{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "e5e44fb3",
   "metadata": {},
   "source": [
    "## Ejemplo 5.3: Estados compuestos separables"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "71cd75b7",
   "metadata": {},
   "source": [
    "$\\newcommand{\\ket}[1]{|{#1}\\rangle}\\newcommand{\\bra}[1]{\\langle{#1}|}$Alice y Bob comparten una fuente de aleatoriedad común, por ejemplo, comprobando en la TV el resultado del sorteo de la lotería nacional  y viendo si ha salido un número par o impar:\n",
    "\n",
    "- Si el resultado es par, Alice y Bob preparan los estados cuánticos $\\rho_0^A$ y $\\sigma_0^B$, respectivamente."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "e51496e6",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "rhoA =\n",
      "\n",
      "   1   0\n",
      "   0   0\n",
      "\n",
      "sigmaB =\n",
      "\n",
      "   0   0\n",
      "   0   1\n",
      "\n",
      "deltaAB =\n",
      "\n",
      "   0   0   0   0\n",
      "   0   1   0   0\n",
      "   0   0   0   0\n",
      "   0   0   0   0\n",
      "\n"
     ]
    }
   ],
   "source": [
    "rhoA    = [[1, 0]; [0, 0]]\n",
    "sigmaB  = [[0, 0]; [0, 1]]\n",
    "deltaAB = kron(rhoA, sigmaB)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "58a03a9d",
   "metadata": {},
   "source": [
    "- Si el resultado es impar, en cambio, preparan los estados $\\rho_1^A$ y $\\sigma_1^B$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "ed3a9d07",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "rhoA_1 =\n",
      "\n",
      "   0.5000   0.5000\n",
      "   0.5000   0.5000\n",
      "\n",
      "sigmaB_1 =\n",
      "\n",
      "   1   0\n",
      "   0   0\n",
      "\n",
      "deltaAB_1 =\n",
      "\n",
      "   0.5000        0   0.5000        0\n",
      "        0        0        0        0\n",
      "   0.5000        0   0.5000        0\n",
      "        0        0        0        0\n",
      "\n"
     ]
    }
   ],
   "source": [
    "rhoA_1    = [[1/2, 1/2]; [1/2, 1/2]]\n",
    "sigmaB_1  = [[1, 0]; [0, 0]]\n",
    "deltaAB_1 = kron(rhoA_1, sigmaB_1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "971bc395",
   "metadata": {},
   "source": [
    "Para un sorteo de la lotería no trucado, la probabilidad de un resultado par o impar es la misma\n",
    "y el estado cuántico compuesto del sistema formado por los sub-sistemas de Alice y Bob está dado por\n",
    "\\begin{align*}\n",
    "        \\delta^{AB} = \\frac{1}{2} \\rho_0^A \\otimes \\sigma_0^B + \\frac{1}{2} \\rho_1^A \\otimes \\sigma_1^B.\n",
    "\\end{align*}\n",
    "Este estado es por tanto un estado separable y no está entrelazado."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "c14c2ac3",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "deltaAB =\n",
      "\n",
      "   0.2500        0   0.2500        0\n",
      "        0   0.5000        0        0\n",
      "   0.2500        0   0.2500        0\n",
      "        0        0        0        0\n",
      "\n"
     ]
    }
   ],
   "source": [
    "deltaAB = 1/2 * deltaAB_0 + 1/2 * deltaAB_1"
   ]
  }
 ],
 "metadata": {
  "celltoolbar": "Raw Cell Format",
  "kernelspec": {
   "display_name": "Octave",
   "language": "octave",
   "name": "octave"
  },
  "language_info": {
   "file_extension": ".m",
   "help_links": [
    {
     "text": "GNU Octave",
     "url": "https://www.gnu.org/software/octave/support.html"
    },
    {
     "text": "Octave Kernel",
     "url": "https://github.com/Calysto/octave_kernel"
    },
    {
     "text": "MetaKernel Magics",
     "url": "https://metakernel.readthedocs.io/en/latest/source/README.html"
    }
   ],
   "mimetype": "text/x-octave",
   "name": "octave",
   "version": "8.2.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}