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Open Systems & Information Dynamics

A Limit Theorem for Quantum Markov Chains Associated to Beam Splittings

Volkmar Liebscher

Published Sep 1, 2001• 0 citations
Publisher

Abstract

We study special quantum Markov chains on a Fock space related to iterated beam splittings as introduced in [23]. Besides a characterization of the position distributions of the chain, we show some kind of weak convergence of such discrete time quantum Markov chains to a kind of continuous time quantum Markov process. Furthermore, we provide existence and uniqueness for the solution of a quantum stochastic differential equation related to this quantum Markov process both on an exponential domain and, on a larger domain, in a pointwise sense.

DOI
10.1023/a:1014029322502

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We develop a rather general framework for constructing quantum Markov processes through Markov operator cocycles (see Chapter 2, Section 3) that satisfy a quantum stochastic differential equation.In order to achieve this goal we first recall the basic facts of Boson Fock quantum stochastic calculus and then give the fundamental results in the theory of quantum stochastic differential equations concerning existence, uniqueness, time reversal, isometricity and coisometricity of solutions. Next we construct the quantum flow associated with a Markov operator cocycle, and give a condition that guarantees that the restriction to a commutative subalgebra is a commutative flow.
Many examples of quantum independent stationary increment processes are solutions of quantum stochastic differential equations. We give a common characterization of these examples by a quantum stochastic differential equation on an abstract *-bialgebra. Specializing this abstract *-bialgebra and the coefficients of the equation, we obtain the equations for the Unitary Noncommutative Stochastic processes of [12], the Quantum Wiener Process [2], the Azéma martingales [11] and for other examples. The existence and uniqueness of a solution of the general equation is shown. Assuming the boundedness of this solution, we prove that it is a continuous and stationary independent increment process.
"Quantum trajectories" are solutions of stochastic differential equations also called Belavkin or Stochastic Schrödinger Equations. They describe random phenomena in quantum measurement theory. Two types of such equations are usually considered, one is driven by a one-dimensional Brownian motion and the other is driven by a counting process. In this article, we present a way to obtain more advanced models which use jump-diffusion stochastic differential equations. Such models come from solutions of martingale problems for infinitesimal generators. These generators are obtained from the limit of generators of classical Markov chains which describe discrete models of quantum trajectories. Furthermore, stochastic models of jump-diffusion equations are physically justified by proving that their solutions can be obtained as the limit of the discrete trajectories.
Abstract Continuous-time quantum walks (CTQWs) are a fundamental tool in the development of quantum algorithms. Recently, it was shown that discretizations of p -adic Schrödinger equations give rise to continuous-time quantum Markov chains (CTQMCs); this type of Markov chain includes the CTQWs constructed using adjacency matrices of graphs as a particular case. Unlike classical continuous-time Markov chains (CTMCs), the CTQMCs studied here do not satisfy the classical Chapman–Kolmogorov equation; this is a genuine quantum effect, due to the fact that the underlying unitary evolution composes at the level of probability amplitudes rather than probabilities, and we make precise the sense in which the term ‘quantum Markov chain’ is used. In this paper, we study a large class of p -adic Schrödinger equations and the associated CTQMCs by comparing them with p -adic heat equations and the associated CTMCs. The comparison is done by a mathematical study of the mentioned equations, which requires, for instance, solving the initial value problems attached to the mentioned equations, and through numerical simulations. We conducted multiple simulations, including numerical approximations of the limiting distribution. Our simulations show that the limiting distribution of quantum Markov chains is greater than the stationary probability of their classical counterparts, for a large class of CTQMCs.
We established a continuous-time framework for entangled quantum Markov chains derived from classical continuous-time Markov processes. Unlike previous studies focused on discrete-time dynamics, we demonstrated that a robust mixing property emerges directly from the structural architecture of the underlying continuous-time process. Our analysis showed that inhomogeneous entangled chains exhibited path-independent convergence to a limiting distribution determined by asymptotic temporal parameters. The results motivated future study of these entangled Markov chains in connection with open quantum system dynamics and quantum master equations.