The geometry of passivity for quantum systems and a novel elementary derivation of the Gibbs state

TL;DR

This paper introduces a geometric framework for understanding passivity in quantum systems, clarifying how passive states lead to the Gibbs state and providing new tools for analyzing non-equilibrium quantum thermodynamics.

Contribution

It reformulates quantum passivity in geometric terms, offering a transparent derivation of the Gibbs state and novel measures for non-equilibrium quantum states.

Findings

Passive states correspond to convex shapes in a 2D plane.

The area of these shapes quantifies deviation from equilibrium.

The approach links passivity to ergotropy and athermality measures.

Abstract

Passivity is a fundamental concept that constitutes a necessary condition for any quantum system to attain thermodynamic equilibrium, and for a notion of temperature to emerge. While extensive work has been done that exploits this, the transition from passivity at a single-shot level to the completely passive Gibbs state is technically clear but lacks a good over-arching intuition. Here, we re-formulate passivity for quantum systems in purely geometric terms. This description makes the emergence of the Gibbs state from passive states entirely transparent. Beyond clarifying existing results, it also provides novel analysis for non-equilibrium quantum systems. We show that, to every passive state, one can associate a simple convex shape in a 2-dimensional plane, and that the area of this shape measures the degree to which the system deviates from the manifold of equilibrium states. This provides a novel geometric measure of athermality with relations to both ergotropy and $\beta$-athermality.