Defining stable phases of open quantum systems

TL;DR

This paper introduces a new stability criterion called uniformity for steady states in open quantum systems, providing a more reliable way to identify stable phases beyond spectral analysis.

Contribution

It proposes the uniformity criterion for stability of steady states, demonstrating its implications and providing evidence of its effectiveness in classical and quantum systems.

Findings

Uniformity implies desirable phase properties.

Numerical evidence links spectral gap to relaxation rates.

Canonical classical automaton satisfies uniformity.

Abstract

The steady states of dynamical processes can exhibit stable nontrivial phases, which can also serve as fault-tolerant classical or quantum memories. For Markovian quantum (classical) dynamics, these steady states are extremal eigenvectors of the non-Hermitian operators that generate the dynamics, i.e., quantum channels (Markov chains). However, since these operators are non-Hermitian, their spectra are an unreliable guide to dynamical relaxation timescales or to stability against perturbations. We propose an alternative dynamical criterion for a steady state to be in a stable phase, which we name uniformity: informally, our criterion amounts to requiring that, under sufficiently small local perturbations of the dynamics, the unperturbed and perturbed steady states are related to one another by a finite-time dissipative evolution. We show that this criterion implies many of the properties one would want from any reasonable definition of a phase. We prove that uniformity is satisfied in a canonical classical cellular automaton, and provide numerical evidence that the gap determines the relaxation rate between nearby steady states in the same phase, a situation we conjecture holds generically whenever uniformity is satisfied. We further conjecture some sufficient conditions for a channel to exhibit uniformity and therefore stability.