Chapter 8 - Two Photon States.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Chapter 8 - Two-photon States"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from qutip import *\n",
    "from numpy import sqrt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "H = basis(2,0)\n",
    "V = basis(2,1)\n",
    "p45 = 1/sqrt(2)*(H + V)\n",
    "m45 = 1/sqrt(2)*(H - V)\n",
    "R = 1/sqrt(2)*(H - 1j*V)\n",
    "L = 1/sqrt(2)*(H + 1j*V)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "hh = tensor(H,H)\n",
    "hv = tensor(H,V)\n",
    "vh = tensor(V,H)\n",
    "vv = tensor(V,V)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2, 2], [1, 1]], shape = (4, 1), type = ket\\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], [1, 1]], shape = (4, 1), type = ket\n",
       "Qobj data =\n",
       "[[ 0.]\n",
       " [ 0.]\n",
       " [ 0.]\n",
       " [ 1.]]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vv"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\n",
       "Qobj data =\n",
       "[[ 0.]]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Show these are all orthogonal:\n",
    "hh.dag()*hv"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\n",
       "Qobj data =\n",
       "[[ 0.]]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vh.dag()*hv"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "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": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qeye(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True\n",
      "True\n",
      "True\n"
     ]
    }
   ],
   "source": [
    "# Example 8.1\n",
    "Phv = H*H.dag() - V*V.dag() # polarization measurement\n",
    "ps_hv = tensor(Phv,qeye(2))\n",
    "pi_hv = tensor(qeye(2),Phv)\n",
    "psi_hv = tensor(Phv,Phv)\n",
    "\n",
    "print(ps_hv*tensor(V,p45) == -tensor(V,p45))\n",
    "\n",
    "print(pi_hv*tensor(V,p45) == tensor(V,m45))\n",
    "\n",
    "print(psi_hv*tensor(V,p45) == -tensor(V,m45))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.2499999999999999"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Example 8.2\n",
    "Pvh = tensor(V,H)*tensor(V,H).dag()\n",
    "expect(Pvh,tensor(R,p45))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.4999999999999998"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Example 8.3\n",
    "Phi = tensor(qeye(2),H)*tensor(qeye(2),H).dag()\n",
    "expect(Phi,tensor(R,p45))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Example 8.4\n",
    "state = tensor(R,p45)\n",
    "expect(Pvh,state) / expect(Phi,state)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Entangled states"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phip = 1/sqrt(2)*(tensor(H,H) + tensor(V,V))\n",
    "Phs = tensor(H,qeye(2))*tensor(H,qeye(2)).dag() \n",
    "Phi = tensor(qeye(2),H)*tensor(qeye(2),H).dag()\n",
    "Phshi = tensor(H,H)*tensor(H,H).dag()\n",
    "\n",
    "expect(Phshi,phip)/ expect(Phi,phip)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [],
   "source": [
    "P_45s45i = tensor(p45,p45)*tensor(p45,p45).dag()\n",
    "P_45i = tensor(qeye(2),p45)*tensor(qeye(2),p45).dag()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9999999999999997"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expect(P_45s45i,phip)/expect(P_45i,phip)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Delete me, this is scratch work from review session"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2, 2], [1, 1]], shape = (4, 1), type = ket\\begin{equation*}\\left(\\begin{array}{*{11}c}0.671\\\\0.224j\\\\-0.224j\\\\0.671\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2, 2], [1, 1]], shape = (4, 1), type = ket\n",
       "Qobj data =\n",
       "[[0.67082039+0.j       ]\n",
       " [0.        +0.2236068j]\n",
       " [0.        -0.2236068j]\n",
       " [0.67082039+0.j       ]]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "psi = 1/sqrt(5)*tensor(L,R) + 2/sqrt(5)*tensor(R,L)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\\begin{equation*}\\left(\\begin{array}{*{11}c}0.450\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\n",
       "Qobj data =\n",
       "[[0.45]]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "psi.dag()*tensor(H,H)*tensor(H,H).dag()*psi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.45"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "9/20"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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