Generic features of the dynamics of complex open quantum systems: Statistical approach based on averages over the unitary group

TL;DR

This paper derives exact formulas for functions over the unitary group and uses ensemble theory to analyze the typical dynamics of complex open quantum systems, including effects like initial correlations and equilibration.

Contribution

It provides a unified analytical framework for understanding generic quantum dynamics using Haar measure averages and random matrix theory.

Findings

Derived exact integral expressions over the Haar measure.

Analyzed typical behaviors of open quantum systems, including entanglement and equilibration.

Compared regular and chaotic systems using eigenvalue distributions.

Abstract

We obtain exact analytic expressions for a class of functions expressed as integrals over the Haar measure of the unitary group in d dimensions. Based on these general mathematical results, we investigate generic dynamical properties of complex open quantum systems, employing arguments from ensemble theory. We further generalize these results to arbitrary eigenvalue distributions, allowing a detailed comparison of typical regular and chaotic systems with the help of concepts from random matrix theory. To illustrate the physical relevance and the general applicability of our results we present a series of examples related to the fields of open quantum systems and nonequilibrium quantum thermodynamics. These include the effect of initial correlations, the average quantum dynamical maps, the generic dynamics of system-environment pure state entanglement and, finally, the equilibration of generic open and closed quantum systems.