{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "e5e44fb3",
   "metadata": {},
   "source": [
    "## Ejercicio 9.1: Ortogonalidad en la codificación supedensa"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "71cd75b7",
   "metadata": {},
   "source": [
    "$\\newcommand{\\ket}[1]{|{#1}\\rangle}\\newcommand{\\bra}[1]{\\langle{#1}|}$Utilizando un entorno de programación, genere los estados $\\delta_m^{BB}$, $m=0,1,2,3$:\n",
    "\n",
    "\\begin{align*}\n",
    "   \\delta_{0}^{BB} = \\dfrac{1}{2} \\left[\\begin{smallmatrix}\n",
    "                 1 & 0 & 0 & 1\\\\\n",
    "                 0 & 0 & 0 & 0\\\\\n",
    "                 0 & 0 & 0 & 0\\\\\n",
    "                 1 & 0 & 0 & 1 \\end{smallmatrix}\\right],\\quad\n",
    "   \\delta_{1}^{BB} = \\dfrac{1}{2} \\left[\\begin{smallmatrix}\n",
    "                 0 & 0 & 0 & 0\\\\\n",
    "                 0 & 1 & 1 & 0\\\\\n",
    "                 0 & 1 & 1 & 0\\\\\n",
    "                 0 & 0 & 0 & 0 \\end{smallmatrix}\\right],\\quad\n",
    "   \\delta_{2}^{BB} = \\dfrac{1}{2} \\left[\\begin{smallmatrix}\n",
    "                 0 & 0 & 0 & 0\\\\\n",
    "                 0 & 1 &-1 & 0\\\\\n",
    "                 0 &-1 & 1 & 0\\\\\n",
    "                 0 & 0 & 0 & 0 \\end{smallmatrix}\\right],\\quad\n",
    "   \\delta_{3}^{BB} = \\dfrac{1}{2} \\left[\\begin{smallmatrix}\n",
    "                 1 & 0 & 0 &-1\\\\\n",
    "                 0 & 0 & 0 & 0\\\\\n",
    "                 0 & 0 & 0 & 0\\\\\n",
    "                -1 & 0 & 0 & 1 \\end{smallmatrix}\\right].\n",
    "\\end{align*}\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "e51496e6",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "# Generamos los estados:\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7625c52a",
   "metadata": {},
   "source": [
    "1. Compruebe que estos estados son ortogonales entre sí: $\\text{Tr}\\bigl[\\delta_i^{BB} \\delta_j^{BB}\\bigr] = 0$ para $i\\neq j$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "746f0b09",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Comprobamos que todos los productos cruzados son 0:\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "94c1b5e6",
   "metadata": {},
   "source": [
    "2. Obtenga las probabilidades de los posibles observables si aplicamos una medida con el POVM  $\\bigl\\{\\Pi_{0}, \\Pi_{1}, \\Pi_{2}, \\Pi_{3}\\bigr\\}$ al estado $\\delta_3^{BB}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "36dc3029",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Definimos el POVM:\n",
    "\n",
    "# Aplicamos la medida:\n"
   ]
  }
 ],
 "metadata": {
  "celltoolbar": "Raw Cell Format",
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.10.9"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}