A quantum central limit theorem for non-equilibrium systems: Exact local relaxation of correlated states

TL;DR

This paper proves that one-dimensional quantum many-body systems with quadratic Hamiltonians locally relax to Gaussian states from a wide class of initial states, advancing understanding of non-equilibrium quantum relaxation.

Contribution

It establishes a quantum central limit theorem for non-equilibrium systems, demonstrating local relaxation to Gaussian states under broad initial conditions.

Findings

Quantum systems relax to Gaussian states locally.

Relaxation occurs for a wide class of initial states.

The proof uses a non-commutative central limit theorem.

Abstract

We prove that quantum many-body systems on a one-dimensional lattice locally relax to Gaussian states under non-equilibrium dynamics generated by a bosonic quadratic Hamiltonian. This is true for a large class of initial states - pure or mixed - which have to satisfy merely weak conditions concerning the decay of correlations. The considered setting is a proven instance of a situation where dynamically evolving closed quantum systems locally appear as if they had truly relaxed, to maximum entropy states for fixed second moments. This furthers the understanding of relaxation in suddenly quenched quantum many-body systems. The proof features a non-commutative central limit theorem for non-i.i.d. random variables, showing convergence to Gaussian characteristic functions, giving rise to trace-norm closeness. We briefly relate our findings to ideas of typicality and concentration of measure.