A short note on passivity, complete passivity and virtual temperatures

TL;DR

This paper provides an intuitive proof that only Gibbs states at positive temperatures are completely passive, using the concept of virtual temperatures to characterize passive states and their properties.

Contribution

It introduces a simple proof linking complete passivity to Gibbs states and employs virtual temperatures to characterize energy transitions in quantum states.

Findings

Only Gibbs states at positive temperatures are completely passive.

Passive states have transitions at positive temperatures.

Complete passivity requires all transitions to share the same positive temperature.

Abstract

We give a simple and intuitive proof that the only states which are completely passive, i.e. those states from which work cannot be extracted even with infinitely many copies, are Gibbs states at positive temperatures. The proof makes use of the idea of virtual temperatures, i.e. the association of temperatures to pairs of energy levels (transitions). We show that (i) passive states are those where every transition is at a positive temperature, and (ii) completely passive states are those where every transition is at the same positive temperature.