Ergodic and mixing quantum channels: From two-qubit to many-body quantum systems

TL;DR

This paper explores the ergodic and mixing properties of quantum channels, especially in many-body systems like the SYK model, providing new analytical constructions and conditions for ergodicity and mixing.

Contribution

It introduces a framework for characterizing ergodic hierarchy levels of quantum channels, including analytical constructions from two-qubit to many-body systems, and links these properties to quantum dynamics.

Findings

Quantum channels can be classified into ergodic, mixing, and non-ergodic based on their construction.

Operator entanglement of unitary operators provides sufficient conditions for mixing.

Many-body systems like the SYK model exhibit mixing behavior within the quantum channel framework.

Abstract

The development of classical ergodic theory has had a significant impact in the areas of mathematics, physics, and, in general, applied sciences. The quantum ergodic theory of Hamiltonian dynamics has its motivations to understand thermodynamics and statistical mechanics. Quantum channel, a completely positive trace-preserving map, represents a most general representation of quantum dynamics and is an essential aspect of quantum information theory and quantum computation. In this work, we study the ergodic theory of quantum channels by characterizing different levels of ergodic hierarchy from integrable to mixing. The quantum channels on single systems are constructed from the unitary operators acting on bipartite states and tracing out the environment. The interaction strength of these unitary operators measured in terms of operator entanglement provides sufficient conditions for the channel to be mixing. By using block diagonal unitary operators, we construct a set of non-ergodic channels. By using canonical form of two-qubit unitary operator, we analytically construct the channels on single qubit ranging from integrable to mixing. Moreover, we also study interacting many-body quantum systems that include the famous Sachdev-Ye-Kitaev (SYK) model and show that they display mixing within the framework of the quantum channel.