Lab 3 - Operators - Solutions.ipynb

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 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Lab 3: Operators\n",
    "An overview of operator properties"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "from numpy import sqrt,cos,sin,pi,arange\n",
    "from qutip import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "H = Qobj([[1],[0]])\n",
    "V = Qobj([[0],[1]])\n",
    "P45 = Qobj([[1/sqrt(2)],[1/sqrt(2)]])\n",
    "M45 = Qobj([[1/sqrt(2)],[-1/sqrt(2)]])\n",
    "R = Qobj([[1/sqrt(2)],[-1j/sqrt(2)]])\n",
    "L = Qobj([[1/sqrt(2)],[1j/sqrt(2)]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 1: the outer product and the projection operator"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We already have the $|H\\rangle$ state represented as a vector in the HV basis, so the $\\hat{P}_H$ operator is the outer product $|H\\rangle\\langle H|$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
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   "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 & 0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[ 1.  0.]\n",
       " [ 0.  0.]]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ph = H*H.dag()\n",
    "Ph"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Same with the $\\hat{P}_V$ operator:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "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 & 0.0\\\\0.0 & 1.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[ 0.  0.]\n",
       " [ 0.  1.]]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Pv = V*V.dag()\n",
    "Pv"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1) Verify Eq. 4.38 for the HV basis states. Repeat for the 45, and ± basis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
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   "outputs": [],
   "source": []
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2) Represent the $\\hat{R}_p(\\theta)$ operator in the HV basis and verify your representation by operating on $|H\\rangle$ and $|V\\rangle$ states."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def Rp(theta):\n",
    "    return Qobj([[cos(theta),-sin(theta)],[sin(theta),cos(theta)]]).tidyup()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isherm = False\\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 = False\n",
       "Qobj data =\n",
       "[[ 0. -1.]\n",
       " [ 1.  0.]]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Rp(pi/2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "V == Rp(pi/2)*H"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3) Using your $\\hat{R}_p(\\theta)$ operator, verify the operator properties described in Sections 4.1 and 4.2. Specifically, verify Eqns. 4.5, 4.6, 4.7, 4.16, 4.18, 4.22, and 4.27"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
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   "outputs": [],
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  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example: similarity transform\n",
    "The following defines a function that creates a similarity transform matrix. It takes the two old basis vectors and the two new basis vectors."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def sim_transform(o_basis1, o_basis2, n_basis1, n_basis2):\n",
    "    a = n_basis1.dag()*o_basis1\n",
    "    b = n_basis1.dag()*o_basis2\n",
    "    c = n_basis2.dag()*o_basis1\n",
    "    d = n_basis2.dag()*o_basis2\n",
    "    return Qobj([[a.data[0,0],b.data[0,0]],[c.data[0,0],d.data[0,0]]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}0.707 & 0.707\\\\0.707 & -0.707\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[ 0.70710678  0.70710678]\n",
       " [ 0.70710678 -0.70710678]]"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Shv45 = sim_transform(H,V,P45,M45)  # as found in Example 4.A.1, Eq. 4.A.10.\n",
    "Shv45"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [1]], shape = [2, 1], type = ket\\begin{equation*}\\left(\\begin{array}{*{11}c}(0.500+0.500j)\\\\(0.500-0.500j)\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [1]], shape = [2, 1], type = ket\n",
       "Qobj data =\n",
       "[[ 0.5+0.5j]\n",
       " [ 0.5-0.5j]]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Shv45 * L  # compare to Eq. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4) Represent $\\hat{P}_H$ in the ±45 basis.\n",
    "Check your answer against Eqns. 4.A.17 and 4.72"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "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 & 0.500\\\\0.500 & 0.500\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[ 0.5  0.5]\n",
       " [ 0.5  0.5]]"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Shv45*Ph*Shv45.dag()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": []
  }
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