On Limiting Distributions of Quantum Markov Chains

Abstract

In a quantum Markov chain, the temporal succession of states is modeled by the repeated action of a "bistochastic quantum operation" on the density matrix of a quantum system. Based on this conceptual framework, we derive some new results concerning the evolution of a quantum system, including its long-term behavior. Among our findings is the fact that the Cesàro limit of any quantum Markov chain always exists and equals the orthogonal projection of the initial state upon the eigenspace of the unit eigenvalue of the bistochastic quantum operation. Moreover, if the unit eigenvalue is the only eigenvalue on the unit circle, then the quantum Markov chain converges in the conventional sense to the said orthogonal projection. As a corollary, we offer a new derivation of the classic result describing limiting distributions of unitary quantum walks on finite graphs (Aharonov et al., 2001 [1]).

Unitary conjugation channels with continuous random phases

“…Assume that Φ U θ ρ = λρΦ U θ , a.s., one can derive that λ = 1 and ρ = I d via the same reasoning as employed in the proof of Theorem 1. Then the conclusion follows upon applying Theorem 7 in [7].…”

Decomposition of quantum Markov chains and its applications

“…A special case of Lemma 14 where E is unital (that is, E(I) = I) was proved in [27]. The following lemma further deals with the case when the limiting state is unique.…”

Quantum Markov chains: Description of hybrid systems, decidability of equivalence, and model checking linear-time properties

“…In the literature, there are several notions of quantum Markov chains defined from different perspectives, but in this paper we focus mainly on the one reported in [9,8]. In the following, we show that the quantum Markov model given in [9,8] is very suited to describe hybrid systems, although it has the same expressive power as the conventional one given in [17,28]. Before that, we recall some necessary definitions below.…”

“…Then it is natural to study the quantum analogue of Markov chains. Actually, the terminology "quantum Markov chains" have appeared many times in the literature [1,2,9,8,17,28], although it does not mean exactly the same thing in different references. A usual approach to defining quantum Markov chains is to view a quantum Markov chain as a pair (E, ρ 0 ) where ρ 0 , a density operator, denotes an initial state of a quantum system, and E is a trace-preserving quantum operation that characterizes the dynamics of the quantum system.…”