{ "metadata": { "name": "density" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Density operator and matrix " ] }, { "cell_type": "code", "collapsed": true, "input": [ "from sympy import *\n", "from sympy.physics.quantum import *\n", "from sympy.physics.quantum.density import *\n", "from sympy.physics.quantum.spin import (\n", " Jx, Jy, Jz, Jplus, Jminus, J2,\n", " JxBra, JyBra, JzBra,\n", " JxKet, JyKet, JzKet,\n", ")\n", "from IPython.core.display import display_pretty" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 59 }, { "cell_type": "code", "collapsed": true, "input": [ "%load_ext sympyprinting" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 60 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Create n density matrix using symbolic states" ] }, { "cell_type": "code", "collapsed": true, "input": [ "psi = Ket('psi')\n", "phi = Ket('phi')" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 61 }, { "cell_type": "code", "collapsed": false, "input": [ "d = Density((psi,0.5),(phi,0.5)); d" ], "language": "python", "metadata": {}, "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": 62, "text": [ "\u03c1((\u2758\u03c8\u27e9, 0.5),(\u2758\u03c6\u27e9, 0.5))" ] } ], "prompt_number": 62 }, { "cell_type": "code", "collapsed": false, "input": [ "display_pretty(d)" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "display_data", "text": [ "\u03c1((\u2758\u03c8\u27e9, 0.5),(\u2758\u03c6\u27e9, 0.5))" ] } ], "prompt_number": 63 }, { "cell_type": "code", "collapsed": false, "input": [ "d.states()" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\begin{pmatrix}{\\left|\\psi\\right\\rangle }, & {\\left|\\phi\\right\\rangle }\\end{pmatrix}$$" ], "output_type": "pyout", "prompt_number": 64, "text": [ "(\u2758\u03c8\u27e9, \u2758\u03c6\u27e9)" ] } ], "prompt_number": 64 }, { "cell_type": "code", "collapsed": false, "input": [ "d.probs()" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\begin{pmatrix}0.5, & 0.5\\end{pmatrix}$$" ], "output_type": "pyout", "prompt_number": 65, "text": [ "(0.5, 0.5)" ] } ], "prompt_number": 65 }, { "cell_type": "code", "collapsed": false, "input": [ "d.doit()" ], "language": "python", "metadata": {}, "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": "iVBORw0KGgoAAAANSUhEUgAAAKoAAAAXCAYAAACbItQpAAAABHNCSVQICAgIfAhkiAAABZhJREFU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\n+vgfqAPSYqWMIZsAAAAASUVORK5CYII=\n", "prompt_number": 66, "text": [ "0.5\u22c5\u2758\u03c6\u27e9\u27e8\u03c6\u2758 + 0.5\u22c5\u2758\u03c8\u27e9\u27e8\u03c8\u2758" ] } ], "prompt_number": 66 }, { "cell_type": "code", "collapsed": false, "input": [ "Dagger(d)" ], "language": "python", "metadata": {}, "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": 67, "text": [ "\u03c1((\u2758\u03c8\u27e9, 0.5),(\u2758\u03c6\u27e9, 0.5))" ] } ], "prompt_number": 67 }, { "cell_type": "code", "collapsed": true, "input": [ "A = Operator('A')" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 68 }, { "cell_type": "code", "collapsed": false, "input": [ "d.apply_op(A)" ], "language": "python", "metadata": {}, "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": 69, "text": [ "\u03c1((A\u22c5\u2758\u03c8\u27e9, 0.5),(A\u22c5\u2758\u03c6\u27e9, 0.5))" ] } ], "prompt_number": 69 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now create a density matrix using spin states" ] }, { "cell_type": "code", "collapsed": true, "input": [ "up = JzKet(S(1)/2,S(1)/2)\n", "down = JzKet(S(1)/2,-S(1)/2)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 70 }, { "cell_type": "code", "collapsed": false, "input": [ "d2 = Density((up,0.5),(down,0.5)); d2" ], "language": "python", "metadata": {}, "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": 71, "text": [ "\u03c1((\u27581/2,1/2\u27e9, 0.5),(\u27581/2,-1/2\u27e9, 0.5))" ] } ], "prompt_number": 71 }, { "cell_type": "code", "collapsed": false, "input": [ "represent(d2)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\left[\\begin{smallmatrix}0.5 & 0\\\\0 & 0.5\\end{smallmatrix}\\right]$$" ], "output_type": "pyout", "prompt_number": 72, "text": [ "\n", "\u23a10.5 0 \u23a4\n", "\u23a2 \u23a5\n", "\u23a3 0 0.5\u23a6" ] } ], "prompt_number": 72 }, { "cell_type": "code", "collapsed": false, "input": [ "d2.apply_op(Jz)" ], "language": "python", "metadata": {}, "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": 73, "text": [ "\n", "\u03c1\u239b\u239bJ \u22c5\u27581/2,1/2\u27e9, 0.5\u239e,\u239bJ \u22c5\u27581/2,-1/2\u27e9, 0.5\u239e\u239e\n", " \u239d\u239d z \u23a0 \u239d z \u23a0\u23a0" ] } ], "prompt_number": 73 }, { "cell_type": "code", "collapsed": false, "input": [ "qapply(_)" ], "language": "python", "metadata": {}, "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": 74, "text": [ "\n", " \u239b\u239b\u210f\u22c5\u27581/2,1/2\u27e9 \u239e \u239b-\u210f\u22c5\u27581/2,-1/2\u27e9 \u239e\u239e\n", "\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\n", " \u239d\u239d 2 \u23a0 \u239d 2 \u23a0\u23a0" ] } ], "prompt_number": 74 }, { "cell_type": "code", "collapsed": false, "input": [ "qapply((Jy*d2).doit())" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$0.5 J_y {\\left|\\frac{1}{2},- \\frac{1}{2}\\right\\rangle }{\\left\\langle \\frac{1}{2},- \\frac{1}{2}\\right|} + 0.5 J_y {\\left|\\frac{1}{2},\\frac{1}{2}\\right\\rangle }{\\left\\langle \\frac{1}{2},\\frac{1}{2}\\right|}$$" ], "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAZ0AAAAlCAYAAACZMTQOAAAABHNCSVQICAgIfAhkiAAACElJREFU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\nURRFUZQa8j/bVKmwXxjmUgAAAABJRU5ErkJggg==\n", "prompt_number": 75, "text": [ "\n", "0.5\u22c5J \u22c5\u27581/2,-1/2\u27e9\u27e81/2,-1/2\u2758 + 0.5\u22c5J \u22c5\u27581/2,1/2\u27e9\u27e81/2,1/2\u2758\n", " y y " ] } ], "prompt_number": 75 }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Evaluate entropy of the density matrices" ] }, { "cell_type": "code", "collapsed": false, "input": [ "entropy(d2)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\frac{1}{2} \\log{\\left (2 \\right )}$$" ], "output_type": "pyout", "png": 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"prompt_number": 76, "text": [ "\n", "log(2)\n", "\u2500\u2500\u2500\u2500\u2500\u2500\n", " 2 " ] } ], "prompt_number": 76 }, { "cell_type": "code", "collapsed": false, "input": [ "entropy(represent(d2))" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\frac{1}{2} \\log{\\left (2 \\right )}$$" ], "output_type": "pyout", "png": 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"prompt_number": 77, "text": [ "\n", "log(2)\n", "\u2500\u2500\u2500\u2500\u2500\u2500\n", " 2 " ] } ], "prompt_number": 77 }, { "cell_type": "code", "collapsed": false, "input": [ "entropy(represent(d2,format=\"numpy\"))" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "pyout", "prompt_number": 78, "text": [ "(0.69314718056-0j)" ] } ], "prompt_number": 78 }, { "cell_type": "code", "collapsed": false, "input": [ "entropy(represent(d2,format=\"scipy.sparse\"))" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "pyout", "prompt_number": 79, "text": [ "(0.69314718056-0j)" ] } ], "prompt_number": 79 }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Density operators with Tensor Products as args" ] }, { "cell_type": "code", "collapsed": false, "input": [ "from sympy.core.trace import Tr\n", "\n", "A, B, C, D = symbols('A B C D',commutative=False)\n", "\n", "t1 = TensorProduct(A,B,C)\n", "\n", "d = Density([t1, 1.0])\n", "d.doit()\n", "\n", "t2 = TensorProduct(A,B)\n", "t3 = TensorProduct(C,D)\n", "\n", "d = Density([t2, 0.5], [t3, 0.5])\n", "d.doit() " ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$0.5 \\left({A A^{\\dagger}}\\right)\\otimes \\left({B B^{\\dagger}}\\right) + 0.5 \\left({C C^{\\dagger}}\\right)\\otimes \\left({D D^{\\dagger}}\\right)$$" ], "output_type": "pyout", "png": 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"prompt_number": 80, "text": [ "\n", " \u239b \u2020\u239e \u239b \u2020\u239e \u239b \u2020\u239e \u239b \u2020\u239e\n", "0.5\u22c5\u239dA\u22c5A \u23a0\u2a02 \u239dB\u22c5B \u23a0 + 0.5\u22c5\u239dC\u22c5C \u23a0\u2a02 \u239dD\u22c5D \u23a0" ] } ], "prompt_number": 80 }, { "cell_type": "code", "collapsed": false, "input": [ "#mixed states\n", "d = Density([t2+t3, 1.0])\n", "d.doit() " ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$1.0 \\left({A A^{\\dagger}}\\right)\\otimes \\left({B B^{\\dagger}}\\right) + 1.0 \\left({A C^{\\dagger}}\\right)\\otimes \\left({B D^{\\dagger}}\\right) + 1.0 \\left({C A^{\\dagger}}\\right)\\otimes \\left({D B^{\\dagger}}\\right) + 1.0 \\left({C C^{\\dagger}}\\right)\\otimes \\left({D D^{\\dagger}}\\right)$$" ], "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAqYAAAAaCAYAAABl7J64AAAABHNCSVQICAgIfAhkiAAAC35JREFU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\n", "prompt_number": 81, "text": [ "\n", " \u239b \u2020\u239e \u239b \u2020\u239e \u239b \u2020\u239e \u239b \u2020\u239e \u239b \u2020\u239e \u239b \u2020\u239e \u239b \u2020\u239e \u239b \n", "1.0\u22c5\u239dA\u22c5A \u23a0\u2a02 \u239dB\u22c5B \u23a0 + 1.0\u22c5\u239dA\u22c5C \u23a0\u2a02 \u239dB\u22c5D \u23a0 + 1.0\u22c5\u239dC\u22c5A \u23a0\u2a02 \u239dD\u22c5B \u23a0 + 1.0\u22c5\u239dC\u22c5C \u23a0\u2a02 \u239dD\u22c5\n", "\n", " \u2020\u239e\n", "D \u23a0" ] } ], "prompt_number": 81 }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Trace operators on Density Operators with Spin States" ] }, { "cell_type": "code", "collapsed": false, "input": [ "from sympy.physics.quantum.density import Density\n", "from sympy.physics.quantum.spin import (\n", " Jx, Jy, Jz, Jplus, Jminus, J2,\n", " JxBra, JyBra, JzBra,\n", " JxKet, JyKet, JzKet,\n", ")\n", "from sympy.core.trace import Tr\n", "\n", "d = Density([JzKet(1,1),0.5],[JzKet(1,-1),0.5]);\n", "t = Tr(d); \n", "\n", "display_pretty(t)\n", "print t.doit()" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "display_data", "text": [ "Tr(\u03c1((\u27581,1\u27e9, 0.5),(\u27581,-1\u27e9, 0.5)))" ] }, { "output_type": "stream", "stream": "stdout", "text": [ "1.00000000000000" ] } ], "prompt_number": 82 }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Partial Trace on Density Operators with Mixed State" ] }, { "cell_type": "code", "collapsed": false, "input": [ "#Partial trace on mixed state\n", "from sympy.core.trace import Tr\n", "\n", "A, B, C, D = symbols('A B C D',commutative=False)\n", "\n", "t1 = TensorProduct(A,B,C)\n", "\n", "d = Density([t1, 1.0])\n", "d.doit()\n", "\n", "t2 = TensorProduct(A,B)\n", "t3 = TensorProduct(C,D)\n", "\n", "d = Density([t2, 0.5], [t3, 0.5])\n", "\n", "\n", "tr = Tr(d,[1])\n", "tr.doit()" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$0.5 A A^{\\dagger} \\mbox{Tr}\\left(B B^{\\dagger}\\right) + 0.5 C C^{\\dagger} \\mbox{Tr}\\left(D D^{\\dagger}\\right)$$" ], "output_type": "pyout", "prompt_number": 83, "text": [ "\n", " \u2020 \u2020 \u2020 \u2020 \n", "0.5\u22c5A\u22c5A \u22c5Tr(B\u22c5B ) + 0.5\u22c5C\u22c5C \u22c5Tr(D\u22c5D )" ] } ], "prompt_number": 83 }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Partial Trace on Density Operators with Spin states" ] }, { "cell_type": "code", "collapsed": false, "input": [ "from sympy.physics.quantum.density import Density\n", "from sympy.physics.quantum.spin import (\n", " Jx, Jy, Jz, Jplus, Jminus, J2,\n", " JxBra, JyBra, JzBra,\n", " JxKet, JyKet, JzKet,\n", ")\n", "from sympy.core.trace import Tr\n", "\n", "tp1 = TensorProduct(JzKet(1,1), JzKet(1,-1))\n", "\n", "#trace out 0 index\n", "d = Density([tp1,1]);\n", "t = Tr(d,[0]); \n", "\n", "display_pretty(t)\n", "display_pretty(t.doit())\n", "\n", "#trace out 1 index\n", "t = Tr(d,[1])\n", "display_pretty(t)\n", "t.doit()" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "display_data", "text": [ "Tr((\u27581,1\u27e9\u2a02 \u27581,-1\u27e9, 1))" ] }, { "output_type": "display_data", "text": [ "\u27581,-1\u27e9\u27e81,-1\u2758" ] }, { "output_type": "display_data", "text": [ "Tr((\u27581,1\u27e9\u2a02 \u27581,-1\u27e9, 1))" ] }, { "latex": [ "$${\\left|1,1\\right\\rangle }{\\left\\langle 1,1\\right|}$$" ], "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAEwAAAAXCAYAAACh3qkfAAAABHNCSVQICAgIfAhkiAAAAedJREFU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\n", "prompt_number": 84, "text": [ "\u27581,1\u27e9\u27e81,1\u2758" ] } ], "prompt_number": 84 }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Examples of qapply() on Density matrices with spin states" ] }, { "cell_type": "code", "collapsed": false, "input": [ "psi = Ket('psi')\n", "phi = Ket('phi')\n", "\n", "u = UnitaryOperator()\n", "d = Density((psi,0.5),(phi,0.5)); d\n", "\n", "display_pretty(qapply(u*d))\n", " \n", "# another example\n", "up = JzKet(S(1)/2, S(1)/2)\n", "down = JzKet(S(1)/2, -S(1)/2)\n", "d = Density((up,0.5),(down,0.5))\n", "\n", "uMat = Matrix([[0,1],[1,0]])\n", "display_pretty(qapply(uMat*d))\n", "\n", "\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "display_data", "text": [ "\u03c1((O\u22c5\u2758\u03c8\u27e9, 0.5),(O\u22c5\u2758\u03c6\u27e9, 0.5))" ] }, { "output_type": "display_data", "text": [ "\n", "\u23a1 0 \u03c1((\u27581/2,1/2\u27e9, 0.5),(\u27581/2,-1/2\u27e9, 0.5))\u23a4\n", "\u23a2 \u23a5\n", "\u23a3\u03c1((\u27581/2,1/2\u27e9, 0.5),(\u27581/2,-1/2\u27e9, 0.5)) 0 \u23a6" ] } ], "prompt_number": 85 }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example of qapply() on Density Matrices with qubits" ] }, { "cell_type": "code", "collapsed": false, "input": [ "from sympy.physics.quantum.gate import UGate\n", "from sympy.physics.quantum.qubit import Qubit\n", "\n", "uMat = UGate((0,), Matrix([[0,1],[1,0]]))\n", "d = Density([Qubit('0'),0.5],[Qubit('1'), 0.5])\n", "\n", "display_pretty(d)\n", "\n", "#after applying Not gate\n", "display_pretty(qapply(uMat*d) )" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "display_data", "text": [ "\u03c1((\u27580\u27e9, 0.5),(\u27581\u27e9, 0.5))" ] }, { "output_type": "display_data", "text": [ "\u03c1((\u27581\u27e9, 0.5),(\u27580\u27e9, 0.5))" ] } ], "prompt_number": 90 }, { "cell_type": "code", "collapsed": true, "input": [], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 86 } ], "metadata": {} } ] }