Chaos, Ergodicity and Equilibria in a Quantum Kac Model

TL;DR

This paper introduces quantum analogs of the Kac equations, analyzes their steady states, and establishes a propagation of chaos, providing insights into quantum kinetic theory and entanglement dynamics.

Contribution

It develops quantum versions of the Kac Master and Boltzmann equations, proves a propagation of chaos theorem, and characterizes the steady states as separable, entanglement-breaking states.

Findings

All steady states are separable and entanglement breaking.

The QKME describes a quantum Markov semigroup with specific properties.

The results facilitate quantitative analysis of approach to equilibrium and entanglement destruction.

Abstract

We introduce quantum versions of the Kac Master Equation and the Kac Boltzmann Equation. We study the steady states of each of these equations, and prove a propagation of chaos theorem that relates them. The Quantum Kac Master Equation (QKME) describes a quantum Markov semigroup, while the Kac Boltzmann Equation describes a non-linear evolution of density matrices on the single particle state space. All of the steady states of the $N$ particle quantum system described by the QKME are separable, and thus the evolution described by the QKME is entanglement breaking. The results set the stage for a quantitative study of approach to equilibrium in quantum kinetic theory, and a quantitative study the rate of destruction of entanglement in a class of quantum Markov semigroups describing binary interactions.