{ "cells": [ { "cell_type": "markdown", "id": "e5e44fb3", "metadata": {}, "source": [ "## Ejercicio 7.2: Verificación experimental de la desigualdad CHSH" ] }, { "cell_type": "markdown", "id": "71cd75b7", "metadata": {}, "source": [ "$\\newcommand{\\ket}[1]{|{#1}\\rangle}\\newcommand{\\bra}[1]{\\langle{#1}|}$Utilizando un entorno de programación matemática, simule el experimento CHSH:\n", "\n", "\"bell-experiment\"\n", "\n" ] }, { "cell_type": "markdown", "id": "21c6d40e", "metadata": {}, "source": [ "1. *Estado entrelazado.*\n", "Construimos el estado conjunto $\\{\\rho,\\sigma\\}$ generado por la fuente de fotones, definido por la matriz de densidad de probabilidad\n", "\\begin{align*}\n", " \\delta_{\\phi} = (\\boldsymbol{I}_2 \\otimes \\boldsymbol{U}_{\\phi}) \\,\\delta_0\\, (\\boldsymbol{I}_2 \\otimes \\boldsymbol{U}_{\\phi})^H,\n", "\\end{align*}\n", "donde $\\phi\\in[0,\\pi]$ es un parámetro ajustable, $\\boldsymbol{I}_2$ denota la matriz identidad, $\\boldsymbol{U}_{\\phi}$ está dada por\n", "\n", "\\begin{align*}\n", " \\boldsymbol{U}_{\\phi} = \\begin{bmatrix} \\cos(\\phi) & \\sin(\\phi)\\\\\n", " -\\sin(\\phi) & \\cos(\\phi)\n", " \\end{bmatrix},\n", "\\end{align*}\n", "\n", "$\\quad\\;\\;$y $\\delta_0$ se corresponde con el estado EPR de máximo entrelazamiento,\n", "\n", "\\begin{align*}\n", " \\delta_0 &= \\frac{1}{2} \\bigl(\\ket{00} + \\ket{11} \\bigr)\\bigl(\\bra{00} + \\bra{11}\\bigr)\n", " = \\frac{1}{2}\\begin{bmatrix}\n", " 1 & 0 & 0 & 1 \\\\\n", " 0 & 0 & 0 & 0 \\\\\n", " 0 & 0 & 0 & 0 \\\\\n", " 1 & 0 & 0 & 1 \\end{bmatrix}.\n", "\\end{align*}" ] }, { "cell_type": "code", "execution_count": 1, "id": "e51496e6", "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "\n", "def generate_delta_phi(phi):\n", " \"\"\"\n", " Generates the density matrix delta_phi as a function of phi.\n", "\n", " Parameters:\n", " phi (float): Angle in radians in the range [0, pi].\n", "\n", " Returns:\n", " numpy.ndarray: 4x4 probability density matrix delta_phi.\n", " \"\"\"\n", "\n" ] }, { "cell_type": "markdown", "id": "7625c52a", "metadata": {}, "source": [ "2. *Preparación de las matrices de medida.* Consideremos los POVM $\\bigl\\{\\Pi_0, \\Pi_1\\bigr\\}$ y $\\bigl\\{\\Pi_+, \\Pi_-\\bigr\\}$, con\n", "\n", "\\begin{align*}\n", " \\Pi_0 = \\left[\\begin{matrix}\n", " 1 & 0\\\\\n", " 0 & 0 \\end{matrix}\\right],\\qquad\n", " \\Pi_1 = \\left[\\begin{matrix}\n", " 0 & 0\\\\\n", " 0 & 1 \\end{matrix}\\right],\\qquad\n", " \\Pi_{+} = \\tfrac{1}{2}\\left[\\begin{matrix}\n", " +1 & +1\\\\\n", " +1 & +1 \\end{matrix}\\right],\\qquad\n", " \\Pi_{-} = \\tfrac{1}{2}\\left[\\begin{matrix}\n", " +1 & -1\\\\\n", " -1 & +1 \\end{matrix}\\right].\n", "\\end{align*}\n", "\n", "Estas medidas se corresponden con las bases horizontal/vertical, y diagonal/diagonal cruzada, respectivamente.\n", "Por tanto, las medidas asociadas a las variables aleatorias $A$, $B$, $X$ e $Y$ del experimento estarían dadas por:\n", "\n", "\\begin{align*} \n", " &\\text{POVM}_A = \\bigl\\{\\Pi_0 \\otimes \\boldsymbol{I}_2,\\;\n", " \\Pi_1 \\otimes \\boldsymbol{I}_2\\bigr\\}, \\quad\\;\\;\\text{(Alice mide horizontal/vertical)}\\\\\n", " &\\text{POVM}_B = \\bigl\\{\\boldsymbol{I}_2 \\otimes \\Pi_0,\\;\n", " \\boldsymbol{I}_2 \\otimes \\Pi_1\\bigr\\}, \\quad\\;\\;\\text{(Bob mide horizontal/vertical)}\\\\\n", " &\\text{POVM}_X = \\bigl\\{\\Pi_{+} \\otimes \\boldsymbol{I}_2,\\;\n", " \\Pi_{-} \\otimes \\boldsymbol{I}_2\\bigr\\},\\quad\\,\\text{(Alice mide diagonal/diagonal cruzada)}\\\\\n", " &\\text{POVM}_Y = \\bigl\\{\\boldsymbol{I}_2 \\otimes \\Pi_{+},\\;\n", " \\boldsymbol{I}_2 \\otimes \\Pi_{-}\\bigr\\}.\\quad\\;\\,\\text{(Bob mide diagonal/diagonal cruzada)}\n", "\\end{align*}" ] }, { "cell_type": "code", "execution_count": 3, "id": "746f0b09", "metadata": {}, "outputs": [], "source": [ "\n" ] }, { "cell_type": "markdown", "id": "bda19e5d", "metadata": {}, "source": [ "3. *Medidas de las variables $(A,B)$, $(A,Y)$, $(X,B)$ y $(X,Y)$:* Determine las probabilidades de obtener\n", "todas las posibles combinaciones $(+1,+1)$, $(+1,-1)$, $(-1,+1)$, $(-1,-1)$, en cada una de estas medidas sobre el estado $\\delta_{\\phi}$.\n", "Tenga en cuenta que tras aplicar una medida Alice, el estado debe colapsar antes de que Bob aplique la medida en su parte del estado.\n", "\n", "$\\quad\\;\\;$Utilice estas probabilidades para calcular las esperanzas:\n", "\n", " \\begin{align*}\n", " \\text{E}\\bigl[ AB \\bigr] &= 1 \\cdot P_{AB}(1,1) - 1 \\cdot P_{AB}(1,-1) - 1 \\cdot P_{AB}(-1,1) + 1 \\cdot P_{AB}(-1,-1),\\\\\n", " \\text{E}\\bigl[ AY \\bigr] &= 1 \\cdot P_{AY}(1,1) - 1 \\cdot P_{AY}(1,-1) - 1 \\cdot P_{AY}(-1,1) + 1 \\cdot P_{AY}(-1,-1),\\\\\n", " \\text{E}\\bigl[ XB \\bigr] &= 1 \\cdot P_{XB}(1,1) - 1 \\cdot P_{XB}(1,-1) - 1 \\cdot P_{XB}(-1,1) + 1 \\cdot P_{XB}(-1,-1),\\\\\n", " \\text{E}\\bigl[ XY \\bigr] &= 1 \\cdot P_{XY}(1,1) - 1 \\cdot P_{XY}(1,-1) - 1 \\cdot P_{XY}(-1,1) + 1 \\cdot P_{XY}(-1,-1).\n", " \\end{align*}" ] }, { "cell_type": "code", "execution_count": 3, "id": "1915594a", "metadata": {}, "outputs": [], "source": [ "\n" ] }, { "cell_type": "markdown", "id": "11f14bbc", "metadata": {}, "source": [ "4. *Comprobación de la desigualdad de CHSH.* Considere la variable aleatoria $Z$ definida como $Z = A B + X B + X Y - A Y$, con valor medio\n", " \\begin{align*}\n", " \\text{E}[Z] = \\text{E}\\bigl[ AB \\bigr] + \\text{E}\\bigl[ XB\\bigr] + \\text{E}\\bigl[ XY \\bigr] - \\text{E}\\bigl[ AY \\bigr].\n", " \\end{align*}\n", " Para comprobar la desigualdad de CHSH, muestre en una gráfica la función $f(\\phi) = \\text{E}[Z]$ para una rejilla de valores $\\phi\\in[0,\\pi]$,\n", " junto con los umbrales de la desigualdad $-2 \\leq f(\\phi) \\leq 2$.\n", " \n", " Determine el valor o valores de $\\phi$ dónde se inclumple esta desigualdad.\n", " \n", " ¿Cuál es el valor máximo de $f(\\phi)$?" ] }, { "cell_type": "code", "execution_count": null, "id": "295b11fa", "metadata": {}, "outputs": [], "source": [ "\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 }