Proposed Semantics for Qobj.__call__.ipynb

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   "source": [
    "# Proposed Semantics for Qobj.\\_\\_call\\_\\_ #"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This Notebook describes proposed semantics for representing the actions of superoperators and operators on operators and states as *functions*. In particular, this will greatly simplify the syntax used to represent evolving a density operator under a quantum channel stored as a ``type=\"super\"`` Qobj."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import qutip as qt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A operator $X$ acting on a Hilbert space $\\mathcal{H}$ is a representation of a function $f_X(\\left|\\psi\\right\\rangle) = X\\left|\\psi\\right\\rangle$ such that $f_X(\\left|\\psi\\right\\rangle + \\left|\\phi\\right\\rangle) = f_X(\\left|\\psi\\right\\rangle) + f_X(\\left|\\phi\\right\\rangle)$ and such that for any scalar $\\alpha$, $f_X(\\alpha \\left|\\psi\\right\\rangle) = \\alpha f_X(\\left|\\psi\\right\\rangle)$. Similarly, given a function $g_Y(\\rho)$ such that $g_Y(\\rho + \\sigma) = g_Y(\\rho) + g_Y(\\sigma)$ and $g_Y(\\alpha \\rho) = \\alpha g_Y(\\rho)$, we can represent the action of $g_Y$ as a superoperator $Y$.\n",
    "\n",
    "In QuTiP, superoperators $Y$ are represented by ``type=\"super\"`` Qobj instances with ``superrep in ['super', 'choi', 'chi']``. For example, a map $g(\\rho) = (\\sigma_x \\rho \\sigma_x + \\rho) / 2$ representing dephasing in the $x$-direction corresponds to the following Qobj:"
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     "data": {
      "text/latex": [
       "Quantum object: dims = [[[2], [2]], [[2], [2]]], shape = [4, 4], type = super, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}0.500 & 0.0 & 0.0 & 0.500\\\\0.0 & 0.500 & 0.500 & 0.0\\\\0.0 & 0.500 & 0.500 & 0.0\\\\0.500 & 0.0 & 0.0 & 0.500\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[[2], [2]], [[2], [2]]], shape = [4, 4], type = super, isherm = True\n",
       "Qobj data =\n",
       "[[ 0.5  0.   0.   0.5]\n",
       " [ 0.   0.5  0.5  0. ]\n",
       " [ 0.   0.5  0.5  0. ]\n",
       " [ 0.5  0.   0.   0.5]]"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dephasing_superop = (qt.to_super(qt.sigmax())  + qt.to_super(qt.qeye(2))) / 2\n",
    "dephasing_superop"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This Qobj then acts on vectorized operators, $|\\rho\\rangle\\!\\rangle$, as given by the ``operator_to_vector`` function. Thus, to apply this superoperator to a density operator requires calling ``operator_to_vector`` and the corresponding ``vector_to_operator``:"
   ]
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  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}0.500 & 0.187\\\\0.187 & 0.500\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[ 0.5         0.18655508]\n",
       " [ 0.18655508  0.5       ]]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rho = qt.rand_dm_ginibre(2)\n",
    "rho_out = qt.vector_to_operator(dephasing_superop * qt.operator_to_vector(rho))\n",
    "rho_out"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is confusing to newcomers to the library, and while it accurately models the Liouvillian representation (that is, ``superrep=\"super\"``), does not model the more general view of a channel as an abstract object. For instance, $\\mathcal{U}(\\rho) = U \\rho U^\\dagger$ (as returned by ``qt.to_super(U)``) is already a quantum channel representing conjugation by a unitary $U$, and need not be written as a superoperator. Using ``__call__``, the group action of superoperators and operators on operators can be modeled in a uniform manner."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here, we prototype by introducing a new method and monkey-patching it into ``Qobj``."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
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   "outputs": [],
   "source": [
    "def qobj_call(self, other):\n",
    "    if not isinstance(other, qt.Qobj):\n",
    "        raise TypeError(\"Only defined for quantum objects.\")\n",
    "    \n",
    "    if self.type == \"super\":\n",
    "        if other.type == \"ket\":\n",
    "            other = qt.ket2dm(other)\n",
    "        elif other.type == \"oper\":\n",
    "            return qt.vector_to_operator(self * qt.operator_to_vector(other))\n",
    "        else:\n",
    "            raise TypeError(\"Can only act super on oper or ket.\")\n",
    "        \n",
    "    elif self.type == \"oper\":\n",
    "        if other.type == \"ket\":\n",
    "            return self * other\n",
    "        elif other.type == \"oper\":\n",
    "            return self * other * self.dag()\n",
    "        else:\n",
    "            raise TypeError(\"Can only act oper on oper or ket.\")\n",
    "\n",
    "qt.Qobj.__call__ = qobj_call"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "psi = qt.rand_ket_haar(2)\n",
    "rho = qt.rand_dm_ginibre(2)\n",
    "X = qt.sigmax()\n",
    "I = qt.qeye(2)\n",
    "S = (qt.to_super(I) + qt.to_super(X)) / 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We finally assert a few usecases of these new semantics."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
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   "source": [
    "assert X(psi) == X * psi\n",
    "assert X(rho) == X * rho * X.dag()\n",
    "assert S(rho) == qt.vector_to_operator(S * qt.operator_to_vector(rho))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
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