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 reformulate 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 β--athermality.
Highlights
The Gibbs state is a cornerstone of equilibrium statistical mechanics, and provides a rigorous no-Loosely speaking, passivity captures the inability of a quantum state to be used to ‘raise a weight’
This simplicity makes it wellsuited to exploring structural aspects of thermodynamics in quantum systems beyond equilibrium, and for elucidating the conceptual ingredients required for a notion of equilibrium to be Accepted in Quantum 2021-02-26, click title to verify
Passivity has received renewed attention. It has played a valuable role in studying the problem of work-extraction from non-equilibrium states [11,12,13,14,15], generalised Gibbs states [16,17,18,19,20] and quantum informationtheoretic approaches to thermodynamics [21,22,23,24,25]
Summary
The Gibbs state is a cornerstone of equilibrium statistical mechanics, and provides a rigorous no-. There have since been shorter and more compact passivity analyses, for example via the concept of ‘virtual temperatures’ [9] While such derivations of the Gibbs state from passivity are technically clear, one might wish for a more intuitive perspective on the structure of passivity that makes the emergence of the Gibbs state entirely inevitable while suggesting natural extensions. This convex shape naturally emerges in the macroscopic regime of many independent copies of the state, and its structure entirely encodes the deviation of the system from the manifold of equilibrium states.
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