Partial order on passive states and Hoffman majorization in quantum thermodynamics

TL;DR

This paper introduces a partial order on passive states based on virtual temperatures, providing a detailed comparison framework that links quantum thermodynamics, majorization, and resource theory to analyze work extraction.

Contribution

It defines a new partial order on passive states, characterizes passivity-preserving operations, and connects Hoffman majorization to quantum thermodynamics for the first time.

Findings

A partial order on passive states based on virtual temperatures.

Necessary and sufficient conditions for state transformations under passivity-preserving operations.

Maximum work extractable equals the ergotropy of pure active states.

Abstract

Passive states, i.e., those states from which no work can be extracted via unitary operations, play an important role in the foundations and applications of quantum thermodynamics. They generalize the familiar Gibbs thermal states, which are the sole passive states being stable under tensor product. Here, we introduce a partial order on the set of passive states that captures the idea of a passive state being virtually cooler than another one. This partial order, which we build by defining the notion of relative passivity, offers a fine-grained comparison between passive states based on virtual temperatures (just like thermal states are compared based on their temperatures). We then characterize the quantum operations that are closed on the set of virtually cooler states with respect to some fixed input and output passive states. Viewing the activity, i.e., non-passivity, of a state as a resource, our main result is then a necessary and sufficient condition on the transformation of a class of pure active states under these relative passivity-preserving operations. This condition gives a quantum thermodynamical meaning to the majorization relation on the set of non-increasing vectors due to Hoffman. The maximum extractable work under relative passivity-preserving operations is then shown to be equal to the ergotropy of these pure active states. Finally, we are able to fully characterize passivity-preserving operations in the simpler case of qubit systems, and hence to derive a state interconversion condition under passivity-preserving qubit operations. The prospect of this work is a general resource-theoretical framework for the extractable work via quantum operations going beyond thermal operations.