Asymptotics of quantum Markov processes: From algebraic structure to characterization of asymptotic states

TL;DR

This paper analyzes the long-term behavior of quantum Markov processes, revealing their algebraic structure and characterizing asymptotic states, with implications for understanding open quantum systems.

Contribution

It generalizes the structure theorem for quantum Markov chains to dynamical semigroups and characterizes all asymptotic and stationary states in quantum processes.

Findings

Asymptotic dynamics preserve a scalar product, acting unitarily within the asymptotic space.

The structure of generators determines attractors and asymptotic states.

Asymptotic states resemble Gibbs states in statistical mechanics.

Abstract

Markov processes play an important role in physics and the theory of open systems in particular. In this paper we study the asymptotic evolution of trace-nonincreasing homogenous quantum Markov processes (both types, discrete quantum Markov chains and continues quantum dynamical semigroups) equipped with a so-called strictly positive T -state in the Schrodinger and the Heisenberg picture. We derive a fundamental theorem specifying the structure of the asymptotic and uncover a rich set of transformations between attractors of quantum Markov processes in both pictures. Moreover, we generalize the structure theorem derived for quantum Markov chains to quantum dynamical semigroups showing how the internal structure of generators of quantum Markov processes determines attractors in both pictures. Based on these results we provide two characterizations of all asymptotic and stationary states, both strongly reminding in form the well-known Gibbs states of statistical mechanics. We prove that the dynamics within the asymptotic space is of unitary type, i.e. quantum Markov processes preserve a certain scalar product of operators from the asymptotic space, but there is no corresponding unitary evolution on the original Hilbert space of pure states. Finally simple examples illustrating the derived theory are given.