phy-521/presentation/Entanglement.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "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": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f22020bd208>"
      ]
     },
     "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": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f22020419e8>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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P3ATcTszw+U67jevu7v77z52dnXR2dra7KUlaIfX09NDT09NU3eHy7o18kMi/bwQ81KDOXcRA78fS68eB04CvlOrNI64ITi6V9/f397fZPEnKU0dHBzSI7e3OgW8nEs8EtiiVbUCkhsq5fUnSKGs34H8AeIrGvfutiYHY2wpllwG7AKsVyqYTg7bXt9kOSVKTmknpXEgMyN5N5PynAwcAnwK+B+wG7A/8hkjbbEEM8i4EtmVgGuYU4B4i1XMCsDGRxvkWcEzFfk3pSFKLhkrpNDNoex9wIJF+6SAC/4eJO2shplauTdxlOwV4mujNH0n9nPu5wM5EHv9iYnbOKcSNV5KkMdbuoO2SYA9fklo0FoO2kqTljAFfkjJhwJekTBjwJSkTBnxJyoQBX5IyYcCXpEwY8CUpEwZ8ScqEAV+SMmHAl6RMGPAlKRMGfEnKhAFfkjJhwJekTBjwJSkTBnxJyoQBX5IyYcCXpEwY8CUpEwZ8ScqEAV+SMmHAl6RMGPAlKRMGfEnKhAFfkjJhwJekTBjwJSkTBnxJyoQBX5IyYcCXpEwY8CUpEwZ8ScrE+KXdAK14Ojt3Y8aMW5bY/lZffTV6e+9jwoQJS2yf0vLIgK9RN2vWAyxYcAmw6RLZ30svTePFF1804EvDMOBrjEwB1lwie+roMDMpNcO/FEnKhAFfkjJhwJekTBjwJSkTBnxJyoQBX5Iy0UzA7wJeqfg6qFTvSOBhYAFwPbBNxba2BK4B5gOPAsc12QZJ0gi1Mg9/J2Bh4fWDhZ+PAI4CvgDMBD4PXA1sDTye6kxNZXcBewKbACcTAf/oNtouSWpBKwF/BtF7L5sAHA58Hfh+KrsF6AX+nYFgfgiwCrA3MI/o6U8CuoETgedbarkkqSWtpFM6GpRvD6wOnF8oWwBcDOxaKNsVuIII9jXnAasCO7bQDklSG1oJ+PcDLxEpm2L+fnPgZWBWqf7MtKxms1RWNJs4OWzWQjskSW1oJqUzh8jP3wqMA/YDTgcmAqcSufl5QH9pvb5UZzywONWbW7H9vrRMkjSGmgn4V6avmiuIvP2XgG+PRaMkSaOv3adl/gLYB3gd0UNfjcjxF3v5U4l0zeL0ug+YXLGtqWnZIN3d3X//ubOzk87OzjabK0krpp6eHnp6epqq22ggdjgfIAZpNwI2JqZbbkZ9Hv9HwJuAf0qvryfm3u9fqLMB8BCwB3BpaR/9/f3lLJGWB+uttwVz5lwEbLFE9rfyypN54onZTJ5c1Z+Q8tLR0QENYnu7Nz19AHiKCNY3Ac8RPf6aiUQQv6xQdhmwC3E1UDOdgRu1JEljqJmUzoXAzcDdqf50Irh/Ki1/ATiemG/fB9wHfC4t+25hO6cDnwYuAk4grgyOBU6hfqqmJGkMNBPw7wMOJNIvHUTg/zDw80Kd44mrhSOIf3M0A3g38GShzlxgZ+A0Yo5+HxHsu0fyBiRJzWk3h78kmMNfTpnDl5aescjhS5KWMwZ8ScqEAV+SMmHAl6RMGPAlKRMGfEnKhAFfkjJhwJekTBjwJSkTBnxJyoQBX1rGTZq0Bh0dHUvsa9KkNZb2W9YYafcfoEhaQp5/vo/B/0F0LPe3LD9iSyNhD1+SMmHAl6RMGPAlKRMGfEnKhAFfkjJhwJekTBjwJSkTBnxJyoQBX5IyYcCXpEwY8CUpEwZ8ScqEAV+SMmHAl6RMGPAlKRMGfEnKhAFfkjJhwJekTBjwJSkTBnxJyoQBX5IyYcCXpEwY8CUpEwZ8ScqEAV+SMmHAl6RMGPAlKRMGfEnKhAFfkjJhwJekTBjwJSkTrQb89YB5wCvAxEJ5byorfs2pWH9L4BpgPvAocFwbbZAktWF8i/VPAp4HVi2V9wM/B75bKFtUqjMVuBq4C9gT2AQ4mQj4R7fYDklSi1oJ+DsAuwBfJwJ/2WPArUOsfwiwCrA3cZVwDTAJ6AZOJE4kkqQx0mw6ZRzRez8OeLpBnY5htrErcAUR7GvOI64WdmyyHZKkNjUb8A8BVgK+N0SdTwAvAnOBC4ANS8s3A2aWymYDC9IySdIYaialsybwZeAA4OUGdX4N3Aw8QgzMHgvcCLwReC7VmUqcDMr60jJJ0hhqJuB/jQjmlw9R57OFn38P3ATcDnQB32m3cZKk0TNcwN8K+BgxYDslldWmY04hZucsrFjvbuA+4M2Fsj5gckXdqWnZIN3d3X//ubOzk87OzmGaK0l56enpoaenp6m6ww20vg+4aIjlPwQOarDsLmAGccIAuJ6Ye79/oc4GwEPAHsClpfX7+/v7h2melkXrrbcFc+ZcBGyxRPa38sqTeeKJ2UyeXNWfWP51dHQQfasltkf821t+xeelOrYP18O/Eegsle0K/Ef6/kCD9bYmBmJPL5RdBhwGrMbATJ3pxKDt9cO0Q5I0QsMF/KeBG0plr0/fbySC9W5Er/03wONEt+4ooud+TmG904FPE1cMJwAbE4O7p1A/VVOSNAZavdO2pni9NxtYm5inP4U4SVwGHEl9IJ8L7AycBlxM5O1PIW68kiSNsXYC/jnU99zvBN7V5Lr3EkFfkrSE+eAyScqEAV+SMmHAl6RMGPAlKRMGfEnKhAFfkjJhwJekTBjwJSkTBnxJyoQBX5IyYcCXpEwY8CUpEwZ8ScqEAV+SMmHAl6RMGPAlKRMGfEnKhAFfkjJhwJekTBjwJSkTBnxJyoQBX5IyYcCXpEwY8CUpEwZ8ScqEAV+SMmHAl6RMGPAlKRMGfEnKhAFfkjJhwJekTBjwJSkTBnxJyoQBX5IyYcCXpEwY8CUpEwZ8ScqEAV+SMmHAl6RMGPAlKRMGfEnKhAFfkjJhwJekTLQa8NcD5gGvABNLy44EHgYWANcD21SsvyVwDTAfeBQ4ro02SJLa0GqwPQl4HugvlR8BHAV8A9idOClcDaxdqDM1lb0M7Al8Gfg8EfQlSWOslYC/A7AL8E2go1A+ATgc+DrwfeBa4IPESeHfC/UOAVYB9iZ6+T8ggv3ngNXba74kqVnNBvxxwHeJAP10adn2RMA+v1C2ALgY2LVQtitwBdH7rzkPWBXYsfkma8XWs7QbsALoWdoN0DKq2YB/CLAS8L2KZZsTaZpZpfKZaVnNZqmsaDZxctisyXZohdeztBuwAuhZ2g3QMmp8E3XWJPLtBxCBvWwq0Wsv5/X7iIHd8cDiVG9uxfp9aZkkaQw108P/GnAzcPkYt0WSNIaG6+FvBXyMGLCdkspq0zGnEL36PmA1YiC32MufSqRrFqfXfcDkin1MTcvK7ujo6Kia2qnlwpYjWLe1iVuLFsGUKVOGr7hc6xi+Sp2RTX7r6Gh1f1qG3NFowXAB/w1E7v7mimWPAD8E/psY1N2E+jz+5sC9hdczgS1K29iAOIGUc/sA2w7TNknSKFqT6N0Xv75B3Hi1C3FCWIXIzX+psN5E4Aki919zODHDZ7VC2ReI/H+xTJK0jOhi8J22hxN3zx4K7AxcSgT8VxfqTAHmAFemOgcRN3EVTwqSpGVIFzFbp51HK2xB3HS1gIFHK5gsXP70ABe0uE4n0VFoNbl/ELBXRXkvcGKL21oWPAUcW3jdQ+vHUpKWmM2BjVtcp5P2Av4fgbMqyrcB1m9xW8uCJ4FjCq+vo/6mRWnMNDMPXyqrGmQfS1VXgQ1nIixnxvIKd1Vg4RhuX8sZn1Spsi7gRQZPod2K6KHvTHUa4p3AH4gA8zfirux/GGZfnwdmEIP+fwN+Q/2VQw/wj8BH075fAT6SlvUSD/Mr2ge4E3iBuIv7q8QMsuJ7ewXYGriKmDBwL/Cvw7SzynapvXPSdv4M7F+qswNxYlpIXKlsX7GdfiLo7w/8D/As8FviybRFE4gU1sPE+7ud+keXQByTbwJHE7Ponk3lryLG2f4nrXsfA8expof4nQ7XjlVTOx5K23qAeI5W0b8Bd6flvcBhFe9b0jJgMvGH2lUq/zIR3F5FBIdiGmIrYBEDz086mLi34rJCnU4Gp3S+lfbTSTxl9VLgcWBSWr4FcE/a7tvS15pp2YPU5/Dfk7Z/dvr5sPQ+/rNQpyvV+QvwSeBdRNB+kcGBbTj7EkH0/6b2H5W2s29avi4xkeGaVOdAIjjOpz6l00OcnH4H7AFMJ05+l5b2dwlxbA5O7T4TeIn6sbIHGZgYsTvwvlT+PWKCxBeIE/PxxP0xuxXWva6JdnSkbT+XtrUT8GHiQYg1hxGfha8QnYP/IH4Pn0TSMulX1AdriF7hd9LPPdQH/HPT8mJ64oNEcH17et3J0Dn8VxG9x+eIIFIzg+ocfjng30IE16LDiMC2bnrdldrQVaizBhE4D27QrmZ0EOnRHxTacCKRr59QqLd/2n854JdvSvxMqrdKer1zev1/Svu9nvrfQy8xGWLlQtkmxCSLD5XW/TFwa4vt2CW93p1qk4irnaNL5ccBj+EEjaXOlI6qnEcEmTXS622Jey7Oa1D/bcAvqb/T+iIi2P7LEPt5O5FaeSrVnU/ck/GGFts7Dngzg9NM5xOf8e1K5VcWfn6GmELcag9/KnECfIjo0S4ievG1tr+NeG8vFNb5VYNtzWAg/QIDNyzW2vQuord9M3FiqX1dC7y1sF4/ccJZVCirnSx+XbHuttQH4eHa8U7iXppLGryP7YjZexeW9nUd8b8xlsdB9hWKg7aqcjHR630/kTqYTuSOf9+g/muJdEPRy0RwWGNwdQA2JALvLcTUyzlpn5dS3ytuxlrEHeHlNtRel9tQfojfojb2eQ7wz0Sq6x7iyuRQ4p/7QByT20vrLKD+8eBDtYdCm9ZK23upYt3FpdflY7AWcUJ8lsH6gXWIY99MO9YkTjyNrJW+391gXxsQnyMtJQZ8VZlHBN7pRMDfh6Hnij9G/X83gwgyaxI96CrvJVI4ezEwk2Q8jU8QQ3mKCIavKZXX2tSoDe2aQOS/DwXOKJQXB4irjslE2rur/BkiVVN1P0JZ+am1zxAnhe2Jnn7Zky2042niBNFI7TjvxuATD8BfW9iXxoABX42cS6Rw9gA2Sq9rykHlD8RMlyMZCCp7E5+v3zXY/qqpbvGR2/sw+DO5KNUdysvAbWn94gDiPmkfVc+CGolViFRRMXWyOtG7r72fGcDHqZ8aWTUbqHwsq1xN/Ge4+cRYSSuuJU5EU9J2Gmm2HYcRAb08qAxxnBcSKaDyGJCWAQZ8NfJbIgXxA2J2yR8Lyzqoz/1+lZiW+CvgdCJXewLxSO0/NNj+NUQgOpsYlN2KmKY5t7TtmcRg4XuIHuQD6Xt5APBY4j+qnUWcqN5IpFvOYCBl0Uirg4nPEgH9GCKV00/M2JnLwAyjU4mZKZcQs5HWTXXK8+LLx7LKVcR7u4o4rvek/WxLnHyOHOJ93Ef8Ts4lBpJvI65QtiLGGw5sox3/RRzbPxM9/ncQ/yRpLtANfBt4HXAjcWLclBi033uY7Utain5K9Fi/Viqvujv0nUQ+fiFxOX8a9Y/f6EzbKs7S+RAx53sBcBMx0FmefbMREWjmUj8Pv1wPokf/F2J65GxiamBxYkIX1Y8FqdrWcDYmerzziNkxXyBOOk8U6uxIzMN/AfgTkVZp5k7bTgYfq5WJYDqLeH+PESfl4lz8od7HZ4C7UlueSPstztxpth0TiPsfavcDPEAc56IDiA7CAuLkfDPw2QbtkiRJkiRJkiRJkiRJkiRJkiRJkiRJkqQx9r+4IPaoHSoqGQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f2202041dd8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f22020419e8>"
      ]
     },
     "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": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f22020737f0>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f2201ef4358>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f22020737f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "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": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f220202b080>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f22020bdb38>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f220202b080>"
      ]
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   "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": 6,
   "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": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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