Do mixed states exhibit deep thermalisation?

TL;DR

This paper investigates the concept of deep thermalisation in quantum systems, revealing its failure for mixed states and proposing a new framework that accounts for initial mixedness, with exact solutions in specific models.

Contribution

The authors introduce a novel paradigm of deep thermalisation for mixed states, extending the concept beyond pure states and providing exact solutions in a chaotic quantum model.

Findings

Deep thermalisation fails for mixed states under unitary evolution.

A new framework for mixed-state deep thermalisation is proposed.

Exact emergence of the mixed-state deep thermal ensemble in the self-dual kicked Ising chain.

Abstract

Deep thermalisation -- where ensembles of pure states on a local subsystem, conditioned on measurement outcomes on its complement, approach universal maximum-entropy ensembles -- represents a stronger form of ergodicity than conventional thermalisation. We show that this framework fails dramatically for mixed initial states, evolved unitarily, even with infinitesimal initial mixedness. To address this, we introduce a new paradigm of deep thermalisation for mixed states, fundamentally distinct from that for pure-state ensembles. In our formulation, the deep thermal ensemble arises by tracing out auxiliary degrees of freedom from a maximum-entropy ensemble defined on an augmented system, with the ensemble structure depending explicitly on the entropy of the initial state. We demonstrate that such ensembles emerge dynamically in generic, locally interacting chaotic systems. For the self-dual kicked Ising chain, which we show to be exactly solvable for a class of mixed initial states, we find exact emergence of the so-defined mixed-state deep thermal ensemble at finite times. Our results therefore lead to fundamental insights into how maximum entropy principles and deep thermalisation manifest themselves in unitary dynamics of states with finite entropy.