Entropy and reversible catalysis

TL;DR

This paper establishes that non-decreasing entropy is both necessary and sufficient for reversible state transformations with a catalyst in quantum and classical systems, resolving a key conjecture and characterizing entropy in single-shot scenarios.

Contribution

It proves the catalytic entropy conjecture for quantum and classical systems, providing a complete single-shot entropy characterization without external randomness.

Findings

Non-decreasing entropy enables reversible transformations with catalysts.

The results apply to von Neumann and Shannon entropy.

Provides a quantitative single-shot characterization of Gibbs states.

Abstract

I show that non-decreasing entropy provides a necessary and sufficient condition to convert the state of a physical system into a different state by a reversible transformation that acts on the system of interest and a further "catalyst" whose state has to remain invariant exactly in the transition. This statement is proven both in the case of finite-dimensional quantum mechanics, where von~Neumann entropy is the relevant entropy, and in the case of systems whose states are described by probability distributions on finite sample spaces, where Shannon entropy is the relevant entropy. The results give an affirmative resolution to the (approximate) "catalytic entropy conjecture" introduced by Boes et al. [PRL 122, 210402 (2019)]. They provide a complete single-shot characterization without external randomness of von Neumann entropy and Shannon entropy. I also compare the results to the setting of phenomenological thermodynamics and show how they can be used to obtain a quantitative single-shot characterization of Gibbs states in quantum statistical mechanics.