Entropic equality for worst-case work at any protocol speed

TL;DR

This paper establishes a universal entropic equality for worst-case work in quantum thermodynamics, applicable at any protocol speed, linking maximum work output to initial state entropy and energy coherences.

Contribution

It introduces a new equality relating worst-case work to a penalty and an optimum term involving max entropy, extending single-shot thermodynamics to dynamic protocols.

Findings

Equality holds for all protocol speeds

Optimum derived by setting penalty to zero

Application demonstrated on an electron box

Abstract

We derive an equality for non-equilibrium statistical mechanics in finite-dimensional quantum systems. The equality concerns the worst-case work output of a time-dependent Hamiltonian protocol in the presence of a Markovian heat bath. It has has the form "worst-case work = penalty - optimum". The equality holds for all rates of changing the Hamiltonian and can be used to derive the optimum by setting the penalty to 0. The optimum term contains the max entropy of the initial state, rather than the von Neumann entropy, thus recovering recent results from single-shot statistical mechanics. Energy coherences can arise during the protocol but are assumed not to be present initially. We apply the equality to an electron box.