{ "cells": [ { "cell_type": "markdown", "metadata": { "tags": [ "hide_input" ] }, "source": [ "$\\newcommand{\\Mn}{M_n(\\mathbb{C})}$\n", "$\\newcommand{\\Mq}{M_2(\\mathbb{C})}$\n", "$\\newcommand{\\Mk}{M_k(\\mathbb{C})}$\n", "$\\newcommand{\\Mm}{M_m(\\mathbb{C})}$\n", "$\\newcommand{\\Mnm}{M_{n \\times m}(\\mathbb{C})}$\n", "$\\newcommand{\\Mnp}{M_n^+(\\mathbb{C})}$\n", "$\\newcommand{\\ra}{\\,\\rightarrow\\,}$\n", "$\\newcommand{\\id}{\\mbox{id} }$\n", "$\\newcommand{\\ot}{ {\\,\\otimes\\,} }$\n", "$\\newcommand{\\Cd}{ {\\mathbb{C}^d} }$\n", "$\\newcommand{\\Rn}{ {\\mathbb{R}^n} }$\n", "$\\newcommand{\\asterisk}{*}$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We have discussed how to describe the quantum evolution of open quantum systems and we have introduced the concept of dynamical map $\\Lambda_t$ and its time-dependent generator $L_t$. The latter quantity is also known as the dissipator and describes the form of the master equation. The connection between the two is given by Eq. ([2.29](preliminaries.html#mjx-eqn-Lambda-t)). A very important problem in the theory of open quantum systems is the following:\n", " \n", "**Problem 1**\n", "\n", "> What are the properties of the local time-dependent generator $L_t$ that guarantee $\\Lambda_t$, as defined by the T-product exponential formula ([2.29](preliminaries.html#mjx-eqn-Lambda-t)), defines a legitimate dynamical map?\n", "\n", "The formulation of our problem is pretty simple, but in general the answer is not known. In fact, it turns out that the answer is only known for the special case of time-independent generators, giving rise to Markovian semigroups. In this case the answer to this important problem is given by the GKSL theorem discussed in **[Chapter 4](markovian_semigroups.html)**. But what about more general types of generators? " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us observe that if we knew a dynamical map $\\Lambda_t$ that was invertible, i.e. there exists $\\Lambda_t^{-1} : \\Mn \\ra \\Mn$ such that $\\Lambda_t^{-1} \\Lambda_t = \\Lambda_t \\Lambda_t^{-1} = \\mathbb{1}_n$, then\n", "\n", "\\begin{equation}\n", " \\dot{\\Lambda}_t = \\dot{\\Lambda}_t \\Lambda_t^{-1} \\Lambda_t = L_t \\, \\Lambda_t\\ ,\n", "\\tag{5.1}\n", "\\end{equation}\n", "\n", "where we defined\n", "\\begin{equation}\n", " L_t := \\dot{\\Lambda}_t \\Lambda_t^{-1} \\ .\n", "\\tag{5.2}\n", "\\end{equation}\n", "\n", "It should be stressed that the inverse of $\\Lambda_t$ need not be CP. One may prove that if $\\Lambda_t$ is CP then $\\Lambda_t^{-1}$ is CP if and only if $\\Lambda_t(\\rho) = U_t \\rho U_t^\\dagger$ with unitary $U_t$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Example 6 " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "> Consider a unitary dynamical map $\\mathcal{U}_t$ defined in ([2.23](preliminaries.html#mjx-eqn-UUU)). It is clear that $\\mathcal{U}_t$ is invertible and $\\mathcal{U}_t^{-1} = \\mathcal{U}_{-t}$ is CP. One finds for the corresponding generator\n", "> \\begin{equation}\n", " L_t(\\rho) = [\\dot{\\mathcal{U} }_t \\mathcal{U}_{-t}](\\rho) = \\dot{U}_t (U_t^\\dagger \\rho U_t) U_t^\\dagger + {U}_t (U_t^\\dagger \\rho U_t) \\dot{U}_t^\\dagger \\ ,\n", "\\tag{5.3}\n", "\\end{equation}\n", "> \n", "> and hence recalling that $U_t$ satisfies the Schrödinger equation $\\dot{U}_t = -iHU_t$, one obtains $L_t(\\rho) = -i[H,\\rho]$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In this course instead of analyzing this problem in full generality we restrict ourselves to study special classes of dynamical maps and corresponding local generators.\n", "Specifically we analyze 3 important classes of generators\n", "\n", "- $C_1$ - a class of time-independent generators giving rise to Markovian semigroups,\n", "- $C_2$ - a class of time-dependent generators giving rise to commutative dynamics,\n", "- $C_3$ - a class of time-dependent generators giving rise to the so-called divisible dynamical maps." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 5.1. Commutative dynamics " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We call a dynamical map $\\Lambda_t$ commutative if $[\\Lambda_t,\\Lambda_u]=0$ for all $t,u\\geq 0$. It means that for each $A\\in \\Mn$ one has\n", "\n", "\\begin{equation}\n", " \\Lambda_t(\\Lambda_u(A)) = \\Lambda_u(\\Lambda_t(A)) \\ .\n", "\\tag{5.4}\n", "\\end{equation}\n", "\n", "It is easy to show that commutativity of $\\Lambda_t$ is equivalent to commutativity of the local generator\n", "\n", "\\begin{equation}\\label{[]=0}\n", " [L_t,L_u]=0\\ ,\n", "\\tag{5.5}\n", "\\end{equation}\n", "\n", "for any $t,u \\geq 0$. Note that in this case the formula ([2.30](preliminaries.html#mjx-eqn-Dyson)) considerably simplifies: the 'T' product drops out and the solution is fully controlled by the integral $\\int_0^t L_u du$:\n", "\\begin{equation}\\label{Dyson-com}\n", " \\Lambda_t = \\exp\\left( \\int_0^t L_u du\\right) = \\mathbb{1}_n + \\int_0^t L_{u}du + \\frac 12 \\left( \\int_0^{t} L_{u}du \\right)^2 + \\ldots \\ .\n", "\\tag{5.6}\n", "\\end{equation}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now, it follows from Theorem 3 that if $\\Lambda = e^M$, then $\\Lambda$ is a quantum channel if $M$ is a GKSL generator. Therefore, one has the following\n", "\n", "**Theorem 4**\n", "\n", "> If $L_t$ satisfies (\\ref{[]=0}), then $L_t$ is a legitimate generator if $\\int_0^t L_\\tau d\\tau$ is a GKSL generator for all $t\\geq 0$.\n", "\n", "Note that, if $L_t = L$ is time independent, then $\\int_0^t L_udu = tL$ and the above theorem reproduces **Theorem 3**." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "It is clear that if $L$ is a legitimate GKSL generator and $f: \\mathbb{R}_+ \\ra \\mathbb{R}$ an arbitrary function, then $L_t = f(t) L$ generates a commutative dynamical map $\\Lambda_t$ iff $\\int_0^t f(u) du \\geq 0$ for all $t\\geq 0$. A typical example of commutative dynamics is provided by\n", "\n", "\\begin{equation}\n", " L_t = \\omega(t) L_0 + a_1(t) L_1 + \\ldots + a_N(t) L_N\\ ,\n", "\\tag{5.7}\n", "\\end{equation}\n", "\n", "where $[L_\\alpha,L_\\beta]=0$ with $L_0(\\rho) = -i[H,\\rho]$, and for $\\alpha>0$ the generators $L_\\alpha$ are purely dephasing, that is, $L_\\alpha(\\rho) = \\Phi_\\alpha(\\rho) - \\frac 12 \\{ \\Phi_\\alpha^*(\\mathbb{I}),\\rho\\}$. One has for the corresponding dynamical map\n", "\n", "\\begin{equation}\n", " \\Lambda_t = e^{\\Omega(t)L_0} \\cdot e^{A_1(t)L_1} \\cdot \\ldots\\cdot e^{A_N(t)L_N}\\ ,\n", "\\tag{5.8}\n", "\\end{equation}\n", "\n", "with\n", "$$\\Omega(t) = \\int_0^t \\omega(u)du\\ ; \\ \\ \\ \\ A_\\alpha(t) = \\int_0^t a_\\alpha(u)du\\ .$$\n", "\n", "It is clear that $\\Lambda_t$ is CP iff $A_\\alpha(t) \\geq 0$ for all $\\alpha=1,\\ldots,N$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Example 7 " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "> Consider a qubit generator $L_0(\\rho) = -i[\\sigma_3,\\rho]$ together with $L_1,L_2,L_3$ defined in ([5.27](#LLL)). One easily proves\n", "> \n", "> \\begin{equation}\n", " [L_0,L_\\alpha] = [L_3,L_\\alpha]=0 \\ ; \\ \\ \\ \\alpha=1,2,3\\ ,\n", "\\tag{5.9}\n", "\\end{equation}\n", "> \n", "> and\n", "> \\begin{equation}\\label{L-L}\n", " [L_1,L_2] = L_1 - L_2 \\ .\n", "\\tag{5.10}\n", "\\end{equation}\n", "> \n", "> Define the time-dependent commutative generator\n", "> \n", "> \\begin{equation}\n", " L_t = \\frac{\\omega(t)}{2} L_0 + \\frac{\\delta(t)}{2} ( \\mu_1 L_1 + \\mu_2 L_2) + \\frac{\\gamma(t)}{2} L_z\\ ,\n", "\\tag{5.11}\n", "\\end{equation}\n", "> \n", "> with $\\mu_1,\\mu_2 \\geq 0$ and $\\mu_1 + \\mu_2=1$. Defining\n", "> \n", "> \\begin{equation}\n", " \\Omega(t) = \\int_0^t \\omega(u)du\\ ; \\ \\ \\Delta(t) = \\int_0^t \\delta(u)du\\ ; \\ \\ \\Gamma(t) = \\int_0^t \\gamma(u)du\\ ,\n", "\\tag{5.12}\n", "\\end{equation}\n", "> \n", "> one finds that if $\\Delta(t) \\geq 0$ and $\\Gamma(t) \\geq 0$, then $L_t$ is a legitimate generator.\n", "The time evolution of $\\rho$ has the following form: the off-diagonal elements evolve according to\n", "> \n", "> $$\\rho_{12} \\ra e^{\\Omega(t) + \\frac 12 \\Delta(t) + \\Gamma(t)} \\rho_{12}\\ ,$$\n", ">\n", "> and diagonal elements\n", "> \n", "> \\begin{eqnarray*}\n", " \\rho_{11} &\\ra & \\rho_{11}\\, e^{-\\Delta(t)} + \\mu_1 \\Big[ 1 - e^{-\\Delta(t)} \\Big] \\ ,\\\\\n", " \\rho_{22} &\\ra & \\rho_{22}\\, e^{-\\Delta(t)} + \\mu_2 \\Big[ 1 - e^{-\\Delta(t)} \\Big] \\ .\n", "\\end{eqnarray*}\n", ">\n", "> If $\\Delta(t) \\ra \\infty$ for $t\\ra \\infty$, then the dynamics possess an equilibrium state\n", ">\n", "> $$\\rho_t \\ \\ \\longrightarrow \\ \\ \\left( \\begin{array}{cc} \\mu_1 & 0 \\\\ 0 & \\mu_2 \\end{array} \\right) \\ .$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Example 8 (Random unitary qubit dynamics) " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "> Consider the following time-dependent generator\n", "> \\begin{equation}\\label{Pauli}\n", " L_t(\\rho) = \\frac 12 \\sum_{k=1}^3 \\gamma_k(t) (\\sigma_k \\rho\\, \\sigma_k - \\rho)\\ ,\n", "\\tag{5.13}\n", "\\end{equation}\n", "> \n", "> where $\\{\\sigma_1,\\sigma_2,\\sigma_3\\}$ are Pauli matrices. It is easy to prove that $[L_t,L_u]=0$ and hence $L_t$ generates a legitimate dynamical map iff\n", "> \n", "> $$\\Gamma_1(t) \\geq 0 \\ ; \\ \\ \\Gamma_2(t) \\geq 0 \\ ; \\ \\ \\Gamma_3(t) \\geq 0 \\ ,$$\n", "> \n", "> where $\\Gamma_k(t) = \\int_0^t \\gamma_k(u)du$. One finds that the corresponding dynamical map $\\Lambda_t$ is given by\n", "> \n", "> \\begin{equation}\\label{RU}\n", " \\Lambda_t(\\rho) = \\sum_{\\alpha=0}^3 p_\\alpha(t) \\sigma_\\alpha \\rho\\, \\sigma_\\alpha \\ ,\n", "\\tag{5.14}\n", "\\end{equation}\n", "> \n", "> where $\\sigma_0 = \\mathbb{I}_2$ and\n", "> \n", "> \\begin{eqnarray*}\n", "p_0(t) &=&\\frac{1}{4}\\, [1+ \\lambda_3(t) + \\lambda_2(t) + \\lambda_1(t)] \\ , \\\\\n", "p_1(t) &=&\\frac{1}{4}\\, [1- \\lambda_3(t) - \\lambda_2(t) + \\lambda_1(t)] \\ ,\\\\\n", "p_2(t) &=&\\frac{1}{4}\\, [1- \\lambda_3(t) + \\lambda_2(t) - \\lambda_1(t)] \\ ,\\\\\n", "p_3(t) &=&\\frac{1}{4}\\, [1+ \\lambda_3(t) - \\lambda_2(t) - \\lambda_1(t)] \\ ,\n", "\\end{eqnarray*}\n", "> \n", "> with\n", "> $$\\lambda_1(t) = e^{-\\Gamma_2(t) - \\Gamma_3(t)}\\ ,$$\n", "> \n", "> and similarly for $\\lambda_2(t)$ and $\\lambda_3(t)$. Interestingly $\\Lambda_t(\\sigma_k)=\\lambda_k(t)\\sigma_k$.\n", "The formula (\\ref{RU}) defines so-called random unitary dynamics. Note that\n", "> \n", "> $$p_0(t) + p_1(t) + p_2(t) + p_3(t)=1 \\ .$$\n", "> \n", "> Moreover, $p_\\alpha(t) \\geq 0$ for $\\alpha=0,1,2,3$ iff $\\Gamma_k(t) \\geq 0$ for $k=1,2,3$. Note that $\\Lambda_t$ is unital. Actually, in the case of qubits any unital dynamical map is random unitary, i.e.\n", "> \n", "> \\begin{equation}\\label{}\n", " \\Lambda_t(\\rho) = \\sum_k p_k(t) U_k(t) \\rho U_k^\\dagger(t)\\ ,\n", "\\tag{5.15}\n", "\\end{equation}\n", "> \n", "> where $p_k(t)$ defines time-dependent probability distribution and $U_k(t)$ is a family of time-dependent unitary matrices. It is no longer true for $n$-level systems with $n>2$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## IBM Q example: Pauli channel " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The most general single-qubit open quantum system model is the time-dependent Pauli channel. The master equation in this case takes the form (see Eq. (\\ref{Pauli}))\n", "\n", "\\begin{equation}\n", "\\frac{d\\rho_{S} }{dt}(t)=\\frac{1}{2}\\sum_i\\gamma_i(t)\\left[\\sigma_i\\rho_{S}(t)\\sigma_i-\\rho_{S}(t)\\right].\n", "\\label{PauliME}\n", "\\end{equation}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Generally, the dynamics described by the master equation above is not phase-covariant [(#1)], except for the case in which $\\gamma_x(t)=\\gamma_y(t)$. Moreover, since the decay rates may take negative values, conditions for complete positivity must be imposed, and they are given in terms of a set of inequalities involving all the three decay rates, as one can see, e.g., from Ref. [(#2)].\n", "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "At a specific time instant $t$, the Pauli channel can be written as\n", "\\begin{equation} \n", " \\mathcal{E} (\\rho) = \\sum_{i=0}^3 p_i \\sigma_i \\rho \\sigma_i,\n", "\\end{equation}\n", " \n", "with $0 \\leq p_i \\leq 1$ and $\\sum_i p_i = 1$. The depolarizing channel is a special case of the Pauli channel where $p_1 = p_2 = p_3 = p/4$. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "It is possible to implement the general Pauli channel with just two ancillary qubits, by preparing them in the suitable entangled state. The first qubit acts as the control for a controlled-$X$ (CNOT) gate, and the second one for a controlled-$Y$. Notice that applying both an X and a Y gates is effectively equivalent to applying a Z gate." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The state $|\\psi \\rangle$ of the ancillae needed for the Pauli channel can be implemented by the following circuit:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "output_type": "execute_result", "data": { "text/plain": [ "