phy-521/presentation/.ipynb_checkpoints/Entanglement-checkpoint.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "# EPR Paradox Example\n",
    "\n",
    "# A neutral pi meson (pi0) decays into an electron/positron (e-/e+) \n",
    "# pair.\n",
    "\n",
    "# pi0 --> e+ + e- (electron-positron pair)\n",
    "# pi0 has angular momentum l = s = 0\n",
    "\n",
    "# Align electron and positron detectors in opposite directions.\n",
    "\n",
    "# |            pi0            |\n",
    "# |e- <------       ------> e+|\n",
    "\n",
    "# Where hbar = 1, the measurement of the spin in some direction is \n",
    "# +/-1 with spin state [1 0] (up) or [0 1] (down).\n",
    "\n",
    "# The Pauli exclusion principle with conserved angular momentum 0\n",
    "# dictates this system must be in the singlet state \n",
    "# chi = [1/sqrt(2) (|up+>|down-> - |down+>|up->)].\n",
    "\n",
    "# In this state, if the positron is measured to have spin [1 0], the \n",
    "# electron must have spin [0 1], or vice versa. There is an equal \n",
    "# probability to find either state during the first measurement.\n",
    "\n",
    "# This view is consistent with the realist view. The realist view could \n",
    "# hold that the electron and position had those angular momenta \n",
    "# from creation.\n",
    "\n",
    "# EPR assumes influences cannot propagate faster than the speed of \n",
    "# light. \"Wave function collapse\" is apparently instantaneous, however.\n",
    "\n",
    "\n",
    "import numpy as np\n",
    "import matplotlib\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib.patches as mpatches\n",
    "%matplotlib inline "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7490162da0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# If information about the measurement of the wave function propagated\n",
    "# at a finite speed, the particles could conceivably be measured such\n",
    "# that both are equally likely to hold either spin up or spin down.\n",
    "\n",
    "# What would happen if the measurements were uncorrelated?\n",
    "\n",
    "plt.hist(np.random.randint(0,2,1000),bins=4)\n",
    "plt.hist(np.random.randint(3,5,1000),bins=4)\n",
    "elepatch = mpatches.Patch(color='blue', label='e-')\n",
    "pospatch = mpatches.Patch(color='orange', label='e+')\n",
    "#plt.legend(handles=[elepatch,pospatch])\n",
    "plt.text(0.5,565,\"e-\",size=20)\n",
    "plt.text(3.5,565,\"e+\",size=20)\n",
    "\n",
    "plt.suptitle(\"Uncorrelated Spins\",fontsize=20)\n",
    "plt.ylim([400,600])\n",
    "plt.xlim([-1,5])\n",
    "plt.xticks([0.125,0.85,3.125,3.85],[\"down\",\"up\",\"down\",\"up\"])\n",
    "plt.tick_params(axis='both',labelsize=15)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7490130940>"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f74897bf128>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7490130940>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# How many violations of angular momentum would be measured?\n",
    "\n",
    "# From running several simulations, it's evident a violation in the \n",
    "# conservation of angular momentum would be measured half of the time.\n",
    "# We can conclude that the information that the entangled particles are in\n",
    "# orthogonal spin states is instantaneously agreed once a measurement\n",
    "# is made.\n",
    "\n",
    "violations = 0\n",
    "\n",
    "for trial in range(0,1000):\n",
    "    elespin = np.random.randint(0,2)\n",
    "    posspin = np.random.randint(0,2)\n",
    "    if elespin == 0:\n",
    "        elespin = np.matrix('0 1')\n",
    "    else:\n",
    "        elespin = np.matrix('1 0')\n",
    "    if posspin == 0:\n",
    "        posspin = np.matrix('0 1')\n",
    "    else:\n",
    "        posspin = np.matrix('1 0')\n",
    "    \n",
    "    elespin.transpose()\n",
    "    posspin.transpose()\n",
    "    chi_squared = elespin*np.matrix('1; 0')*posspin*np.matrix('0; 1') - posspin*np.matrix('1; 0')*elespin*np.matrix('0; 1')\n",
    "    \n",
    "    if chi_squared == 0:\n",
    "        violations = violations + 1\n",
    "\n",
    "zeroes = np.zeros(violations,dtype=int)\n",
    "ones = np.full((1000-violations),1,dtype=int)\n",
    "result = np.concatenate((ones,zeroes))\n",
    "\n",
    "plt.ylim([400,600])\n",
    "plt.xlim([-1,2])\n",
    "plt.xticks([0.125,0.85],[\"violation\",\"adherence\"])\n",
    "plt.tick_params(axis='both',labelsize=15)\n",
    "plt.suptitle(\"Conservation Violations\",fontsize=20)\n",
    "plt.hist([result],bins=4)\n",
    "plt.figure()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fb39afb7128>"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Bell's Experiment took this a step further, to rule out locality \n",
    "# completely. Establish the detectors to \"float\" such that they \n",
    "# measure the components of the spins of the electron and positron \n",
    "# along a unit vector a or b, with angles phi_a and phi_b, \n",
    "# respectively. Compute a product P of the spins in units of hbar/2. \n",
    "# This will give +/-1. \n",
    "\n",
    "# /            pi0            \\\n",
    "# /e- <------       ------> e+\\\n",
    "\n",
    "# QM predicts P(a,b) = -a dot b, the expectation value of the product \n",
    "# of the spins.\n",
    "\n",
    "# In 1964, Bell derived the Bell inequality for a local hidden variable\n",
    "# theory:   abs(P(a,b) - P(a,c)) <= 1 + P(b,c)\n",
    "\n",
    "# For any local hidden variable theory, the Bell inequality must hold. \n",
    "# It introduces a third unit vector c, which is any other unit vector \n",
    "# than a or b.\n",
    "\n",
    "# Does the quantum mechanical prediction violate the Bell inequality?\n",
    "# Testing several detector configurations in a plane, systematically \n",
    "# from 0 to pi, we determine whether the QM prediction is consistent \n",
    "# with a local hidden variable theory.\n",
    "\n",
    "violations = 0\n",
    "trials = 0\n",
    "for step_a in range(0,6):\n",
    "    for step_b in range (0,6):\n",
    "        phi_a = step_a/6*np.pi\n",
    "        phi_b = step_b/6*np.pi\n",
    "        phi_c = phi_a - 0.5*phi_b\n",
    "        P_ab = -1 * np.cos(phi_a - phi_b)\n",
    "        P_ac = -1 * np.cos(phi_a - phi_c)\n",
    "        P_bc = -1 * np.cos(phi_b - phi_c)\n",
    "        \n",
    "        bell_lhs = np.abs(P_ab - P_ac)\n",
    "        bell_rhs = 1 + P_bc\n",
    "        \n",
    "        if bell_lhs > bell_rhs:\n",
    "            violations = violations + 1\n",
    "            \n",
    "        trials = trials + 1\n",
    "                \n",
    "zeroes = np.zeros(violations,dtype=int)\n",
    "ones = np.full((trials-violations),1,dtype=int)\n",
    "result = np.concatenate((ones,zeroes))\n",
    "\n",
    "plt.ylim([trials/2-10,trials/2+10])\n",
    "plt.xlim([-1,2])\n",
    "plt.xticks([0.125,0.85],[\"violation\",\"adherence\"])\n",
    "plt.tick_params(axis='both',labelsize=15)\n",
    "plt.suptitle(\"Hidden Locality Violations\",fontsize=20)\n",
    "plt.hist([result],bins=4)\n",
    "plt.figure()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fb39beac240>"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# It is seen that the QM prediction disagrees with a local hidden \n",
    "# variable theory in a significant number of configurations.\n",
    "\n",
    "# On average, for random orientations between 0 and pi, how often? \n",
    "# After running several trials, it appears to be about half of the, \n",
    "# which is what one would expect from the quantum mechanical\n",
    "# prediction.\n",
    "\n",
    "violations = 0\n",
    "trials = 0\n",
    "for rand_a in range(0,10):\n",
    "    for rand_b in range (0,10):\n",
    "        phi_a = np.random.rand(1)*np.pi\n",
    "        phi_b = np.random.rand(1)*np.pi\n",
    "        phi_c = phi_a - 0.5*phi_b\n",
    "        P_ab = -1 * np.cos(phi_a - phi_b)\n",
    "        P_ac = -1 * np.cos(phi_a - phi_c)\n",
    "        P_bc = -1 * np.cos(phi_b - phi_c)\n",
    "        \n",
    "        bell_lhs = np.abs(P_ab - P_ac)\n",
    "        bell_rhs = 1 + P_bc\n",
    "        \n",
    "        if bell_lhs > bell_rhs:\n",
    "            violations = violations + 1\n",
    "            \n",
    "        trials = trials + 1\n",
    "\n",
    "zeroes = np.zeros(violations,dtype=int)\n",
    "ones = np.full((trials-violations),1,dtype=int)\n",
    "result = np.concatenate((ones,zeroes))\n",
    "\n",
    "plt.ylim([trials/2-10,trials/2+10])\n",
    "plt.xlim([-1,2])\n",
    "plt.xticks([0.125,0.85],[\"violation\",\"adherence\"])\n",
    "plt.tick_params(axis='both',labelsize=15)\n",
    "plt.suptitle(\"Hidden Locality Violations\",fontsize=20)\n",
    "plt.hist([result],bins=4)\n",
    "plt.figure()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "# The prediction (Bell's inequality) made by assuming a local hidden\n",
    "# variable is violated by a significant number of the possibile \n",
    "# bborientations. This simulation cannot determine which theory is\n",
    "# correct, but the QM prediction has been confirmed through experiment.\n",
    "# No hidden local variable holds actionable information about the \n",
    "# state. Entangled states retain their entanglement in a non-local \n",
    "# nature."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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