Asymptotic properties of quantum Markov chains

TL;DR

This paper analyzes the long-term behavior of quantum Markov chains, showing they converge to an attractor space and providing a basis construction method that simplifies understanding their asymptotic dynamics.

Contribution

It introduces a basis construction for the attractor space of quantum Markov chains, linking asymptotic behavior to fixed points of quantum operations, with potential computational benefits.

Findings

Quantum Markov chains are confined to an attractor space.

A basis for the attractor space can be constructed when a positive invariant state exists.

The construction offers computational advantages for large-dimensional systems.

Abstract

The asymptotic dynamics of quantum Markov chains generated by the most general physically relevant quantum operations is investigated. It is shown that it is confined to an attractor space on which the resulting quantum Markov chain is diagonalizable. A construction procedure of a basis of this attractor space and its associated dual basis is presented. It applies whenever a strictly positive quantum state exists which is contracted or left invariant by the generating quantum operation. Moreover, algebraic relations between the attractor space and Kraus operators involved in the definition of a quantum Markov chain are derived. This construction is not only expected to offer significant computational advantages in cases in which the dimension of the Hilbert space is large and the dimension of the attractor space is small but it also sheds new light onto the relation between the asymptotic dynamics of quantum Markov chains and fixed points of their generating quantum operations.