Ergodic theory of diagonal orthogonal covariant quantum channels

TL;DR

This paper investigates the ergodic properties of a class of quantum channels covariant under diagonal orthogonal transformations, linking their behavior to classical stochastic matrices and applying findings to quantum chaos models.

Contribution

It introduces a framework connecting quantum channel ergodicity with classical stochastic matrices and applies this to analyze symmetric dual unitary circuits in quantum chaos.

Findings

Quantum channels' ergodic behavior is governed by classical stochastic matrices.

Symmetric dual unitary circuits show diverse ergodic behaviors.

The approach bridges quantum ergodicity with classical ergodic theory.

Abstract

We analyze the ergodic properties of quantum channels that are covariant with respect to diagonal orthogonal transformations. We prove that the ergodic behaviour of a channel in this class is essentially governed by a classical stochastic matrix. This allows us to exploit tools from classical ergodic theory to study quantum ergodicity of such channels. As an application of our analysis, we study dual unitary brickwork circuits which have recently been proposed as minimal models of quantum chaos in many-body systems. Upon imposing a local diagonal orthogonal invariance symmetry on these circuits, the long-term behaviour of spatio-temporal correlations between local observables in such circuits is completely determined by the ergodic properties of a channel that is covariant under diagonal orthogonal transformations. We utilize this fact to show that such symmetric dual unitary circuits exhibit a rich variety of ergodic behaviours, thus emphasizing their importance.