as the plot thus carries complete information about the channel.\n",
    "\n",
    "As an example, conjugation by $\\sigma_z$ leaves $\\mathbb{1}$ and $\\sigma_z$ invariant, but flips the sign of $\\sigma_x$ and $\\sigma_y$. This is indicated in Hinton diagrams by a negative-valued square for the sign change and a positive-valued square for a +1 sign."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x118a03400>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "qt.visualization.hinton(qt.to_super(qt.sigmaz()));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As a couple more examples, we also consider the supermatrix for a Hadamard transform and for $\\sigma_z \\otimes H$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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PPPHEpMuYGhs3buTqq6+edBmaIX2/uKCq9jIfioPrdg7MF/C21n3HZWBKHcPy6fw+NCpf\nvi5J0hC+S1aSpEYGpiRJDQxMSZIaGJiSJA1RVd70I0lSCztMSZIaGJiSJDUwMCVJamBgSpI0hC8u\nkCSpkYEpSVIDA1OSpAYGpiRJDQxMSZKG8E0/kiQ1ssOUJKmBgSlJUgMDU5KkIXxxgSRJjQxMSZIa\nrPfAPGnSBUiS1odjx441T+NK8vwktyR5sPv3eYuMOSPJV5N8M8n9Sa4e2Pa+JAeT3N1Nlw47poEp\nSRrb8WuYrVMPdgC3VdUW4LZueaEjwDur6izgFcDbkpw1sP0jVXV2N+0ddkADU5LUizUOzG3A9d38\n9cBrF6nnUFV9o5v/IfAt4LSVHtDAlCT1Yi1PyQKnVtWhbv4R4NTlBid5CfDvgL8dWP32JPckuXax\nU7oLGZiSpF6M2GFuSrJvYNq+8POS3JrkvkWmbQuOW8CSbWuSHwN2A79bVT/oVn8S+CngbOAQ8OFh\nf593yUqSxraCU62Hq+qcIZ950VLbkjyaZHNVHUqyGXhsiXHPZD4s/2tVfWHgsx8dGPNnwJeGFWyH\nKUnqxRpfw9wDXNHNXwHctHBAkgCfAr5VVX+yYNvmgcXXAfcNO+DQwEzyme6W25MXrL8wyb8keeWw\nz5AkrX9rHJgfAC5O8iBwUbdMkhcnOX7H6y8BbwL+/SKPj/xxknuT3ANcALxj2AFbTsleBdwLvBd4\nT1fQjwPXAh+sqjua/zxJ0rq1li8uqKrHgQsXWf9d4NJu/mtAltj/TaMec2iHWVX/CLwFeHeSc7vV\nHwG+B7xv1ANKktanNe4w11zTTT9VdWuSTwLXJ7kGeCPwsqp6clWrkyTNhFkOwlaj3PTz+8y3tp8H\nrqmqe5camGT7crcKS5LWHzvMTlXNJfkQ8DGGPK9SVbuAXWPWJkmaIbMahK1GfQ7zCHCsqnp5TYMk\naf0wMCVJamBgSpI0xCxfm2xlYEqSerHeA3OkV+NV1XVV9WOrVYwkaXZ5l6wkSQ1mNQhbGZiSpF4Y\nmJIkDTHLp1pbGZiSpF4cO7a+H9E3MCVJvbDDlCSpgYEpSdIQXsOUJKmRgSlJUgMDU5KkBgamJElD\nVJWPlUiS1MIOU5KkBmsZmEmeD3weeAnwHeA3qup7i4z7DvBD4ChwpKrOGWX/QSP9WokkSUtZ418r\n2QHcVlVbgNu65aVcUFVnHw/LFewPGJiSpJ6scWBuA67v5q8HXrva+xuYkqSxjRKWPQXmqVV1qJt/\nBDh1qdKAW5PclWT7CvZ/itcwJUm9GDEINyXZN7C8q6p2DQ5IcivwokX2fc+C41aSpQ5+XlUdTPJC\n4JYk366q20fY/ykGpiSpFyMG5uEF1xQX+7yLltqW5NEkm6vqUJLNwGNLfMbB7t/HktwInAvcDjTt\nP8hTspKkXqzxKdk9wBXd/BXATQsHJNmY5DnH54FXAfe17r+QHaYkaWwTeHHBB4C/TPJbwN8DvwGQ\n5MXAn1fVpcxfl7wxCczn3Wer6ivL7b8cA1OS1Iu1fA6zqh4HLlxk/XeBS7v5h4CXjrL/cgxMSVIv\nfDWeJElD+HuYkiQ1ssOUJKmBHaYkSUNUFUePHp10GavKwJQk9cIOUzpBbNy4kSeeeGLSZUyNjRs3\nTroEzRivYUoniKuvvnrSJUgzy7tkJUlqZGBKktTAU7KSJDWww5QkaYgJvHx9zRmYkqRe2GFKktTA\nwNTMuuyyyzjllFMmXcbUmJubY/fu3ZMuQ1qXPCWrmWZYPp3fh7S67DAlSWpghylJUgM7TEmShvAa\npiRJjewwJUlqYGBKkjTEiXBK9qRJFyBJWh+O/8RXyzSuJM9PckuSB7t/n7fImH+b5O6B6QdJfrfb\n9r4kBwe2XTrsmAamJKkXx44da556sAO4raq2ALd1y09TVQ9U1dlVdTbwC8CPgBsHhnzk+Paq2jvs\ngAamJKkXa9lhAtuA67v564HXDhl/IfC/q+rvV3pAA1OSNLaq4ujRo81TD06tqkPd/CPAqUPGXw58\nbsG6tye5J8m1i53SXcjAlCT1YsQOc1OSfQPT9oWfl+TWJPctMm1bcNwClmxbk5wM/BrwVwOrPwn8\nFHA2cAj48LC/z7tkJUm9GPFU6+GqOmfI51201LYkjybZXFWHkmwGHlvmoy4BvlFVjw589lPzSf4M\n+NKwgu0wJUljG6W77Oka5h7gim7+CuCmZca+gQWnY7uQPe51wH3DDmhgSpJ6scaB+QHg4iQPAhd1\nyyR5cZKn7nhNshG4GPjCgv3/OMm9Se4BLgDeMeyAnpKVJPViLV9cUFWPM3/n68L13wUuHVh+AnjB\nIuPeNOoxDUxJUi98NZ4kSUP0eKp1ahmYkqReGJiSJDUwMCVJamBgSpI0xInw814GpiSpF3aYkiQ1\nMDAlSWpgYEqS1MDAlCRpCF9cIElSI++SlSSpwXrvMIf+vFeS85PUMtNX16JQSdJ0W+Of91pzLR3m\nHcDmRdb/GrAT+ESvFUmSZs4sB2GroYFZVU8CjwyuS/IzwIeA91fVX61SbZKkGXLCB+ZCSZ4L3AT8\nDXBN3wVJkmbT0aNHJ13Cqhp6DXNQkpOAzwJHgDfWEv93Isn2JPu6aXsPdUqSptgo1y9ntRMdtcN8\nP/CLwLlV9cOlBlXVLmDXOIVJkmbLrAZhq+bATHI58C7g1VX14OqVJEmaRQYmkORs4FPAjqq6eXVL\nkiTNohM+MJNsAr7I/E0+n0nyooVjquqRheskSSeW9R6YLTf9vBr4SeBS4NASkyTpBHb8B6Rbp3El\n+fUk9yc5luScZcZtTfJAkv1Jdgysf36SW5I82P37vGHHHBqYVXV9VWW5qf1PlCStV2t8l+x9wH8A\nbl9qQJINwMeBS4CzgDckOavbvAO4raq2ALd1y8sa6bESSZKWspaBWVXfqqoHhgw7F9hfVQ91L+G5\nAdjWbdsGXN/NXw+8dtgxffm6JKkXU3gN8zTg4YHlA8DLu/lTq+r4JcVHgFOHfZiBKUnqw83AphHG\nPyvJvoHlXd0z/E9Jcivwr240Bd5TVTetoMZFVVUlGZr2BqYkaWxVtXUVPvOiMT/iIHDGwPLp3TqA\nR5NsrqpDSTYDjw37MK9hSpLWqzuBLUnOTHIycDmwp9u2B7iim7+C+XekL8vAlCTNnCSvS3KA+de1\n/rckN3frX5xkL0BVHQGuYv508beAv6yq+7uP+ABwcZIHgYu65WV5SlaSNHOq6kbgxkXWf5f59wYc\nX94L7F1k3OPAhaMc0w5TkqQGBqYkSQ0MTEmSGhiYkiQ1MDAlSWpgYEqS1MDAlCSpgYEpSVIDA1OS\npAYGpiRJDQxMSZIaGJiSJDUwMCVJamBgSpLUwMCUJKmBgSlJUgMDU5KkBgamJEkNDExJkhoYmJIk\nNTAw17G5ublJlzBV/D4kjeMZky5Aq2f37t2TLkGS1g07TEmSGhiYkiQ1mPbAfHTSBUjSCvnfr3Um\nVTXpGiRJmnrT3mFKkjQVDExJkhoYmJIkNTAwJUlqYGBKktTAwJQkqYGBKUlSAwNTkqQGBqYkSQ0M\nTEmSGhiYkiQ1MDAlSWpgYEqS1MDAlCSpgYEpSVIDA1OSpAYGpiRJDQxMSZIaGJiSJDUwMCVJavD/\nAIi4rHw15E2rAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1188cde48>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "qt.visualization.hinton(qt.to_super(qt.hadamard_transform()));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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mAjQzs86SeFTwBmDvkvW98m2VzKXsMnBEbMh/Pg3cTHZpuSkpLgU/AUyvsBxHNsrq8gTH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\nAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x118e9d048>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "qt.visualization.hinton(qt.to_super(qt.tensor(qt.sigmaz(), qt.hadamard_transform())));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Reduced Channels"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As an example of tensor contraction, we now consider the map $S(\\rho) = \\Tr_2[\\cnot (\\rho \\otimes \\ket{0}\\bra{0}) \\cnot^\\dagger]$.\n",
    "We can think of the $\\cnot$ here as a system-environment representation of an open quantum process, in which an environment register is prepared in a state $\\rho_{\\text{anc}}$, then a unitary acts jointly on the system of interest and environment. Finally, the environment is traced out, leaving a *channel* on the system alone. In terms of [Wood diagrams](http://arxiv.org/abs/1111.6950), this can be represented as the composition of a preparation map, evolution under the system-environment unitary, and then a measurement map."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](files/sprep-wood-diagram.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The two tensor wires on the left indicate where we must take a tensor contraction to obtain the measurement map. Numbering the tensor wires from 0 to 3, this corresponds to a ``tensor_contract`` argument of ``(1, 3)``."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[[2], [2]], [[2, 2], [2, 2]]], shape = (4, 16), type = other\\begin{equation*}\\left(\\begin{array}{*{11}c}1.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 1.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 1.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 1.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[[2], [2]], [[2, 2], [2, 2]]], shape = (4, 16), type = other\n",
       "Qobj data =\n",
       "[[ 1.  0.  0.  0.  0.  1.  0.  0.  0.  0.  0.  0.  0.  0.  0.  0.]\n",
       " [ 0.  0.  1.  0.  0.  0.  0.  1.  0.  0.  0.  0.  0.  0.  0.  0.]\n",
       " [ 0.  0.  0.  0.  0.  0.  0.  0.  1.  0.  0.  0.  0.  1.  0.  0.]\n",
       " [ 0.  0.  0.  0.  0.  0.  0.  0.  0.  0.  1.  0.  0.  0.  0.  1.]]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s_meas = qt.tensor_contract(qt.to_super(qt.identity([2, 2])), (1, 3))\n",
    "s_meas"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Meanwhile, the ``super_tensor`` function implements the swap on the right, such that we can quickly find the preparation map."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[[2, 2], [2, 2]], [[2], [2]]], shape = (16, 4), type = other\\begin{equation*}\\left(\\begin{array}{*{11}c}1.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 1.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0\\\\\\vdots & \\vdots & \\vdots & \\vdots\\\\0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[[2, 2], [2, 2]], [[2], [2]]], shape = (16, 4), type = other\n",
       "Qobj data =\n",
       "[[ 1.  0.  0.  0.]\n",
       " [ 0.  0.  0.  0.]\n",
       " [ 0.  1.  0.  0.]\n",
       " [ 0.  0.  0.  0.]\n",
       " [ 0.  0.  0.  0.]\n",
       " [ 0.  0.  0.  0.]\n",
       " [ 0.  0.  0.  0.]\n",
       " [ 0.  0.  0.  0.]\n",
       " [ 0.  0.  1.  0.]\n",
       " [ 0.  0.  0.  0.]\n",
       " [ 0.  0.  0.  1.]\n",
       " [ 0.  0.  0.  0.]\n",
       " [ 0.  0.  0.  0.]\n",
       " [ 0.  0.  0.  0.]\n",
       " [ 0.  0.  0.  0.]\n",
       " [ 0.  0.  0.  0.]]"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q = qt.tensor(qt.identity(2), qt.basis(2))\n",
    "s_prep = qt.sprepost(q, q.dag())\n",
    "s_prep"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For a $\\cnot$ system-environment model, the composition of these maps should give us a completely dephasing channel. The channel on both qubits is just the superunitary $\\cnot$ channel:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(<matplotlib.figure.Figure at 0x11882c7f0>,\n",
       " <matplotlib.axes._subplots.AxesSubplot at 0x118b74898>)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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iBYiIO4AryHqlHwROBj4SES/l+58HZgNnADcAbwVOG46Azcyss7VronNJe0i6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\nNTMzS8iJ1czMLCEnVjMzs4T+P+49YTr6zX5yAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11882c7f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "qt.visualization.hinton(qt.to_super(qt.cnot()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now complete by multiplying the superunitary $\\cnot$ by the preparation channel above, then applying the partial trace channel by contracting the second and fourth index indices. As expected, this gives us a dephasing map."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[[2], [2]], [[2], [2]]], shape = (4, 4), type = super, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}1.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 1.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[[2], [2]], [[2], [2]]], shape = (4, 4), type = super, isherm = True\n",
       "Qobj data =\n",
       "[[ 1.  0.  0.  0.]\n",
       " [ 0.  0.  0.  0.]\n",
       " [ 0.  0.  0.  0.]\n",
       " [ 0.  0.  0.  1.]]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qt.tensor_contract(qt.to_super(qt.cnot()), (1, 3)) * s_prep"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x11922d128>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "qt.visualization.hinton(qt.tensor_contract(qt.to_super(qt.cnot()), (1, 3)) * s_prep);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Epilouge"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table><tr><th>Software</th><th>Version</th></tr><tr><td>QuTiP</td><td>4.3.0.dev0+6e5b1d43</td></tr><tr><td>Numpy</td><td>1.13.1</td></tr><tr><td>SciPy</td><td>0.19.1</td></tr><tr><td>matplotlib</td><td>2.0.2</td></tr><tr><td>Cython</td><td>0.25.2</td></tr><tr><td>Number of CPUs</td><td>2</td></tr><tr><td>BLAS Info</td><td>INTEL MKL</td></tr><tr><td>IPython</td><td>6.1.0</td></tr><tr><td>Python</td><td>3.6.2 |Anaconda custom (x86_64)| (default, Jul 20 2017, 13:14:59) \n",
       "[GCC 4.2.1 Compatible Apple LLVM 6.0 (clang-600.0.57)]</td></tr><tr><td>OS</td><td>posix [darwin]</td></tr><tr><td colspan='2'>Thu Jul 20 22:59:16 2017 MDT</td></tr></table>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "version_table()"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}