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   "source": [
    "## Ejemplo 5.5: Medida parcial de un estado independiente"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "71cd75b7",
   "metadata": {},
   "source": [
    "$\\newcommand{\\ket}[1]{|{#1}\\rangle}\\newcommand{\\bra}[1]{\\langle{#1}|}$Consideremos dos estados cuánticos independientes $\\rho^A$ y $\\sigma^B$,\n",
    "con una matriz de densidad de probabilidad asociada\n",
    "\\begin{align*}\n",
    " \\delta^{AB} = \\rho^A \\otimes \\sigma^B.\n",
    "\\end{align*}\n",
    "Asumamos que Alice realiza una medida con respecto a $\\bigl\\{ \\Pi_0^A, \\Pi_1^A \\bigr\\}$ en su parte del estado.\n",
    "\n",
    "Entonces, el POVM a aplicar al estado compuesto\n",
    "estaría dado por\n",
    "\\begin{align*}\n",
    "  \\bigl\\{ \\Pi_0^A \\otimes \\boldsymbol{I}_2^B,\\; \\Pi_1^A \\otimes \\boldsymbol{I}_2^B \\bigr\\}.\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "e51496e6",
   "metadata": {},
   "outputs": [
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     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Definimos un estado compuesto independiente:\n",
      "\n",
      "rhoA =\n",
      "\n",
      "   0.7500        0\n",
      "        0   0.2500\n",
      "\n",
      "sigmaB =\n",
      "\n",
      "   0.5000   0.5000\n",
      "   0.5000   0.5000\n",
      "\n",
      "deltaAB =\n",
      "\n",
      "   0.3750   0.3750        0        0\n",
      "   0.3750   0.3750        0        0\n",
      "        0        0   0.1250   0.1250\n",
      "        0        0   0.1250   0.1250\n",
      "\n",
      "\n",
      "Proyectores de medida:\n",
      "\n",
      "Pi0A =\n",
      "\n",
      "   1   0\n",
      "   0   0\n",
      "\n",
      "Pi1A =\n",
      "\n",
      "   0   0\n",
      "   0   1\n",
      "\n",
      "I2B =\n",
      "\n",
      "   1   0\n",
      "   0   1\n",
      "\n",
      "Pi0AB =\n",
      "\n",
      "   1   0   0   0\n",
      "   0   1   0   0\n",
      "   0   0   0   0\n",
      "   0   0   0   0\n",
      "\n",
      "Pi1AB =\n",
      "\n",
      "   0   0   0   0\n",
      "   0   0   0   0\n",
      "   0   0   1   0\n",
      "   0   0   0   1\n",
      "\n",
      "\n",
      "Probabilidades de la medida del sistema compuesto:\n",
      "\n",
      "PrA0 = 0.7500\n",
      "PrA1 = 0.2500\n"
     ]
    }
   ],
   "source": [
    "fprintf('\\nDefinimos un estado compuesto independiente:\\n\\n')\n",
    "rhoA    = [[3/4, 0]; [0, 1/4]]\n",
    "sigmaB  = [[1/2, 1/2]; [1/2, 1/2]]\n",
    "deltaAB = kron(rhoA, sigmaB)\n",
    "\n",
    "fprintf('\\nProyectores de medida:\\n\\n')\n",
    "Pi0A = [[1, 0]; [0, 0]]\n",
    "Pi1A = [[0, 0]; [0, 1]]\n",
    "I2B  = [[1, 0]; [0, 1]]\n",
    "\n",
    "Pi0AB = kron(Pi0A, I2B)\n",
    "Pi1AB = kron(Pi1A, I2B)\n",
    "\n",
    "fprintf('\\nProbabilidades de la medida del sistema compuesto:\\n\\n')\n",
    "PrA0 = trace(Pi0AB*deltaAB)\n",
    "PrA1 = trace(Pi1AB*deltaAB)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7625c52a",
   "metadata": {},
   "source": [
    "De acuerdo a las ecuaciones de medida de un sistema compuesto, se tiene que\n",
    "\\begin{align*}\n",
    "   p_{0}\n",
    "   &= \\text{Tr}\\bigl[\\bigl(\\Pi_0^A \\otimes \\boldsymbol{I}_2^B\\bigr)\\delta^{AB}\\bigr]\\\\\n",
    "   &= \\text{Tr}\\bigl[\\bigl(\\Pi_0^A \\rho^A\\bigr) \\otimes \\bigl(\\boldsymbol{I}_2^B \\sigma^B\\bigr)\\bigr]\\\\\n",
    "   &= \\text{Tr}\\bigl[\\Pi_0^A \\rho^A\\bigr]\n",
    "   \\text{Tr}\\bigl[\\boldsymbol{I}_2^B \\sigma^B\\bigr]\\\\\n",
    "   &= \\text{Tr}\\bigl[\\Pi_0^A \\rho^A\\bigr]\n",
    "\\end{align*}\n",
    "donde hemos utilizado la propiedad del producto mixto, la propiedad $\\text{Tr}[\\boldsymbol{A} \\otimes \\boldsymbol{B}] = \\text{Tr}[\\boldsymbol{A}]\\text{Tr}[\\boldsymbol{B}]$; y que $\\text{Tr}\\bigl[\\boldsymbol{I}_2^B \\sigma^B\\bigr] = \\text{Tr}\\bigl[\\sigma^B\\bigr]=1$ ya que $\\sigma^B$ se corresponde a un estado cuántico.\n",
    "\n",
    "De forma análoga, se tiene que $p_{1} = \\text{Tr}\\bigl[\\Pi_1^A \\rho^A\\bigr]$.\n",
    "Concluimos que esta medida sobre estado conjunto independiente modela la medida sobre el sub-sistema $A$, como podríamos esperar."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "746f0b09",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "PrA0 = 0.7500\n",
      "PrA1 = 0.2500\n"
     ]
    }
   ],
   "source": [
    "% Comprobamos que el desarrollo es correcto\n",
    "PrA0 = trace(Pi0A*rhoA)\n",
    "PrA1 = trace(Pi1A*rhoA)"
   ]
  }
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