Concentration of ergotropy in many-body systems

TL;DR

This paper demonstrates that in large many-body quantum systems, ergotropy, the maximum extractable work, concentrates around a typical value for most states, with both theoretical proofs and numerical evidence supporting this phenomenon.

Contribution

The paper proves measure concentration of ergotropy in many-body systems and compares it with von Neumann entropy, extending understanding of quantum battery energetics.

Findings

Ergotropy is almost constant for most states in large systems.

Lipschitz continuity of ergotropy enables measure concentration proof.

Numerical evidence suggests concentration also under minimal prior information.

Abstract

Ergotropy -- the maximal amount of unitarily extractable work -- measures the ``charge level'' of quantum batteries. We prove that in large many-body batteries ergotropy exhibits a concentration of measure phenomenon. Namely, the ergotropy of such systems is almost constant for almost all states sampled from the Hilbert--Schmidt measure. We establish this by first proving that ergotropy, as a function of the state, is Lipschitz-continuous with respect to the Bures distance, and then applying Levy's measure concentration lemma. In parallel, we showcase the analogous properties of von Neumann entropy, compiling and adapting known results about its continuity and concentration properties. Furthermore, we consider the situation with the least amount of prior information about the state. This corresponds to the quantum version of the Jeffreys prior distribution -- the Bures measure. In this case, there exist no analytical bounds guaranteeing exponential concentration of measure. Nonetheless, we provide numerical evidence that ergotropy, as well as von Neumann entropy, concentrate also in this case.