{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#Lab 4 - Measurements\n",
    "First our standard definitions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "from numpy import sqrt,pi,cos,sin,arange,random,exp\n",
    "from qutip import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "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": "code",
   "execution_count": 3,
   "metadata": {},
   "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": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def Delta(state, op):\n",
    "    \"\"\"Calculate std. dev. of an observable in a given state\"\"\"\n",
    "    eO2 = state.dag()*op*op*state\n",
    "    eO = state.dag()*op*state\n",
    "    return sqrt(eO2.data[0,0] - (eO.data[0,0])**2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "###Q: Define the $\\hat{\\mathscr{P}}_{HV}$ operator"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "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\n",
       "Qobj data =\n",
       "[[ 1.  0.]\n",
       " [ 0. -1.]]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Phv = H*H.dag() - V*V.dag()\n",
    "Phv"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "###Q: What is the expectation value $\\langle \\hat{\\mathscr{P}}_{HV}\\rangle$ for state $|\\psi\\rangle = \\frac{1}{\\sqrt{5}}|H\\rangle + \\frac{2}{\\sqrt{5}}|V\\rangle$? Interpret this result given the amplitudes in the state."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "psi = 1/sqrt(5)*H + 2/sqrt(5)*V"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[1], [1]], shape = [1, 1], type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}-0.600\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[1], [1]], shape = [1, 1], type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[-0.6]]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "psi.dag()*Phv*psi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "###Q: What is the variance of $\\mathscr{P}_{HV}$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[1], [1]], shape = [1, 1], type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}1.000\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[1], [1]], shape = [1, 1], type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[ 1.]]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "psi.dag()*Phv*Phv*psi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.64"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1.0 - (-0.6)**2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "###Ex: Use the random function to generate a mock data set for the state $|\\psi\\rangle$.\n",
    "    random.choice([1,-1],size=10,p=[0.2,0.8])\n",
    "gives a list of 10 numbers, either 1 or -1 with the associated probability p:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [],
   "source": [
    "data = random.choice([1, -1],size=20,p=[0.2,0.8])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.69999999999999996"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data.mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.51000000000000001"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data.var()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "###Q: Verify the mean and variance of the mock data set match your QM predictions. How big does the set need to be for you to get ±5% agreement?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [],
   "source": [
    "data = random.choice([1, -1],size=10000,p=[0.2,0.8])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.59840000000000004"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data.mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.64191744000000017"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data.var()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "10,000 does pretty well for getting to the predictions. \"There is no substitute for an adequate sample size.\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "###Q: Answer problems 5.11, 5.12, 5.13, 5.14, 5.17, 5.18, 5.19 from the textbook. These are an opportunity to practice with a new operator $\\hat{\\mathscr{P}}_{C}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 5.11\n",
    "P_45 = P45*P45.dag() - M45*M45.dag()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [],
   "source": [
    "#5.12\n",
    "P_c = L*L.dag() - R*R.dag()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#5.13\n",
    "(Phv*P_45 - P_45*Phv) == 2j * P_c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#5.14\n",
    "(Phv*P_c - P_c*Phv) == -2j * P_45"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [],
   "source": [
    "#5.15\n",
    "P_p45 = P45*P45.dag()\n",
    "P_v = V*V.dag()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "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 & 0.500\\\\-0.500 & 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.   0.5]\n",
       " [-0.5  0. ]]"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P_p45*P_v - P_v*P_p45"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##5.19 - this one is tricky, but is easier with our custom function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "psi = 1/sqrt(3)*H + sqrt(2/3.0)*exp(1j*pi/3.0)*V\n",
    "psi.norm()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.83147941928309788+0j)"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Delta(psi,P_45)*Delta(psi,Phv)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.81649658092772592+0j)"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1/2j*(psi.dag()*(Phv*P_45 - P_45*Phv)*psi).data[0,0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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