Ergodicity of quantum cellular automata

TL;DR

This paper introduces a class of quantum cellular automata, establishes criteria for their ergodicity, and links disorder in local operators to unique invariant states and decay of correlations.

Contribution

It develops a quantum analogue of classical oscillation norms and provides new ergodicity criteria for quantum cellular automata.

Findings

Ergodicity is achieved with sufficiently large local disorder.

Unique invariant states exhibit exponential decay of correlations.

Quantum oscillation norms are effective tools for analyzing ergodicity.

Abstract

We define a class of dynamical maps on the quasi-local algebra of a quantum spin system, which are quantum analogues of probabilistic cellular automata. We develop criteria for such a system to be ergodic, i.e., to possess a unique invariant state. Intuitively, ergodicity obtains if the local transition operators exhibit sufficiently large disorder. The ergodicity criteria also imply bounds for the exponential decay of correlations in the unique invariant state. The main technical tool is a quantum version of oscillation norms, defined in the classical case as the sum over all sites of the variations of an observable with respect to local spin-flips.