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   "source": [
    "## Ejercicio 10.3: Protocolo de doble llave"
   ]
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    "$\\newcommand{\\ket}[1]{|{#1}\\rangle}\\newcommand{\\bra}[1]{\\langle{#1}|}$Considere un protocolo de doble llave para la transmisión de un estado\n",
    "binario. Alice y Bob aplican las transformaciones unitarias \n",
    "$U_A = \\bigl[\\begin{smallmatrix} \\cos(\\alpha)& -\\sin(\\alpha)\\\\\\sin(\\alpha)& \\cos(\\alpha) \\end{smallmatrix}\\bigr]$\n",
    "y $U_B = \\bigl[\\begin{smallmatrix} \\cos(\\beta)& -\\sin(\\beta)\\\\\\sin(\\beta)& \\cos(\\beta) \\end{smallmatrix}\\bigr]$,\n",
    "que se corresponden a rotaciones con dos ángulos secretos $\\alpha$ y $\\beta$, respectivamente.\n",
    "\n",
    "- Demuestre que estas transformaciones son conmutativas, es decir, que $U_A U_B = U_B U_A$.\n"
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    "\n",
    "\n"
   ]
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    "- Asuma ahora que $\\alpha = \\pi/4$ y que $\\beta = -\\pi/3$. Determine los estados intermedios\n",
    "del protocolo de doble llave  para la transmisión segura de un estado $\\rho_A = \\bigl[\\begin{smallmatrix} 3/4& 0\\\\ 0 & 1/4 \\end{smallmatrix}\\bigr]$."
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    "# Definimos las matrices de rotación\n",
    "\n",
    "# Definimos el estado inicial\n",
    "\n",
    "# Realizamos los pasos de protocolo de doble llave\n"
   ]
  }
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