examples/notebooks/density.ipynb

{
 "metadata": {
  "name": "density"
 }, 
 "nbformat": 2, 
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown", 
     "source": [
      "# Density operator and matrix "
     ]
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "from sympy import *", 
      "from sympy.physics.quantum import *", 
      "from sympy.physics.quantum.density import *", 
      "from sympy.physics.quantum.spin import (", 
      "    Jx, Jy, Jz, Jplus, Jminus, J2,", 
      "    JxBra, JyBra, JzBra,", 
      "    JxKet, JyKet, JzKet,", 
      ")", 
      "from IPython.core.display import display_pretty"
     ], 
     "language": "python", 
     "outputs": [], 
     "prompt_number": 1
    }, 
    {
     "cell_type": "code", 
     "collapsed": true, 
     "input": [
      "%load_ext sympyprinting"
     ], 
     "language": "python", 
     "outputs": [], 
     "prompt_number": 2
    }, 
    {
     "cell_type": "markdown", 
     "source": [
      "Create n density matrix using symbolic states"
     ]
    }, 
    {
     "cell_type": "code", 
     "collapsed": true, 
     "input": [
      "psi = Ket('psi')", 
      "phi = Ket('phi')"
     ], 
     "language": "python", 
     "outputs": [], 
     "prompt_number": 3
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "d = Density((psi,0.5),(phi,0.5)); d"
     ], 
     "language": "python", 
     "outputs": [
      {
       "latex": [
        "$$\\rho\\left(\\begin{pmatrix}{\\left|\\psi\\right\\rangle }, & 0.5\\end{pmatrix},\\begin{pmatrix}{\\left|\\phi\\right\\rangle }, & 0.5\\end{pmatrix}\\right)$$"
       ], 
       "output_type": "pyout", 
       "prompt_number": 4, 
       "text": [
        "\u03c1((\u2758\u03c8\u27e9, 0.5),(\u2758\u03c6\u27e9, 0.5))"
       ]
      }
     ], 
     "prompt_number": 4
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "display_pretty(d)"
     ], 
     "language": "python", 
     "outputs": [
      {
       "output_type": "display_data", 
       "text": [
        "\u03c1((\u2758\u03c8\u27e9, 0.5),(\u2758\u03c6\u27e9, 0.5))"
       ]
      }
     ], 
     "prompt_number": 5
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "d.states()"
     ], 
     "language": "python", 
     "outputs": [
      {
       "latex": [
        "$$\\begin{pmatrix}{\\left|\\psi\\right\\rangle }, & {\\left|\\phi\\right\\rangle }\\end{pmatrix}$$"
       ], 
       "output_type": "pyout", 
       "prompt_number": 6, 
       "text": [
        "(\u2758\u03c8\u27e9, \u2758\u03c6\u27e9)"
       ]
      }
     ], 
     "prompt_number": 6
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "d.probs()"
     ], 
     "language": "python", 
     "outputs": [
      {
       "latex": [
        "$$\\begin{pmatrix}0.5, & 0.5\\end{pmatrix}$$"
       ], 
       "output_type": "pyout", 
       "prompt_number": 7, 
       "text": [
        "(0.5, 0.5)"
       ]
      }
     ], 
     "prompt_number": 7
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "d.doit()"
     ], 
     "language": "python", 
     "outputs": [
      {
       "latex": [
        "$$0.5 {\\left|\\phi\\right\\rangle } {\\left\\langle \\phi\\right|} + 0.5 {\\left|\\psi\\right\\rangle } {\\left\\langle \\psi\\right|}$$"
       ], 
       "output_type": "pyout", 
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKoAAAAXCAYAAACbItQpAAAABHNCSVQICAgIfAhkiAAABXZJREFU\naIHt2musHGUdx/EP57RUKVAuAootKMXK5QCBgkCigfgGXqDBEkqi8Q0NxhpaNURISNRgiOFFUy7S\nlCDaA0UKL0oEDbS8IEtCFKNEIMhFisYQ6gUVqEDAI+CL/4xnds6zs7M7e86emv0mm515LvN75jfP\nPNdhxIgRSbYPuwA9sDeVtRt7671sh7FS4FJsw0Z8F9fi0JoX3I2bcBaOwdV4EseV0tW9HtyQCLsJ\nC2rmv7GBDr2VtQnzzXfYkAibDe+r9EiUexy78JVC2LXYaWaFTvF+6fcmPp9I16pxLTgBlyfCV+GS\nGvnPxWUNdKhf1ibMN9/hEPHClBm09930SJT7S3gHiwthnxA3f2kNsT/hVtyDb+DYusId+ComEuHj\nuKtG/tvwwQY69F9RT1a/5ZlvvsMXcE4ifNDed9MjUe4n8HAi4Yu4t4bYjAs2THcj9ukQ9018qiLv\ncnxvADqtmtcoM4mP1Uw733yH67Bvh7hBel9Hr8V017JAtAIvJBLuEk35XJN3ZSl+hDUVeddi8wB0\nZpv56DssxL87xA3S+zp6mO6ejhCtyhuJNG/iYCwSXVQnFuFbOAj7i5dgI/7YQ4G/iNNFt/FxbMWv\nsQnvFtLtwb/wUbxcusYS0Y3+eQA6s80wfT8cX8NKcd87svAP4W/Z8Tgexd2mJ0f9et+vXhsrRauS\narK3ZnFHpDIWeAHLCudr8bT2sRfpLmipmDysE0avxSniId4p/YYuF5OOMleISpiiV51UWeswqV7X\nP0zfN4iKdaWoHDkX4czseCGexwOlvP14369ei+muP39j30sILKyIK/JJvFQ4v00sl3ynS779cD9+\nhh9kOifhKfGgvi8mPOVJwos4MsufM45T8ZsB6swmw/L9RPwBr+Mz+Ech7jTT/k3hKvy1lL9X75vq\n/a+iPq/zGGFxdoG/d4jPKRs6hVd0H2ddJ27y5uw8n9jk48a80KmZ+R34cuH8Qvx0FnRmi2H5/k9s\nwYdxvljDzRnTPvz5ixgWlenF+8Z6+Rh1Cs+JMVGZ/cUYomrCcZ948OX1u0VYUZGPuMHieGQCvyuc\n513JEx3yF2fsYxXlbKqT4nYxdChzlJgZpyrhGjyeHQ/L93wMeSn+I3oaOExUlCInaq9YRep6Pyg9\nxJiovBwyjtfwYFVGsTvySCnscFHwnaXwVuH4wCzNBYWw9Ti+cL5ZdM8ptpjZ/fwkka5fnZb+mFR/\neWoYvudsK+W/WPuLdwjO66Bd1/umei3adz6eFIuuxYXaM8QAuLy9tayU7udi0bbIadl/ldl7Mt0j\nC2HH49ns+NNiJ2R1Iu9y8aa+VQh7F78Vk5RB6cw2w/A95z3hTc6E6Zd1gdgteyiRrxfvB6HXxsH4\nvZgR59yKx0rpVmaF2lEIO0G0IvkbNiZaibvM3AZslc5XiTHJkiztpiz8c/iFqEQpNmif7eYskV4l\n6EenXNa6TKrfog7Ld6KHeUu0ZHBN9r8UPxY7ZCl69b6JXov2bb5Xswt9Gz/MwhaLB1nkFbFt93gh\n7Bnx4DeKN34Kv8T1us9a7xVrhhtE93G0GEvuxmfxdiLPgdnvpUTc68KMj2hfz+tHZy4Ylu9Ei7wa\nt4iW7WzTnnxdrJeW6cf7JnpDo1URt173yRexjXdmRfyxqrfx6uq0aqRJMal+izpXtLrEr1NvxaOp\n973qtaj3dc5cskJ0g1WMiRn6ryrS7BI7Jx9ooNOEPYbXQvdLeRUkxSC870WvTXiumaqI6/RxSJEL\nTS9vVLFV+zpfrzpUl7WK9WYuuwybOr53++ZhEN73okf/z6AxB3QIP0p0K91o+vFuXR06l3VvpOpe\nJsQ+fDcG9eF0XT3+v57BiBEjRowYMaIe/wURMbsOqkJiMgAAAABJRU5ErkJggg==\n", 
       "prompt_number": 8, 
       "text": [
        "0.5\u22c5\u2758\u03c6\u27e9\u22c5\u27e8\u03c6\u2758 + 0.5\u22c5\u2758\u03c8\u27e9\u22c5\u27e8\u03c8\u2758"
       ]
      }
     ], 
     "prompt_number": 8
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "Dagger(d)"
     ], 
     "language": "python", 
     "outputs": [
      {
       "latex": [
        "$$\\rho\\left(\\begin{pmatrix}{\\left|\\psi\\right\\rangle }, & 0.5\\end{pmatrix},\\begin{pmatrix}{\\left|\\phi\\right\\rangle }, & 0.5\\end{pmatrix}\\right)$$"
       ], 
       "output_type": "pyout", 
       "prompt_number": 9, 
       "text": [
        "\u03c1((\u2758\u03c8\u27e9, 0.5),(\u2758\u03c6\u27e9, 0.5))"
       ]
      }
     ], 
     "prompt_number": 9
    }, 
    {
     "cell_type": "code", 
     "collapsed": true, 
     "input": [
      "A = Operator('A')"
     ], 
     "language": "python", 
     "outputs": [], 
     "prompt_number": 10
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "d.operate_on(A)"
     ], 
     "language": "python", 
     "outputs": [
      {
       "latex": [
        "$$\\rho\\left(\\begin{pmatrix}A {\\left|\\psi\\right\\rangle }, & 0.5\\end{pmatrix},\\begin{pmatrix}A {\\left|\\phi\\right\\rangle }, & 0.5\\end{pmatrix}\\right)$$"
       ], 
       "output_type": "pyout", 
       "prompt_number": 11, 
       "text": [
        "\u03c1((A\u22c5\u2758\u03c8\u27e9, 0.5),(A\u22c5\u2758\u03c6\u27e9, 0.5))"
       ]
      }
     ], 
     "prompt_number": 11
    }, 
    {
     "cell_type": "markdown", 
     "source": [
      "Now create a density matrix using spin states"
     ]
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "up = JzKet(S(1)/2,S(1)/2)", 
      "down = JzKet(S(1)/2,-S(1)/2)"
     ], 
     "language": "python", 
     "outputs": [], 
     "prompt_number": 12
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "d2 = Density((up,0.5),(down,0.5)); d2"
     ], 
     "language": "python", 
     "outputs": [
      {
       "latex": [
        "$$\\rho\\left(\\begin{pmatrix}{\\left|\\frac{1}{2},\\frac{1}{2}\\right\\rangle }, & 0.5\\end{pmatrix},\\begin{pmatrix}{\\left|\\frac{1}{2},- \\frac{1}{2}\\right\\rangle }, & 0.5\\end{pmatrix}\\right)$$"
       ], 
       "output_type": "pyout", 
       "prompt_number": 13, 
       "text": [
        "\u03c1((\u27581/2,1/2\u27e9, 0.5),(\u27581/2,-1/2\u27e9, 0.5))"
       ]
      }
     ], 
     "prompt_number": 13
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "represent(d2)"
     ], 
     "language": "python", 
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{smallmatrix}0.5 & 0\\\\0 & 0.5\\end{smallmatrix}\\right]$$"
       ], 
       "output_type": "pyout", 
       "prompt_number": 14, 
       "text": [
        "", 
        "\u23a10.5   0 \u23a4", 
        "\u23a2        \u23a5", 
        "\u23a3 0   0.5\u23a6"
       ]
      }
     ], 
     "prompt_number": 14
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "d2.operate_on(Jz)"
     ], 
     "language": "python", 
     "outputs": [
      {
       "latex": [
        "$$\\rho\\left(\\begin{pmatrix}J_z {\\left|\\frac{1}{2},\\frac{1}{2}\\right\\rangle }, & 0.5\\end{pmatrix},\\begin{pmatrix}J_z {\\left|\\frac{1}{2},- \\frac{1}{2}\\right\\rangle }, & 0.5\\end{pmatrix}\\right)$$"
       ], 
       "output_type": "pyout", 
       "prompt_number": 15, 
       "text": [
        "", 
        "\u03c1\u239b\u239bJ \u22c5\u27581/2,1/2\u27e9, 0.5\u239e,\u239bJ \u22c5\u27581/2,-1/2\u27e9, 0.5\u239e\u239e", 
        " \u239d\u239d z               \u23a0 \u239d z                \u23a0\u23a0"
       ]
      }
     ], 
     "prompt_number": 15
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "qapply(_)"
     ], 
     "language": "python", 
     "outputs": [
      {
       "latex": [
        "$$\\rho\\left(\\begin{pmatrix}\\frac{1}{2} \\hbar {\\left|\\frac{1}{2},\\frac{1}{2}\\right\\rangle }, & 0.5\\end{pmatrix},\\begin{pmatrix}- \\frac{1}{2} \\hbar {\\left|\\frac{1}{2},- \\frac{1}{2}\\right\\rangle }, & 0.5\\end{pmatrix}\\right)$$"
       ], 
       "output_type": "pyout", 
       "prompt_number": 16, 
       "text": [
        "", 
        " \u239b\u239b\u210f\u22c5\u27581/2,1/2\u27e9     \u239e \u239b-\u210f\u22c5\u27581/2,-1/2\u27e9     \u239e\u239e", 
        "\u03c1\u239c\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, 0.5\u239f,\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, 0.5\u239f\u239f", 
        " \u239d\u239d     2          \u23a0 \u239d      2           \u23a0\u23a0"
       ]
      }
     ], 
     "prompt_number": 16
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "qapply((Jy*d2).doit())"
     ], 
     "language": "python", 
     "outputs": [
      {
       "latex": [
        "$$0.25 \\hbar \\mathbf{\\imath} {\\left|\\frac{1}{2},- \\frac{1}{2}\\right\\rangle } {\\left\\langle \\frac{1}{2},\\frac{1}{2}\\right|} - 0.25 \\hbar \\mathbf{\\imath} {\\left|\\frac{1}{2},\\frac{1}{2}\\right\\rangle } {\\left\\langle \\frac{1}{2},- \\frac{1}{2}\\right|}$$"
       ], 
       "output_type": "pyout", 
       "png": "iVBORw0KGgoAAAANSUhEUgAAAToAAAAgCAYAAACCeLQVAAAABHNCSVQICAgIfAhkiAAACFdJREFU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\naYlORiKRSCQSiUQikUgkEolEIn3MB5i8Sl6iaHkQAAAAAElFTkSuQmCC\n", 
       "prompt_number": 17, 
       "text": [
        "0.25\u22c5\u210f\u22c5\u2148\u22c5\u27581/2,-1/2\u27e9\u22c5\u27e81/2,1/2\u2758 - 0.25\u22c5\u210f\u22c5\u2148\u22c5\u27581/2,1/2\u27e9\u22c5\u27e81/2,-1/2\u2758"
       ]
      }
     ], 
     "prompt_number": 17
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "m = represent(_, format='sympy')"
     ], 
     "language": "python", 
     "outputs": [], 
     "prompt_number": 18
    }, 
    {
     "cell_type": "markdown", 
     "source": [
      "Compute the entropy $-Tr[\\rho ln \\rho]$."
     ]
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "entropy(m)"
     ], 
     "language": "python", 
     "outputs": [
      {
       "latex": [
        "$$0.25 \\hbar \\mathbf{\\imath} \\pi$$"
       ], 
       "output_type": "pyout", 
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEIAAAATCAYAAAAkhtu6AAAABHNCSVQICAgIfAhkiAAAA51JREFU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\n3zajGMUoKvEPo/HTT4ZldQMAAAAASUVORK5CYII=\n", 
       "prompt_number": 19, 
       "text": [
        "0.25\u22c5\u210f\u22c5\u2148\u22c5\u03c0"
       ]
      }
     ], 
     "prompt_number": 19
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "entropy(represent(d2))"
     ], 
     "language": "python", 
     "outputs": [
      {
       "latex": [
        "$$0.693147180540511 + 2.40152712165347 \\times 10^{-13} \\mathbf{\\imath}$$"
       ], 
       "output_type": "pyout", 
       "png": "iVBORw0KGgoAAAANSUhEUgAAAZ8AAAAZCAYAAADufCPlAAAABHNCSVQICAgIfAhkiAAACvtJREFU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\niUQikUgkEolEIl3EfwE+IaAu1CMRKAAAAABJRU5ErkJggg==\n", 
       "prompt_number": 24, 
       "text": [
        "0.693147180540511 + 2.40152712165347e-13\u22c5\u2148"
       ]
      }
     ], 
     "prompt_number": 24
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "entropy(represent(d2, format='numpy'))"
     ], 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 21, 
       "text": [
        "(0.69314718056-0j)"
       ]
      }
     ], 
     "prompt_number": 21
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "entropy(d2)"
     ], 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 22, 
       "text": [
        "(0.69314718056-0j)"
       ]
      }
     ], 
     "prompt_number": 22
    }
   ]
  }
 ]
}