densitymatrix-3level.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "c0d9ee5b-48a8-46b3-a556-097da2cd30a8",
   "metadata": {},
   "source": [
    "密度行列は量子力学の基本変数です。そして密度行列は$\\rho^\\dagger=\\rho, \\rm{tr} \\rho =1$ および$\\rho \\succeq 0$を満たす必要があります。ただ、最後の$\\rho \\succeq 0$、つまり半正定値条件は行列の固有値が0以上というもので、取り扱いが面倒です。同値な条件はいくつか知られています。その一つに$\\rho \\succeq 0 \\Leftrightarrow$ すべての主小行列式がゼロ以上、というのがあります(佐竹線形代数p.163 §4, 定理6 とその脚注参照)\n",
    "これを使うと、密度行列の要素から有限個の不等式で書き下すことができます。\n",
    "\n",
    "今回は3準位系の密度行列$\\rho$に対する条件を、密度行列の要素の不等式で書き下す、ということをしてみます。ただ、無駄に面倒なので、雑に調べて終わりにしました。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fe5afdfc-6b3c-436c-b438-56a969d499c6",
   "metadata": {},
   "source": [
    "三準位系では、$3\\times 3$の密度行列になり、主小行列式は$2^3=8$個でてきます($n \\times n$だと$2^n$個となります)。今後のことを考えると、減らしたいところです。ところで、半正定値条件でなく、正定値条件$\\rho \\succ 0$は、$\\rho$のすべての主座小行列式がゼロと同値なので、3個、$n \\times n$の密度行列でも$n$個の行列式を調べるだけでokです。これに還元してやるという手法を使えばよいです。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7943e197-1b7f-4039-8ac5-e827db4f59ed",
   "metadata": {},
   "source": [
    "半正定値条件を正定値条件で書くには、$\\rho$に単位行列×微小数の正定値条件を調べればokです。\n",
    "もう少し丁寧に書くと$\\rho$の半正定値条件は、$\\rho\\succeq 0 \\Leftrightarrow\\forall{}\\varepsilon>0,\\quad \\rho+\\varepsilon I \\succ0$となります。よって、$\\rho+\\varepsilon I$の首座行列式たちを考えて、$\\varepsilon$のべきでまとめ、各項を精査するという方針をとります。$I$は$n\\times n$の単位行列です。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1bfb5b86-1bc8-4e05-b1b5-85d5b09a2f46",
   "metadata": {},
   "source": [
    "この証明は$\\mathbb{R} \\ni x, \\varepsilon$ について$x\\geq 0 \\Leftrightarrow \\forall{}\\varepsilon>0, x+\\varepsilon > 0$について考えてみるとすぐわかります。(なお対偶をとると、$\\exists \\varepsilon >0, x+\\varepsilon \\leq 0 \\Leftrightarrow x < 0$なので、自明だと思います)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "47ce3208-3dd3-4e21-b87e-2b274f4cc07d",
   "metadata": {},
   "source": [
    "堀田量子本には、3準位系の密度行列$\\rho$に対する条件は露わには書いてありませんでした。Phys. Lett. A 314, 339 (2003)を参照せよとあります。今回は、そのarXiv版であるhttps://arxiv.org/abs/quant-ph/0301152 を参照しつつ、見てゆきましょう。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2fec75fa-3fcc-4ff9-876b-bec0b5c2d6ed",
   "metadata": {},
   "source": [
    "この場合、式(3.15)は変更され、\n",
    "$\\hat{\\lambda}_{n}^{\\dagger}=\\hat{\\lambda}_{n}, \\operatorname{Tr}\\left[\\hat{\\lambda}_{n}\\right]=0, \\operatorname{Tr}\\left[\\hat{\\lambda}_{n} \\hat{\\lambda}_{n^{\\prime}}\\right]=N \\delta_{n n^{\\prime}}$\n",
    "から、\n",
    "$\\hat{\\lambda}_{n}^{\\dagger}=\\hat{\\lambda}_{n}, \\operatorname{Tr}\\left[\\hat{\\lambda}_{n}\\right]=0, \\operatorname{Tr}\\left[\\hat{\\lambda}_{n} \\hat{\\lambda}_{n^{\\prime}}\\right]=2 \\delta_{n n^{\\prime}}$\n",
    "と定数が変わります。\n",
    "それに伴い、密度行列も\n",
    "$\\hat{\\rho}=\\frac{1}{N}\\left(\\hat{I}+\\sum_{n=1}^{N^{2}-1}\\left\\langle\\lambda_{n}\\right\\rangle \\hat{\\lambda}_{n}\\right)$\n",
    "から、\n",
    "$\\hat{\\rho}=\\frac{1}{N} \\hat{I}+\\frac{1}{2} \\left ( \\sum_{n=1}^{N^{2}-1}\\left\\langle\\lambda_{n}\\right\\rangle \\hat{\\lambda}_{n} \\right )$という式を使うように変更します。これは単なる規格化係数を変更するだけで、物理は何も変わりません。念のため。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "51122b09",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "ab404ea8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(λ1, λ2, λ3, λ4, λ5, λ6, λ7, λ8, λ9, λ10, λ11, λ12, λ13, λ14, λ15, λ16, λ17)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "var('λ1:18', real=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "15d6def2",
   "metadata": {},
   "outputs": [],
   "source": [
    "e=Symbol('epsilon', real=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "86406ec1-4c9e-4b4d-8b78-bfe1484d7081",
   "metadata": {},
   "source": [
    "Gell-Mann行列を入力します。文献に従い、正規化の定数は$2$とします。$\\hat \\lambda_8$には、$\\frac{1}{\\sqrt{3}}$という定数がかかります。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "258c8690",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{\\sqrt{3}}{3} & 0 & 0\\\\0 & \\frac{\\sqrt{3}}{3} & 0\\\\0 & 0 & - \\frac{2 \\sqrt{3}}{3}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[sqrt(3)/3,         0,            0],\n",
       "[        0, sqrt(3)/3,            0],\n",
       "[        0,         0, -2*sqrt(3)/3]])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_λ1=Matrix([[0,1,0],[1,0,0],[0,0,0]])\n",
    "_λ2=Matrix([[0,-I,0],[I,0,0],[0,0,0]])\n",
    "_λ3=Matrix([[1,0,0],[0,-1,0],[0,0,0]])\n",
    "_λ4=Matrix([[0,0,1],[0,0,0],[1,0,0]])\n",
    "_λ5=Matrix([[0,0,-I],[0,0,0],[I,0,0]])\n",
    "_λ6=Matrix([[0,0,0],[0,0,1],[0,1,0]])\n",
    "_λ7=Matrix([[0,0,0],[0,0,-I],[0,I,0]])\n",
    "_λ8=1/sqrt(3)*Matrix([[1,0,0],[0,1,0],[0,0,-2]])\n",
    "_λ8"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "2eea2e42",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2$"
      ],
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(_λ8*_λ8).trace()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5ccd4e78-a385-41b3-b99e-a71a81a34cbc",
   "metadata": {},
   "source": [
    "密度行列$\\rho = \\frac{1}{3}\\hat I + \\frac{1}{2}(\\hat \\lambda_1 \\lambda_1+ \\hat \\lambda_2 \\lambda_2 + \\hat \\lambda_3 \\lambda_3+ \\hat \\lambda_4 \\lambda4 + \\hat \\lambda_5 \\lambda_5+ \\hat \\lambda_6 \\lambda_6+ \\hat \\lambda_7 \\lambda_7+ \\hat \\lambda_8 \\lambda_8)$ を代入します"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "95589b78",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{λ_{3}}{2} + \\frac{\\sqrt{3} λ_{8}}{6} + \\frac{1}{3} & \\frac{λ_{1}}{2} - \\frac{i λ_{2}}{2} & \\frac{λ_{4}}{2} - \\frac{i λ_{5}}{2}\\\\\\frac{λ_{1}}{2} + \\frac{i λ_{2}}{2} & - \\frac{λ_{3}}{2} + \\frac{\\sqrt{3} λ_{8}}{6} + \\frac{1}{3} & \\frac{λ_{6}}{2} - \\frac{i λ_{7}}{2}\\\\\\frac{λ_{4}}{2} + \\frac{i λ_{5}}{2} & \\frac{λ_{6}}{2} + \\frac{i λ_{7}}{2} & - \\frac{\\sqrt{3} λ_{8}}{3} + \\frac{1}{3}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[λ3/2 + sqrt(3)*λ8/6 + 1/3,              λ1/2 - I*λ2/2,       λ4/2 - I*λ5/2],\n",
       "[            λ1/2 + I*λ2/2, -λ3/2 + sqrt(3)*λ8/6 + 1/3,       λ6/2 - I*λ7/2],\n",
       "[            λ4/2 + I*λ5/2,              λ6/2 + I*λ7/2, -sqrt(3)*λ8/3 + 1/3]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ρ3= Rational(1,3)*eye(3) + Rational(1,2)*(_λ1*λ1 + _λ2*λ2 + _λ3*λ3 + _λ4*λ4 + _λ5*λ5 + _λ6*λ6 + _λ7*λ7 + _λ8*λ8)\n",
    "ρ3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "aede14e6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ρ3.trace()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ce4b10bf-8198-4f86-9f84-d49e796c7c9e",
   "metadata": {},
   "source": [
    "まずは、手始めに$\\rho$の行列式を求めてみましょう。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "9d36083a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 18 \\sqrt{3} λ_{1}^{2} λ_{8} - 18 λ_{1}^{2} + 54 λ_{1} λ_{4} λ_{6} + 54 λ_{1} λ_{5} λ_{7} + 18 \\sqrt{3} λ_{2}^{2} λ_{8} - 18 λ_{2}^{2} - 54 λ_{2} λ_{4} λ_{7} + 54 λ_{2} λ_{5} λ_{6} + 18 \\sqrt{3} λ_{3}^{2} λ_{8} - 18 λ_{3}^{2} + 27 λ_{3} λ_{4}^{2} + 27 λ_{3} λ_{5}^{2} - 27 λ_{3} λ_{6}^{2} - 27 λ_{3} λ_{7}^{2} - 9 \\sqrt{3} λ_{4}^{2} λ_{8} - 18 λ_{4}^{2} - 9 \\sqrt{3} λ_{5}^{2} λ_{8} - 18 λ_{5}^{2} - 9 \\sqrt{3} λ_{6}^{2} λ_{8} - 18 λ_{6}^{2} - 9 \\sqrt{3} λ_{7}^{2} λ_{8} - 18 λ_{7}^{2} - 6 \\sqrt{3} λ_{8}^{3} - 18 λ_{8}^{2} + 8$"
      ],
      "text/plain": [
       "18*sqrt(3)*λ1**2*λ8 - 18*λ1**2 + 54*λ1*λ4*λ6 + 54*λ1*λ5*λ7 + 18*sqrt(3)*λ2**2*λ8 - 18*λ2**2 - 54*λ2*λ4*λ7 + 54*λ2*λ5*λ6 + 18*sqrt(3)*λ3**2*λ8 - 18*λ3**2 + 27*λ3*λ4**2 + 27*λ3*λ5**2 - 27*λ3*λ6**2 - 27*λ3*λ7**2 - 9*sqrt(3)*λ4**2*λ8 - 18*λ4**2 - 9*sqrt(3)*λ5**2*λ8 - 18*λ5**2 - 9*sqrt(3)*λ6**2*λ8 - 18*λ6**2 - 9*sqrt(3)*λ7**2*λ8 - 18*λ7**2 - 6*sqrt(3)*λ8**3 - 18*λ8**2 + 8"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ρ3.det()*54*4\n",
    "ρ3_det=ρ3.det()*54*4\n",
    "ρ3_det"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "37ec12a1-172c-4e57-93c6-060686e52fc5",
   "metadata": {},
   "source": [
    "上式はゼロ以上です(必要ないですが、216を掛けました。比較しやすくするためです)。以下に、文献の(31)式を引用しました。"
   ]
  },
  {
   "attachments": {
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M155Ae6XMErr6ChTniGvJK0FVvZ1vqN2MsU++B+39HZQChfpyFEh1tCGMO3z8\ncN+lSpy/phTc2Q53iyuqvnhJKWgr6lFekIPio5dhtbbt/4R+/z6JEz8qx+o70U78On++Vj5uQ84c\nyEFeSRWsNs0dPvC77xIqmwSLdneLdq2+CFmyd8JXcxDjQxeivDkK1FroyrEwdDwOanxNbr/fWO1S\nX15gp/+3x5/6/RsnmyorqbxUWbR5bc8csNu/2/+pH/598gSaRKuvLNpetC7Qgzt9oTk4HqELy+3a\nTFW35/Fagg8ub5qPz8qUQoZhLKM4ZG6mhvalBFG7gGTaeaaBGo5nUpR3d0q8PjVzDv1qWEgGVSnH\n0i2msoxICp28no5caiBt3UFaEKAir6AU2ufgV+XGitk3omit1WUH6TvCaUBcJh04foS2vhpEXipv\nCs8oszyT1JZRasADlFxieZ7ZsG8G+UJFAWkVSoltpGuGKoQymi667WjYRzN8QaqANHKkttrCBFJD\nQ4lFblhW0lZTZoSKoImjPEcWD6vSKVi0Q4iJYGuyIsjb7beaaoReelNElpVa3EjtQg20b4av0OEA\nckjltYWUIK28JBY5tFLjWrRUnRlBKiGrOIeUSEuFCWqhc4nkHtG6Qg9qKCvC26FbwRWp/p67+scw\nHoRHOGIVaQGkUkWQ4TiiNwItcSi0RZSoMTEih1PId8ASOmVoV/S38kBqMZg6YkrtOmLiOx6IyDIY\nmLXiK8TAYsVYS8bK9ndXUXqw+E51AhXaMd7626Ze3hTnxngZvSMINSXYq2xNLsVqvCgozYqD2gbI\nAw0oON2OkumdZS/yjrPkcFVQqr9aDEyOaE/rUJcbS2r/VJuD643ULrLTK/pkQqGdOkgOqEZMpNLc\nFxOpOIIITjeY8FmmRrSTxiuI0twYZ+gSPahIJX91LNlTeXbEGMYx3O+INRqykAw6pxTpaWGcirRS\noTIxjrJhEAZezJ6bEAZHlEEtZqlKiS1sO2LSjFeKXzExXcrAYn6Njj1I4JDxlAxnrzBK1weXO8M5\nWh0dTauNKugEjgyikmMT1IsS8+WAa2fY83ogvb5HOXAWhwZRabDvRWHppWRNsnVro0jlthglyRF0\nICawjdrl3Opoim65Ejkw+dBSWWoQ9UrMlwPhneHcaoqOXm3cJ5uNssqFYLLlz0uTpKBeiZTvfGU9\nqH/W0doolV3bzI4YwziG+2PEju7Gtnrxu8Mf4COXyKi7oqcoqMrbixNKkcPoDmLJ3H14IWUCuihF\nEn7j3sFf45Pw9ug+Somgugbnpd/3/QH36gtawlHs3qbBbXTGOA6jSz8Mki5u1y58bRBcUV/wCTZ0\nnIjh/ZQCK3SJiEUI6pGZtd1ybEbtZkwI34ph+Vswpa8XdJ+PxLA1klCbw0+oPXkStT8ph87SJQKx\nIeLaMrOw3VJlpVimwS8BH5RgyZAOUJW/hadmH1TedJz6swdxtrmXaIrqSUSPUQP7VyLnmFJmRC02\nTwjH1mH52DKlL7x0n2PksDWiJYxRR43DiPOZ2Nj8y2g5pdlYUTkC46KMoiDNaaN2+an2JE62XIkQ\nIVcWWZYri/KFg/ESPkDJkiHooCrHW0/NRrNr+1MtTp6sFZrfElR4MnoM1NiPlZaVSIh2IWTRLsGQ\nDiqUv/UUmi9aT+qfakSNG4HzmRubL3OGYcxwvyNWW41q5U9j5OBoHD6KSrnAYeo3LcSiO/6MKU+a\n5Hxq/yRmrUjDpMCmQUt3ZD8OiN++0YPRWy5qAXWovXQQiwfH4ePTSpEe5VrElVSd0hcIdNibn4/f\nhw+En1JiFZ8ReCFC/MeGT7DJ1AuQnLB+z+Nrn/9AwfxETJw4EVGzD6NnVzsDc6vhgxFyZfGJaWWF\nkV8cGoR03I9TGUn6uo6e9B5UD3RXPtD2hI5OEDUuw4rsUqWkEckJ64fnv/bBfxTMR6Ko68So2Tjc\ns6vxgx8Sqr4IDqpG3l7LA3FrcqwoF1WDwzDQbnqzG6tdfEa8gAjRRzZ8ssnE8RVO2OJQBKUD95/K\nQJLULqMn4T3VA3BbbUNHI0FMtMpWZMNUi3TlixEqVxYZSaKuE0dj0nsquE+0rtEDVd9gBFXnwQ0q\nzzC/OdzviHXuYcUR+T/8+9/S7+M4bbAkpjv/BV7q0QEvFVtcGxKU4v3Xd2PEvHj7Do4YbLMzNwDe\nk7HkZTvLUg7xOObu24bsPasRb7gUJ77n/DnxS9Ub3bvKJcAJHDuiQx+HLLIa4cMl45mPDQXGxrO+\n9jY89d4KzPnfMISFya8X3lyKGPeNoVCHD9cPovkbCowH0R8a4JO0GmlTn75e12enf4hZwe5yGgWB\nzyBWWnlds9V4EK2vxW1PvYcVc/73el3DXngTSy0K1gf+jwp3bv9Rs9Wy1kWHyiNl8LHkHFrghmoX\ndThkld8AY5X/AQ0+SVidNhVPN7bLs9Px4axgh2TQOgTiGVmJsNXEE/uhwQdJq9Mw9WmlrmHPYvqH\ns+Be0bpAD3z88ahPGfYfbVuNZ5jfJMotSjci5QWyFD8lx4gB/pSqj0LWUnX+LBoc3Is0otz4acgm\n9IHLduJ1akq3UXb2x5QS7k0+0R/TcYfDqqR4HFGnZsYDNQaFGz/ZJcWmqckoXM0WijxcmRPNGOna\nGmXdUpTcTqooaq10ZlKsXmyuctBCSpJ9WhzLUpkWQLATMO96ZH0MyXA0cKj120WKC/J3jRIpObda\nMSeeFHTuqjZTHvppvdxrntY/KyktADbbWitsscpTcxoyjAfhAXnEQjF7SQRUuzZhp0FundOfZaLA\n5HaLV+/p+PzLXKQEi5nyrnfwqel9ABzDypQNeOzNl2Frfat933BER4/H3PVfYukf/4rARxPw+Ulr\nK2wKuvPYM3s85pzqhTfXvGbz/EboDmJ+wjL8EpSGne+ENm2FdOwEytERHWwlIzOgdm8hSlUq6Dat\nRZ6nT0Jr96KwVJhg3Sas9fTK6sqxd3edaJcqZLYgyKt7lweAsgpcv/PcJpxCRZkUXikFIDrAjdQu\nqMXewlKoVDpsWpvXxiuNzUWH8r27USc6d9WNEDflEj3oDlnlrWu8atg7WB93L/JfisOHR9o+Jx3D\n3Cg4t9dk7RHk7TuNfymH9vk9/J8ZiG5W41iEIVv+HCKX/gF//WQOup18B2+dfgh91o3D7LJY5NKn\nGKZ8UkK340X8MWwZEJuLqk+HXb8loS9P+D12nJrbDEdpB178YxiW3RqHvBOrMNTCKnx1XiKi/ncz\n7p39MTJefBz3O7yNWi02j+2F+No3sHPLJPQ2vP5jC/Fgz88wuuIbzLRzD7V281j0e7Mn1r18EE/E\nbkJQxjkUT3Jw8DWjFkfy9uG0WeNdxMZp7wAz0vDsH5Wi6/wOXQaEo68jTqOUnLbfm+i57mUcfCIW\nm4IycK54EpyrrQ4n925FmYXctIfeG479wdl45SGlwIDf+z+DgdaVrQkpMLn/UFS8vhpdpj+JeXWJ\nKKpbIqYGTiDa6JZImOmqEbqT2Lu1TEkI6wj25C5kLX8pRDewjUvbRZzuSB72mSsRLm6chncwA2nm\nSoTfdRmAcMeUSIizH97suQ4vH3wCsZuCkHGuGM6rvBV7dXEjZJV/FuYq3wUDwvvaTNosIz040B9D\nK17H6i7T8eS8OiQW1WGJU0okceP0z81jb0EkckE2lU+H+oovsOilV/FZu1eQ/elU9LO34S/D3GR4\nSGZ9FXpP2obTh95B8D1iIH3yXWT/+V6clqL0Q4LRV/7QdVRPxuElX6A+6x18dj0o/jRWzcvEgLkJ\nVp0wafDIKa40fqJRNRCPh4nflz/CuxssPzbgE/5X5Hw+FfT+c4h+dQ3KHZpEyoPJlP9ehiOmTlgz\nqN08Af2mAEu3zUT/4eMwQpxn1/p8iw84SNd34Ixy4A70Dw7oK4uZ/YdjnFxZ5BtVVjLMxcjJsZPl\nu7XRByYPRPbwfKwc9gSGT/QXCrUGucXK+3pEXatKkOfoDgmeikPt0oTu5F4UOKbkrYD8kMQULMW2\nmf0xfNwIYR12Yb1hZc8cQHFF044YV78vwV631Fd+cGBg9nDkrxyGJ4ZPhD/qscZYiWSufo+SPKFH\nOcWodJfS29UDOfu+1DeNXwfgrFmp/fJ9vBAxHqt/PxtbMuPZCWMYS+hvUHoiUvyGqJ61ZJv63E1S\nTEZyiT7uSluUSBqbGavzKE4lxZxpaJpJ/ik5NxgoIK1SKbFCTRZFiHNo4vLsxGk1UOmCIApaYJx7\n6tDKt2j7D8qBNodi4EOi+lapyY0jby/Djaa1oq4qUVfDnEV1VLb9Y0p7MVgfO+d83FQLY1BqcinO\n24sCUpsSQMoxIoZtWEO5kyMpZXc1aRuOU2aU88k4WxQjpi2jtCAv482XFX1ryq8k5akKoMB5R/Rt\nqN9cWuVNk61kUNfHM/ktoDLluG0oowV+oIgsG9roULsYoE+EbDv2xxYtixET+hHnTV6Gm9ZrcylW\n6reGud5E/aXYsXb9ImhUaGfy8gqiD5z5yhbFiAn9SAsiL6NNsZUYUpP8ZzX7UiiocxRlHm8g7dev\nUzencs61Rf8Ubd89kEbFx1O88orsdaeVXQvkODO/BTY0viJN3pdXsdEMw1jGAxyxc7Q63IvgFW6Q\nrFDJRu8vBjZrPbgxGac+4LSKMkK8zLafMeYwpfjeSb0Sckw2apY3c3ZsWyAt5cRIn42hHFtbHKWG\nW9jSSAya/nHCHWxEfkghJsfyBeqdMJXGbCsR69ngpeB/NzlieiOvMnZsJEwTpko7GwS+Q8cbL1lJ\nLmnN2baF046Y4oSpDAd7PaZJRM/RR2EqUoV9pDxEIstXY+rFK5Qt8CNEZNlx0F3NOaH3NgZDR9vl\nOloqSuxOavFe2ztishOmMnJsJCwkTBUOhE/IC3pHISl1K31bZ7kP2cVpR0xxwlSGkyQZs+TL+smb\n7/UtzM5ljyO/Jz5w4jvboH9WplGY4dNDNTkUG2baTxqxPwnQP8BiZ7LJMIxHOGJ1lBenofYRH8sD\ntLaadqeGkbd3nIlBNkc2emIgj40lX4e2AMqgyNAxlL7zFOltt/RdydL2So5vCySvnlnPrF+WEUYa\nVTvy7tSJOhm+vNvp93hrWnOzPqOUnTCV0ey1CeUpU7MBxE2OmGLkzR0bmaJEaRBVzrtnGmmMViTl\nOsOJSjvliClOGMwGe5lzGSGiripxXgsXIjmRZvvuNSI76D5uGHH0T3zG5JjrSXPaRUFbNI3CU3PE\nhKGtHTHFCbPg2OgpStRPPq6fVzhiztbPCKccMcUJE7pgcb/FcxkUIuqqEsopXYk+u7xPspjutZQ2\n6J9GiMltWJj1yakDK/r6a+fM+gxjFw+IEVNj6NIT2PnyNaxNnIiJiStxLGARvq1ahWF2gk+7jJ6K\nKBWwPysLf3jjVZjmbzVF1XsSNuUvwaBbCvCe/rvmo9g7FYcvVmGVvS9ziG/xdWknRI4dgaGNOY4a\nX0NHYOykAQZJJ9UICgvGsZ0lZvFe1+CNEYt2YtvM3k1PWV4nFLO3pCPp4R/x9+Zmum0NrgHeIxZh\n57aZFuPgQmdvQXrSw/hRquzARbhEl7BooPJm6VEcEFcYNbSxoJW5dg23PzIdOftXWtQtn/HLkTNn\nLG47+xWuKGW684eQl/MhJvzvOvT+ZCfeCbWkZPuxp1CNyMcDleO2o89jT0NduEfUwITmtItEfT5e\nSWuPt17ugVvlkjZEX1ks2rkNMy1XFlvSk/Dwj3+/ntz5XEYypqSvxuq5UXg0aiG+bLOH8q7h2u2P\nYHrOfqy0rERYnjMHY287i6+u1OPo/jJRWIJFI6fhw9VzEfVoFJYeacMgsebqgUL9ulfx/qNvYZJR\nPkQD9u9BoToSblB5hvntoThkNyjKLUwHNqB1FbZXxJqJ/tac7T3qHMeNtyadQdrPLsDC7RIHcfrW\nZHNpuERnz56lI5lR5O0dQ1nG97X16G8Xqx3bq9Tl6G8vObCJs03qKG9yuLIaJemBu2LEHOT0ftpz\nvR3kuEmn8uu1KEbMEWRZGubrknXFGXvVlv1Tuk1va7VLvl1svGevObwixjCO4SFPTTqLCoHJJTh7\nYhmGWUg74fF0mYD5id9hsXlCNDfghyk7dmCK/e0IXEAtNsdHYnP4Tnz76TAHUgSY89T7l/D+U8pB\na+KlQceOHdHn+Xfx2v3rMWHq30xWMOuxadmnePDtV5xLe9FSVE/i1bcfxKfLTLcCcpzazTPwYee3\n8LKlJZNm4jdlB3a0qhLpcLL6B+VvCRW69uwBXf5OlCglDuM3BTt2THFgBw5nuQ/3dRK/egSil2Kf\nVPfdj471G7C1+ZVtu/5ZvQOf7++Kjt7KsSn1m7Ds0wfx9itu0XiG+c1xgztiAmmg1Dic2MvDUCF0\n9koErZqPNR6Q79BLo0HrS1JKTxCOtWEFKJ7bH+1PZGD0/GaPSlCpNVC33G+wTn0+Env2RGJ+o3vT\nBR26CDegcDeOKCUSuuIUvFo9B++Nt3YPp/XpMv49zKmej3cPOpde45rmKYzv8h226lMVFOObOqDu\nm2LkFJQ337kT/bF1u+N2vDFoOKIzvlKOgf+T9kL7fXvot3NtFl7QtGpl1egT7A9pr7b/U0pkbsft\ntyl/NoO26Z9Cp7/ehV3oAMt5gnUoTnkV1XPeEzqjFDEM0yJufEfsRqf9MKzMD8ayZxei/AZNU+U4\ntdjx0iDMpnhE31amH/hXL8lATfsOyvuexb8u/YJ299wuH1zdgYLtgCZuJK5HtNVuRvwr/9TneHPB\nYpLzqHpj5ra3UT0xHpudcOg7B0UjOrrx9RA6i7K7HwxF9ODebty/0RoDMTIuDOOjlGjL2p1YnnkJ\nsR+8BE8MV/KLfwtxlzKx5kspLkyH8s2f4YeopfDkxaRTVdaDT2s3x+OVf0o53izFrzIM4wzOZda/\niTm9PBT/b3IFxuSdwCpLafid5OqRj7Hk+4F47VlfpcRRLmLP8pXYc7Ye538Uxt7rXnRQd8Kg+EkY\nZJ7c3L0cXY0pyw7gV+WwkYcnrsCEAOXAk7haieIvSlDx92rU/HIb+oSPwNBH71dWJa7gwIercCVi\nKoZ08IwhSdoQP31LO0x4MQjtlLJmUfE55mcfRO3FK/hV0qMegxE/aZB51nm3cxWVxX/Dtj0XUPuf\n7TFkxDiE9miLtSInEXq0bekaHPgncFefYRgd8RA8RGUsoiv/FK998UfMmvWkcdjAlQP4cNUVREwd\n4kD9T2Npf19M/WYyChqWQcqZzTCMZdgRaza1+HJhHEbOP4WnPsjC3NGebVQZhmHakqvfF2HppFGY\nf3IwMgpWYYwj240xzE0MO2LOIm1Zkr8B2af8MeO1wR64asAwDNPGSKveudfw3PBRCO6p5tuXDOMA\n7IgxDMMwDMO4CQ7WZxiGYRiGcRPsiDEMwzAMw7gJdsQYhmEYhmHcBDtiDMMwDMMwboIdMYZhGIZh\nGDfBjhjDMAzDMIybYEeMYRiGYRjGTbAjxjAMwzAM4ybYEWMYhmEYhnET7IgxDMMwDMO4CXbEGIZh\nGIZh3AQ7YgzDMAzDMG6CHTGGYRiGYRg3wY4YwzAMwzCMm2BHjGEYhmEYxk2wI8YwDMMwDOMmbl5H\nrPYICr46rxwwDMPciJzHVwVHUKscMQzTXHQ4ubcIf/9ZOXQl579CwRH7vdODHbF6lO89hPM65VBB\nd/4Q9pbXK0eW0Z3ciwJbn6ndjLFPvgft/R2UAoX6chTkFOPo5atKQUsQ9S/IQfHRy3DJ2coLkFN8\nFC6pWmtz5gBy8kpQVW/SeM7g0jaRqpaDvJIquKJqrY1L29yDdbvV+U3rYwfcr30PT47d7DnO2NXL\nOFqcg5ycPJRU1YthjmE8FR3KF4Zi/EENfO9UihR09RdkPRa2o9Jef9edxN6CcmEZTehwP7TvPYmx\nm+30TvJYKijVH6QJS6Xd1VpxrKW6g+kUptFQYpF0bAVtCSX7gvxTK5QCE7RllBrwACWXWD5Hw74Z\n5AsVBaRZ+f/m0LCPZoi6qALSxNW0lAbaN8OXoAogV1StddFSdWYEqaChuLw6pcx5XNom2mrKjFAR\nNHHkgqq1Mq5tc8/V7dbmt66PWipJfoACUsvEX+6lpnAa9eocRZnf1pG24QztTA6izuEZVObuijGM\nBWpyY8k7IotqlGOZGiqcFkqRqTvpVJ2WGs6sp1iNirxjsug7S3os+nBOrIbgn2rZFko+yQMBlGqj\nE3i8Iyb5ilC1I2+Nl/jtTTFZ39kwNloqW+Cv/x9rjlhFqj+pY3PJus2rovRg8Z3qBCp0gfGoSg8W\n9VFTgmtORsHi2tQJhW43uHbRFlKCWsgxOF1ItKW4tk20hQmkFnIMTm95zVodl7a5B+t2a/Nb18e6\nXIpV+5O1+acZP/yNosWktt/kTPpaP9F1ATVZFKFSU2yuoXWV7bhvconn2yzm5sJKn6lbG0U+E7cb\n+Qh1WRHmfsXp/fRx6kQK7aYhL5WwB9YcMUGdcPjUNt73cEcsmF5MTaL4+HhKSt1g32BUpFFYgD/5\nmAqsEb3gfSnlsHJsBc91nlw7AFjl3GqKjl5N55RD59BSYYJayDGYXOHvuLRNXDooW+fc6miKXt0y\nKbq6zW/aicFNoI+HU3xJFbXWxiTTBG0dfbt+Gg3wbke9xqTS1uMNyhvOIcskgrJMKlC2wE9MomMp\n1/OVhLmJkBZlLPWX3FjRF4VdC8kwsN0VqeQvyhCS0TQuNlyis2fPU51WWTSy4WiJ3kkpviqKWmu5\nd3q4IyY6r3JknxrKigqjjEJZYJYcMcnTVdkUlsK5DAoR51DF5rpggDlHGSGikVxkiM5lhAglUYlZ\nZytaNUnpHJGTPYoS9TN9q7eJm4NL20SqmjQoN2MFwQmkju6Ka3dpm3uwbrc6v3V91A8W5o6QfRro\n+NZUGtOrHXkPmEbrpduKyjuOU0dZEUIXfJKpRClpRF5N8KFk0zcYxm3IjpElm6ot+4SS4udQ7mml\nQEKxHYjJsdA3HHHE5PEAEVkWJ0q/macmaz+fgawBaZh0n1Jghg578/Px+/CB8FNKrOIzAi9EiP/Y\n8Ak22X4uwAF8MEI+GT5p+clE1V5AhLiWDZ9sMg8M9DRCRyPBByhbkY1SpchpXNomUtUSRMuUYUV2\ni2vW6ri0zT1Yt1ud37o++gVgoLoAhXubGx7vhR5Pz0RWeS0OLuqJXeM7o33vWCwt+r4ZD2NU41yV\n+HV3O5jEPEN9ZzvxsxpVFzhsn/EQjhUht2owwgaqlIImVL3HIm3FPAzrrBQIjh3eK2yvClGRoeKn\nc/gFDIS6oBCWuqeHO2K/4Oz2xZgSG4c//zkWz45fij2mj1FK1Odj9ooeeOdFWy7WCRw7okOfB7or\nx7ZQI3y4ZGXzsaGg5VZWHT5cP5Dmbyho+UCqDodctQ1wQdVamUA8EytGvqo12Nrikc+1bYLAZyBX\nbWvLB+XWxqVt7sG63er81vWxDx4eoMPur48qx81FhQ4PPY9lh2tx5rMYXE4PRecuj2P6hgoHnuo8\ng8pjyp9WqKw6pfzFMO5FV3kEZT490VWtFNhCdxCfrSiDV9C7eGOkI/9ghT4PY4BuNyx1Tw93xIqx\n+8enkZb1Ef761ywsi/oKw7v2x8JyQ6ugw8F3F+D2WdPR26arKhkKNTrd55ggPdd5cvEA0MoEPhMr\nZvpVWNPykc/Fg74rB+XWhicGruK3rY9q3K1Robq6pYksVFD3fBpzN53GmeKZuDfjCfyx/WNIt+No\nMcyNwqmqSqBrR3grx+ZIucWkFCwfIuGhCBSO3InK4qn276bZQn03NKpqWOqeHuyIdUX8jhNYO6bb\n9aXA9sNexStdD2PWi6twWinTlS9CSu1szAu1s2B47ATK0REd2ivHdqjdW4hSlQq6TWuRZ8XK1h7J\nw4EzyoEtaveisFQFlW4T1lo7mcBu/jM9tdhbWAqVSodNa/MsDgBXvy9BXo5QouJKN+d50qF8727U\niaapytyIg0qpMZLCF8DuZQust4mUc64E3zderK4eFcUHhOttA1059u6uE7pVhcyNxjWTcjvlSPIz\nejlWx9bBdpvr6itQrG9vx3KOuVSOdnX7Kr4vydPLsLjSvdoo6ZpNfdRfr9TWjuUJc8RGSDhkJ1yk\nj53+1AMoP4GW+0w6nD/0MV6NjkHqj09i/poMPN+iUYhhPIdTFWVAhz+ISZk1VOg2MBrR0S/ivT1b\nMf7QOAQ+txBftmiO0wly9zTvnS50xGpxJM+SwbD+2nvS1nq3mJVp1NedMJne8Osrfu1fjc0npONj\n+HBGJSYuGCrmgq6jdvNY9HuzJ9atGiq+fwsyN1Qr70hIySxXY/GU/ujV72lklCvF1pCSx/Z7Ez3X\nrcJQcTFbMjfA8GzX0RVjRvAgzNr+g1JgiVpsHtsPb/Zch1XyyWBUNfH+l3OD0WtaJXyHPINOuyLg\nP9fG9Lr2iOywmb6Kv0Fd3Tfy4G76ynM0i7eUKK8/hlbMRO6ffYHqZfisWHlLzxkcELONvwzrhz6D\nZsHmZQtst8kP2P5SEDq374LHRw9DQHsfPJX9T9ylvGuGGPQW9h+Kipm5kKv2GZqqVo3Cea8js7AQ\nhcprW/o0jPnwe9xhRcmkgdZMTuJV/E0d6r4ptvhe3hFHe7TtNq/dnICR7/8Df3pmCHqefQN+7YOx\n2GjF2BiXytGebtd+ibnBvTCt0hdDnumEXRH+sKWO+qSIFmRl/ZUHh8VoTx/FtSSMfB//+NMzGNLz\nLN7wa4/gxeXivyxjW44GnF6OUUF27ISL9bFlXEXltoWI7d0ePRO+xsPLKlBbnoWZT/eAl/IJy7SH\nj/VRTU+n+6wG8DKMx6JSP4Ipy17D/Xmz8MTTi10wyTHHY1fErlaKAczqis5hHK0UJipzGj7qEAza\nZ2CcJSdCfKJxALS/wmRM7eYJ6DcFWLptJvoPH4cRYoDZtT7fYIBRo/fg8UhaOgNPKSVWEcZ9gnwy\nzOw/HOPkkyHfzGbrUDzjRaz5t3JoETEgT+iHKViKbTP7Y/i4EWIA2IX1BierXTMRT6x5DJ+tfR49\nvC7hbPV/o/O9tynvtiVi0FscioHZw5G/chieGD4R/sKBXZNrOPJ1RpCYbbzx1lh0VUqsYb9NJPwR\nNfEJ+N7xIKZ8UYETy6w452LQWxw6ENnD87Fy2BMYPtFf+NZr0FS1b3EISVi+YgVWSK8PpuJPqsew\nfu0kdFE+0XbYa/NSLJv+DYJHBqKDygs9nn8XKQ8ewDSDFWNDXCpHu7pdizXiPGse+wxrnxeD+KWz\nqP7vznCLOjqgj6XLpuOb4JEI7KCCV4/n8W7Kgzgw7UWssiBIx+QocRrLX/0LvrE13/QUfZRWAzdM\nx8AO3hj60TXEbLiAfx5ehucf6iB0zhG80VHqyHVXYLpTTP3PV8RPx8NCGMZ9yLck8w6dN56EdRmA\noX7i3cNvYLnhMOYqlKcnPQzlcVALj3PLOT78aEGZlAbnPJ09e9b4tWc2CXmR3+w9+uPzdcrDptoc\nirHzCHVNbhx5exlmwNWK71OJ77OUeyiXYsX3xFrLr1GTS3HeXkbZrrW5sSSMmlniRm3RNApPzaEF\n4potP1pfQ7lx3uQVkEpNVRPfLyWRu557SJaZjyueEW9R+gotlaUFkZcmlnKvpytW2tNSLiz9I/fW\nH9t3rE0cTHWiLaO0IC/SiEZrqpqc7sRyLiwtFSX2sr2Tgw1alr7CkTbfQ9M0IM20PfojCbl/mMvC\npXJ0RLf1cvWElAWO6eOeaRqCZpqQqIK4Hsk8mvbv5tiIqowRFJebad1OtII+liT7WHnE3gpS9vv0\nMdRPSe564IzzucT0340YyjH5cn0eMafSajBM66DXSUupJCrTKED0PyCMPvpBKdOj2AyoKC5PKbqO\n8p7dMbOEkn1AMaYdROChjphk3HwoevVxMjYL8oXYvGDFkJkPgEWUqLYsBAm9gVVphME03uzAetZr\nG46YfqBSGRtYCUuJG+vyaHK4NNjKjWleb3lAVhkNJBImCSrrsihC1McnJJpikpbSxymRFBiZTqXO\n2FVJhk45YsqgpzLfzsFqAkx9e1l2xBxvE0l2GgqMnUfLs5fS5MBAmpxjsgODMuipDB0bPTYSph5O\noQeakyDTBOcdMQfb3AwpNw7MkhS6VI4O6nZj7qiQ6BhKWvoxpUQGUmR6qUl/bm2c0EcFKTkqVFFk\nmH+xWTaiIo3C4yQZWbETraKPWsqJcXAypv2OvhA2orOmM4VOW0/fNk5YW4K+L5vKVL4e491MGujM\ngaIWOX0M0xL0uRn9FlCZcnydurUUpfKh8MX7jO2bMr5Ck0jm8yAHHTEbi0Gem9BVGPzJoZMp88AZ\nvfHW1n1L6+O6k5d3DGVZ3PBJpiEnRj8z95tbajyACDMgJRz0k5bSTJANrMrKXm2yA2cuZCsGVhmo\nzA2sjHHixjrKmxyuDBKWHDFlQLYwkOgxTFCpOKBNg7A8aNvezskKTjliyqAH84FKj7UEmFYcsea1\nSR2V7Slt6jj6cxrsoKAMejBzbGQsJ0yV5RfRgmm8c45YM9rcCCH/1AAz582lcmyGbuuTFwqZXs8k\nrXfUTLe/aU2c1EeBtiyVAkwcrubJsYLSwuOUdrBgJ1pNH8togR8c01nhiO3ZdoBc6wvJMm8XsECZ\nAGqpen0MabwbZaHQmBxTLQY1pYhh2pSSZPKxsHorIe2XGho6jdYfuSRPHBuOU2aUhuAVRGmWDF/D\nbkqSFoh8kmi3rf5UtoD8rKwMe+5Tk+2HYVn+bPQ8vwF/mTgRiW/k49roHFRXrcOYbhaiFqrzMF98\nLqngfzA2Ph5BFzKQKI5XHVbehxpBYcE4trPELJbjGrwxYtFObJvZ20I8RChmb0lH0sM/4u+VSpEt\nrgHeIxZh57aZFtNphM7egvSkh/GjOFnt5hn4sPNbeNlq3g39ybBo5zbMtHwybElPwsM//h2V992H\nTqKoR2AvJaZHhfvu74j6DVtRoj9uba7h2u2PYHrOfqwcZuHRVJ/xWJ4zB2NvO4uvpJAROzSrTWrP\noNowMMWvO3qjCnl79U90iJNdw+2PTEfO/pWwXLXlyJkzFred/QpNVduPreukAGNZmm1HM9pcKZKo\n3RyPyM3h2Pntp0bX6Fo5Oq7b992n10YE9lLkp7oP93esx4atbaONUmWd0sfazYiP3Izwnd/iU4P/\nc1yO0kMBs3AsfoFFXdPTWvpYXYKdx0IwdJADOqvqhoHhj+J+2xH4zUSF3knFOPNxIA4umoGU5Dfw\n2S0v4+sTq4yvMzgRK+fEI+m9CQhWihimTenzGJ5WF2LPfuXYgPZPLkLRlkT4nPpY73tMTFqBy6M2\n40ztfiQZGj7F55iY9Dc0PBWP+Kca8LckcTxxPvLMA0ZF99yJYyFDYbF7Kg7ZzYF+bzxrt3Wai41b\nkw5yen82ZWc3vpbQKOFV+4xaQtnby5q/iiVm4dKKmuGKn35VQu3ErNOpFTEnsbIi1hz016kKo48a\nNwHTLwE7uDJgDf3speX1cu7WZPOQVmv6xWSRvFBcSctGzaMD+neah0vlqG9XOZZTRtZPdaIHr4FI\nK379mlbcK5eNonnNFmQdlW037Ncz9ftxBs8Uf+833DOlmTigj/rbrYZ74TEMYwXljlGb7ZMr36I3\n2r/SgFukH4pPdhOgQ/FLPph49xc4NbefUuYsmzH2lkggl8TMWSlqEQcxu+OjyJtagW9mOpewpz5/\nArq/dDc2li1E/1vL8dYjT+DwrIPYOKq5z1ddxeXLgEbj0umyZY4txIM9P8Poim/g5GWLcyzG4L/c\nhfS1z6ObSoeTHw5Bn4+fwoEvLa/cOMTmsbglshypLanX1cu4DA1aU4y1O17CoKSrSJo7BPdIBQ27\n8OanvbG1eJKNHDlWcKkc65E/oTteunsjyhb2x63lb+GRJw5j1sGNaLY6tgW1O/DSoCRcTZqLIbIg\nsevNT9F7azEmNVuQBtSvwbC7Y9GupXbCrj6WYm7XIahbUY0l9nIqSkipQraW4ZJyaJ/focuAcPS1\ntsrHMDcap5cjtHcBJlZvREsS5jtE6Vx0HVKHFdVLYKl73mSOmECf+ygTYaWbMcYpo3IRe5avxJ6z\n9Tj/41XA6150UHfCoPhJGPRH5SPNpeJzzM8+iNqLV/CrdL4egxE/aRCcOd3Vym1YuuYApOxPfYaN\nRoTDj5+3NRX4fP5GlP9ci4tXfsV/tfsj2t/ZG8/OeQ49lU80B935Q9iSuwPHf7iG2/oMw+iIh9Ch\nJRcuBubU1IsY8vZY5525VucoVk9ZhgO/KoeNPDwRKyYEKAfNw7VylHJSLcWaA/8E7uqDYaMj8FCL\nGqX1OLp6CpaZCxITV0yAc5IUlmLPcqwsqNTbCb1+Bw7HnOec0W6BHX08vTwUA/e/jFLh7Tlk1tgR\nY2565NyCI698gKMLAltxnDyN5aEDsf/lUqNwB0NuPkdMoCtfiNAEYFlxC1ZMGIZhPAC2ZwzjLFLC\n7Cfx+XM7rDpJLUNy9kKRgGUothhfKuPhe022DqreM1H4gQZ5eVVKCcMwzI1IlbBjGnxQyE4YwzSf\n9hi2Mh+jTn2OAw48RNZsqvKQp/kAhTacMImbckWMYRiGYRjGE7gpV8QYhmEYhmE8AXbEGIZhGIZh\n3AQ7YgzDMAzDMG6CHTGGYRiGYRg3wY4YwzAMwzCMm2BHjGEYhmEYxk2wI8YwDMMwDOMWgP8P+OON\nnxE/8PUAAAAASUVORK5CYII=\n"
    }
   },
   "cell_type": "markdown",
   "id": "372a0db1-b2f4-40be-8a65-4f68c1ec0043",
   "metadata": {},
   "source": [
    "![スクリーンショット 2022-01-10 131248.png](attachment:4f9b9fab-457f-4311-8993-97d130c6754b.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9ad8694e-658f-4b1c-88a2-1966d112a6f7",
   "metadata": {},
   "source": [
    "これらを比較すると、(i) 式全体の符号が違います。(ii) $-9 \\sqrt{3}\\left\\{2\\left(\\lambda_{1}^{2}+\\lambda_{2}^{2}+\\lambda_{3}^{2}\\right)-\\left(\\lambda_{4}^{2}+\\lambda_{5}^{2}+\\lambda_{6}^{2}+\\lambda_{7}^{2}\\right)\\right \\}$ ではなく、$-9 \\sqrt{3}\\left\\{2\\left(\\lambda_{1}^{2}+\\lambda_{2}^{2}+\\lambda_{3}^{2}\\right)-\\lambda_8\\left(\\lambda_{4}^{2}+\\lambda_{5}^{2}+\\lambda_{6}^{2}+\\lambda_{7}^{2}\\right)\\right\\}$と$\\lambda_8$が入ってきてます。おそらく論文のミスでしょう。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "97cffca0-a719-475d-af07-f1b82c66fe4f",
   "metadata": {},
   "source": [
    "論文では条件式がこれで終わってます。我々の場合はこれだと不十分で、主行列式を調べつくす必要があります。おそらく、独立な不等式ではなく、徒労に終わると思われますが、調べる必要はあります。まずは、$\\rho + \\varepsilon I$を求めてみます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "db818feb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\epsilon + \\frac{λ_{3}}{2} + \\frac{\\sqrt{3} λ_{8}}{6} + \\frac{1}{3} & \\frac{λ_{1}}{2} - \\frac{i λ_{2}}{2} & \\frac{λ_{4}}{2} - \\frac{i λ_{5}}{2}\\\\\\frac{λ_{1}}{2} + \\frac{i λ_{2}}{2} & \\epsilon - \\frac{λ_{3}}{2} + \\frac{\\sqrt{3} λ_{8}}{6} + \\frac{1}{3} & \\frac{λ_{6}}{2} - \\frac{i λ_{7}}{2}\\\\\\frac{λ_{4}}{2} + \\frac{i λ_{5}}{2} & \\frac{λ_{6}}{2} + \\frac{i λ_{7}}{2} & \\epsilon - \\frac{\\sqrt{3} λ_{8}}{3} + \\frac{1}{3}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[epsilon + λ3/2 + sqrt(3)*λ8/6 + 1/3,                       λ1/2 - I*λ2/2,                λ4/2 - I*λ5/2],\n",
       "[                      λ1/2 + I*λ2/2, epsilon - λ3/2 + sqrt(3)*λ8/6 + 1/3,                λ6/2 - I*λ7/2],\n",
       "[                      λ4/2 + I*λ5/2,                       λ6/2 + I*λ7/2, epsilon - sqrt(3)*λ8/3 + 1/3]])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ρ3_e= Rational(1,3)*eye(3) + Rational(1,2)*(_λ1*λ1 + _λ2*λ2 + _λ3*λ3 + _λ4*λ4 + _λ5*λ5 + _λ6*λ6 + _λ7*λ7 + _λ8*λ8) + eye(3)*e\n",
    "ρ3_e"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5db53228-9570-4729-be6e-230f5b90c426",
   "metadata": {},
   "source": [
    "見やすくするために、216をかけてみました。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "8b714830-0057-41b6-903c-eff50f7ed0d5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 216 \\epsilon^{3} + 216 \\epsilon^{2} + \\epsilon \\left(- 54 λ_{1}^{2} - 54 λ_{2}^{2} - 54 λ_{3}^{2} - 54 λ_{4}^{2} - 54 λ_{5}^{2} - 54 λ_{6}^{2} - 54 λ_{7}^{2} - 54 λ_{8}^{2} + 72\\right) + 18 \\sqrt{3} λ_{1}^{2} λ_{8} - 18 λ_{1}^{2} + 54 λ_{1} λ_{4} λ_{6} + 54 λ_{1} λ_{5} λ_{7} + 18 \\sqrt{3} λ_{2}^{2} λ_{8} - 18 λ_{2}^{2} - 54 λ_{2} λ_{4} λ_{7} + 54 λ_{2} λ_{5} λ_{6} + 18 \\sqrt{3} λ_{3}^{2} λ_{8} - 18 λ_{3}^{2} + 27 λ_{3} λ_{4}^{2} + 27 λ_{3} λ_{5}^{2} - 27 λ_{3} λ_{6}^{2} - 27 λ_{3} λ_{7}^{2} - 9 \\sqrt{3} λ_{4}^{2} λ_{8} - 18 λ_{4}^{2} - 9 \\sqrt{3} λ_{5}^{2} λ_{8} - 18 λ_{5}^{2} - 9 \\sqrt{3} λ_{6}^{2} λ_{8} - 18 λ_{6}^{2} - 9 \\sqrt{3} λ_{7}^{2} λ_{8} - 18 λ_{7}^{2} - 6 \\sqrt{3} λ_{8}^{3} - 18 λ_{8}^{2} + 8$"
      ],
      "text/plain": [
       "216*epsilon**3 + 216*epsilon**2 + epsilon*(-54*λ1**2 - 54*λ2**2 - 54*λ3**2 - 54*λ4**2 - 54*λ5**2 - 54*λ6**2 - 54*λ7**2 - 54*λ8**2 + 72) + 18*sqrt(3)*λ1**2*λ8 - 18*λ1**2 + 54*λ1*λ4*λ6 + 54*λ1*λ5*λ7 + 18*sqrt(3)*λ2**2*λ8 - 18*λ2**2 - 54*λ2*λ4*λ7 + 54*λ2*λ5*λ6 + 18*sqrt(3)*λ3**2*λ8 - 18*λ3**2 + 27*λ3*λ4**2 + 27*λ3*λ5**2 - 27*λ3*λ6**2 - 27*λ3*λ7**2 - 9*sqrt(3)*λ4**2*λ8 - 18*λ4**2 - 9*sqrt(3)*λ5**2*λ8 - 18*λ5**2 - 9*sqrt(3)*λ6**2*λ8 - 18*λ6**2 - 9*sqrt(3)*λ7**2*λ8 - 18*λ7**2 - 6*sqrt(3)*λ8**3 - 18*λ8**2 + 8"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "collect(ρ3_e.det()*54*4,e)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "521a4379-ce84-41ae-a395-45e13e257895",
   "metadata": {},
   "source": [
    "[8]のセルで求めた式がゼロならば、$\\left(- λ_{1}^{2} - λ_{2}^{2} - λ_{3}^{2} - λ_{4}^{2} - λ_{5}^{2} - λ_{6}^{2} - λ_{7}^{2} - λ_{8}^{2} + \\frac{4}{3}\\right) > 0$ である必要があります。ただこれは独立ではなさそうです。この式を見やすくすると、\n",
    "$ λ_{1}^{2} + λ_{2}^{2} + λ_{3}^{2} + λ_{4}^{2} + λ_{5}^{2} + λ_{6}^{2} + λ_{7}^{2} + λ_{8}^{2} < \\frac{4}{3}$となり、半径$\\frac{2}{\\sqrt{3}}$の球の中にあると見れます。これはブロッホ球の拡張のような式です。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "0bfe2abd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{λ_{3}}{2} + \\frac{\\sqrt{3} λ_{8}}{6} + \\frac{1}{3} & \\frac{λ_{1}}{2} - \\frac{i λ_{2}}{2}\\\\\\frac{λ_{1}}{2} + \\frac{i λ_{2}}{2} & - \\frac{λ_{3}}{2} + \\frac{\\sqrt{3} λ_{8}}{6} + \\frac{1}{3}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[λ3/2 + sqrt(3)*λ8/6 + 1/3,              λ1/2 - I*λ2/2],\n",
       "[            λ1/2 + I*λ2/2, -λ3/2 + sqrt(3)*λ8/6 + 1/3]])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ρ3_2=ρ3\n",
    "ρ3_2.col_del(2)\n",
    "ρ3_2.row_del(2)\n",
    "ρ3_2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "7407b356-bfa3-45e7-a436-c7dfc15768a7",
   "metadata": {},
   "outputs": [],
   "source": [
    "ρ3_2_e = ρ3_2 + eye(2)*e"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "f9079248",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\epsilon^{2} + \\frac{\\sqrt{3} \\epsilon λ_{8}}{3} + \\frac{2 \\epsilon}{3} - \\frac{λ_{1}^{2}}{4} - \\frac{λ_{2}^{2}}{4} - \\frac{λ_{3}^{2}}{4} + \\frac{λ_{8}^{2}}{12} + \\frac{\\sqrt{3} λ_{8}}{9} + \\frac{1}{9}$"
      ],
      "text/plain": [
       "epsilon**2 + sqrt(3)*epsilon*λ8/3 + 2*epsilon/3 - λ1**2/4 - λ2**2/4 - λ3**2/4 + λ8**2/12 + sqrt(3)*λ8/9 + 1/9"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ρ3_2_e.det()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "16eb995e-2867-4148-b24c-9e2245cab302",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 36 \\epsilon^{2} + \\epsilon \\left(12 \\sqrt{3} λ_{8} + 24\\right) - 9 λ_{1}^{2} - 9 λ_{2}^{2} - 9 λ_{3}^{2} + 3 λ_{8}^{2} + 4 \\sqrt{3} λ_{8} + 4$"
      ],
      "text/plain": [
       "36*epsilon**2 + epsilon*(12*sqrt(3)*λ8 + 24) - 9*λ1**2 - 9*λ2**2 - 9*λ3**2 + 3*λ8**2 + 4*sqrt(3)*λ8 + 4"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "collect(ρ3_2_e.det()*4*9,e)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a3dec98c-114f-4872-9612-305b7e519777",
   "metadata": {},
   "source": [
    "上の式ですべての$\\varepsilon$で成立させるには、$-9 \\lambda_{1}^{2}-9 \\lambda_{2}^{2}-9 \\lambda_{3}^{2}+3 \\lambda_{8}^{2}+4 \\sqrt{3} \\lambda_{8}+4 > 0$が必要、または$-9 \\lambda_{1}^{2}-9 \\lambda_{2}^{2}-9 \\lambda_{3}^{2}+3 \\lambda_{8}^{2}+4 \\sqrt{3} \\lambda_{8}+4 =0$のときは、$12\\sqrt{3}\\lambda_8 + 24 >0$が必要となります。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e24e5766-e188-4910-b34f-532d0af521c8",
   "metadata": {},
   "source": [
    "これらが独立であるかは簡単にはわかりません。実際、一番初めに求めた式の$\\lambda_4, \\lambda_5, \\lambda_6, \\lambda_7$に$0$を代入してみると、"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "6ae9a3dd-a84d-4768-a3b9-210bbc02591f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 9 \\sqrt{3} λ_{1}^{2} λ_{8} - 9 λ_{1}^{2} + 9 \\sqrt{3} λ_{2}^{2} λ_{8} - 9 λ_{2}^{2} + 9 \\sqrt{3} λ_{3}^{2} λ_{8} - 9 λ_{3}^{2} - 3 \\sqrt{3} λ_{8}^{3} - 9 λ_{8}^{2} + 4$"
      ],
      "text/plain": [
       "9*sqrt(3)*λ1**2*λ8 - 9*λ1**2 + 9*sqrt(3)*λ2**2*λ8 - 9*λ2**2 + 9*sqrt(3)*λ3**2*λ8 - 9*λ3**2 - 3*sqrt(3)*λ8**3 - 9*λ8**2 + 4"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ρ3_det.subs(λ4,0).subs(λ5,0).subs(λ6,0).subs(λ7,0)/2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4df36787-456f-462a-b29e-be048f1a5ac3",
   "metadata": {},
   "source": [
    "となり、かなり似た式が出てきて、多分独立ではなさそうな雰囲気ですが、確固としたことは言えません。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "24c72a0b-61c4-4da0-9d31-d95a7f571cf1",
   "metadata": {},
   "source": [
    "最後の首座行列式は一つの不等式です。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "9a2edaaf-c82a-474d-a745-e87626b9fcd9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{λ_{3}}{2} + \\frac{\\sqrt{3} λ_{8}}{6} + \\frac{1}{3}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([[λ3/2 + sqrt(3)*λ8/6 + 1/3]])"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ρ3_2\n",
    "ρ3_2.col_del(1)\n",
    "ρ3_2.row_del(1)\n",
    "ρ3_2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "74e52e93-f5a3-476c-bc72-45a8f201dad6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{λ_{3}}{2} + \\frac{\\sqrt{3} λ_{8}}{6} + \\frac{1}{3}$"
      ],
      "text/plain": [
       "λ3/2 + sqrt(3)*λ8/6 + 1/3"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ρ3_2.det()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7926b4ae-a1e5-4060-8115-769234cf3a3b",
   "metadata": {},
   "source": [
    "これも0以上である必要があります。ただ、これも、独立ではなさそうな気がしますが、わかりません。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c0f963a3-4d3e-4d5e-8ee8-211f5404e62b",
   "metadata": {},
   "source": [
    "これは学術論文ではないので、とりあえず簡単にここまでにしておきます。いずれにせよ、わかるのは、2準位系のときの単なる球ではなく、3準位系での密度行列が満たすべき条件、特に半正定値条件は複雑な$\\lambda$たちの多項式の不等式になるということです。さらに、求めた不等式たちは正しいのですが、独立かどうかはわからないということです。おそらく独立ではないと思います。いずれにせよ、どの不等式も満たす必要はありますし、満たした暁にはその密度行列は、3準位系の密度行列であるということではあります。"
   ]
  }
 ],
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