On limiting distributions of quantum Markov chains

TL;DR

This paper investigates the long-term behavior of quantum Markov chains, proving the existence of Cesàro limits and convergence conditions, and offers a new derivation of classical results on limiting distributions of quantum walks.

Contribution

It establishes the existence of Cesàro limits for quantum Markov chains and characterizes their convergence properties, providing a new derivation of known results on quantum walk distributions.

Findings

Cesàro limit always exists and equals the orthogonal projection onto the eigenspace of the eigenvalue 1.

Convergence occurs if the eigenvalue 1 is the only eigenvalue on the unit circle.

Provides a new derivation of classical results on limiting distributions of quantum walks.

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 Ces$\grave{a}$ro 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 \cite{AAKV01}.