{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Chapter 4 in-class problems\n",
    "## Using what you learned in Lab, answer questions 4.7, 4.8, 4.10, 4.11, 4.12, and 4.13"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from numpy import sqrt, pi, cos, sin\n",
    "from qutip import *"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Remember, these states are represented in the HV basis:"
   ]
  },
  {
   "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": "markdown",
   "metadata": {},
   "source": [
    "The sim_transform function creates the matrix $\\bar{\\mathbf{S}}$ that can convert from one basis to another. As an example, it will create the tranform matrix to convert from HV to ±45 if you run:\n",
    "\n",
    "    Shv45 = sim_transform(H,V,P45,M45)    #  creates the matrix Shv45\n",
    "    \n",
    "Then you can convert a ket from HV to ±45 by applying the Shv45 matrix:\n",
    "\n",
    "    Shv45*H    #  will convert H from the HV basis to the ±45 basis\n",
    "    \n",
    "To convert operators, you have to sandwich the operator between $\\bar{\\mathbf{S}}$ and $\\bar{\\mathbf{S}}^\\dagger$:\n",
    "\n",
    "    Shv45*Ph*Shv45.dag()     #  converts Ph from HV basis to the ±45 basis."
   ]
  },
  {
   "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": [
    "ShvLR = sim_transform(H,V,L,R)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\\begin{equation*}\\left(\\begin{array}{*{11}c}0.707\\\\0.707\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\n",
       "Qobj data =\n",
       "[[ 0.70710678]\n",
       " [ 0.70710678]]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ShvLR*H"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\\begin{equation*}\\left(\\begin{array}{*{11}c}-0.707j\\\\0.707j\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\n",
       "Qobj data =\n",
       "[[ 0.-0.70710678j]\n",
       " [ 0.+0.70710678j]]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ShvLR*V"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4.11: Express $\\hat{R}_p(\\theta)$ in ±45 basis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def Rp(theta):\n",
    "    return Qobj([[cos(theta),-sin(theta)],[sin(theta),cos(theta)]]).tidyup()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = False\\begin{equation*}\\left(\\begin{array}{*{11}c}0.267 & -0.964\\\\0.964 & 0.267\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = False\n",
       "Qobj data =\n",
       "[[ 0.26749883 -0.96355819]\n",
       " [ 0.96355819  0.26749883]]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Rp(1.3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "Shv45 = sim_transform(H,V,P45,M45)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4.12: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "Rp45 = Shv45*Rp(pi/4)*Shv45.dag()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Rp45*Shv45*P45 == Shv45*V   # convert P45 to the ±45 basis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\\begin{equation*}\\left(\\begin{array}{*{11}c}0.707\\\\-0.707\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\n",
       "Qobj data =\n",
       "[[ 0.70710678]\n",
       " [-0.70710678]]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Rp45* Qobj([[1],[0]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "ShvLR = sim_transform(H,V,L,R)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = False\\begin{equation*}\\left(\\begin{array}{*{11}c}(0.707-0.707j) & 0.0\\\\0.0 & (0.707+0.707j)\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = False\n",
       "Qobj data =\n",
       "[[ 0.70710678-0.70710678j  0.00000000+0.j        ]\n",
       " [ 0.00000000+0.j          0.70710678+0.70710678j]]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ShvLR*Rp(pi/4)*ShvLR.dag()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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