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    "# Maximally Entangled States\n",
    "$$\n",
    " \\newcommand{\\ul}[1]{\\underline{#1}}\n",
    " \\newcommand{\\rvx}[0]{{\\ul{x}}}\n",
    "\\newcommand{\\rvy}[0]{{\\ul{y}}}\n",
    "$$\n",
    "\n",
    " The purpose of this notebook is to construct a maximally entangled state vector and its\n",
    " corresponding density matrix, and then to calculate the \"entanglement profile\"\n",
    " of that state. \n",
    " \n",
    " In Entaglish, density matrices are stored in the class DenMat.\n",
    " That class contains attributes: num_rows, row_shape and arr.\n",
    " arr is a numpy array of shape=(num_rows, num_rows).\n",
    " row_shape is a tuple such that the product of its components is num_rows.\n",
    " For example, a state with row_shape=(2,3,4) consists of 3 qudits with\n",
    " d=2,3,4 and num_rows=24.\n",
    "\n",
    " \n",
    " See Entanglish-Original-Ref for an explicit definition of the maximally entangled states that we use. The basic requirement for a density matrix $\\rho_{\\rvx, \\rvy}$\n",
    " to be maximally entangled is for its partial trace $\\rho_{\\rvx}={\\rm tr}_\\rvy \\rho_{\\rvx, \\rvy}$ to be a diagonal matrix with all terms in the diagonal equal to the same constant. The sum of the diagonal elements must of course be one. For example, $\\rho_{\\rvx}={\\rm diag}(0.25, 0.25, 0.25, 0.25)$. (If $\\rvx$ and $\\rvy$ have different numbers of possible values, this assumes that\n",
    " $\\rvx$ is the one with the smaller number of values.)\n",
    " \n",
    "Given a state with num_row_axes qudits, one can define\n",
    "a (bipartite) entanglement for each possible bi-partitions of range(\n",
    "num_row_axes). By a bi-partition we mean two nonempty disjoint subsets\n",
    "whose union is range(num_row_axes). An entanglement profile \n",
    "is a dictionary mapping bi-partition half-size to a dictionary that \n",
    "maps each bi-partition of that half-size to its entanglement.\n",
    "\n",
    "**Entanglish-Original-Ref**\n",
    "* \"A New  Algorithm for Calculating\n",
    "Squashed Entanglement and a Python Implementation Thereof\", by R.R.Tucci"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First change your working directory to the entanglish directory in your computer, and add its path to the path environment variable."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "/home/rrtucci/PycharmProjects/Entanglish/entanglish/jupyter_notebooks\n",
      "/home/rrtucci/PycharmProjects/Entanglish\n"
     ]
    }
   ],
   "source": [
    "import os\n",
    "import sys\n",
    "print(os.getcwd())\n",
    "os.chdir('../../')\n",
    "print(os.getcwd())\n",
    "sys.path.insert(0,os.getcwd())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "from entanglish.DenMat import *\n",
    "from entanglish.MaxEntangState import *\n",
    "from entanglish.EntangCase import *\n",
    "from entanglish.PureStEnt import *"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create a DenMat object called dm_max with num_rows=24 and row_shape=(2, 2, 3 ,2).Then\n",
    "set its arr to a maximally entangled state."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "st_vec=\n",
      " [0.57735027+0.j 0.        +0.j 0.        +0.j 0.57735027+0.j\n",
      " 0.        +0.j 0.        +0.j 0.        +0.j 0.        +0.j\n",
      " 0.        +0.j 0.        +0.j 0.57735027+0.j 0.        +0.j\n",
      " 0.        +0.j 0.        +0.j 0.        +0.j 0.        +0.j\n",
      " 0.        +0.j 0.        +0.j 0.        +0.j 0.        +0.j\n",
      " 0.        +0.j 0.        +0.j 0.        +0.j 0.        +0.j]\n",
      "entang= 1.0986122886681098\n"
     ]
    }
   ],
   "source": [
    "dm_max = DenMat(24, (2, 2, 3, 2))\n",
    "max_ent_st = MaxEntangState(dm_max.num_rows, dm_max.row_shape,\n",
    "                            [0, 1, 3], [2])\n",
    "EntangCase.check_max_entang_st(max_ent_st)\n",
    "st_vec = max_ent_st.get_st_vec()\n",
    "entang = max_ent_st.get_known_entang()\n",
    "dm_max.set_arr_from_st_vec(st_vec)\n",
    "\n",
    "print('st_vec=\\n', st_vec)\n",
    "print(\"entang=\", entang)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Class PureStEnt is a child of class EntangCase.\n",
    "\n",
    "All objects with EntangCase as a parent\n",
    "calculate entanglement numerically, from\n",
    "an algorithm, not from a known analytical formula. \n",
    "For a pure state, that algo is the von Neumann entropy of a partial trace of dm.\n",
    "\n",
    "All objects with EntangCase as a parent\n",
    "inherit methods for calculating and printing entanglement profiles"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "bi-partition half-size=1\n",
      "(0 | 1, 2, 3) :\t-0.00000, max-entang= 0.69315\n",
      "(1 | 0, 2, 3) :\t 0.63651, max-entang= 0.69315\n",
      "(2 | 0, 1, 3) :\t 1.09861, max-entang= 1.09861\n",
      "(3 | 0, 1, 2) :\t 0.63651, max-entang= 0.69315\n",
      "bi-partition half-size=2\n",
      "(0, 1 | 2, 3) :\t 0.63651, max-entang= 1.38629\n",
      "(0, 2 | 1, 3) :\t 1.09861, max-entang= 1.38629\n",
      "(0, 3 | 1, 2) :\t 0.63651, max-entang= 1.38629\n",
      "(1, 2 | 0, 3) :\t 0.63651, max-entang= 1.38629\n",
      "(1, 3 | 0, 2) :\t 1.09861, max-entang= 1.38629\n",
      "(2, 3 | 0, 1) :\t 0.63651, max-entang= 1.38629\n",
      "\n"
     ]
    }
   ],
   "source": [
    "ecase = PureStEnt(dm_max, 'eigen')\n",
    "pf = ecase.get_entang_profile()\n",
    "ecase.print_entang_profiles([pf], dm_max.row_shape)"
   ]
  }
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