Equivariant relative submajorization

TL;DR

This paper introduces a group-equivariant extension of relative submajorization, providing new tools to analyze thermodynamic transformations under symmetry constraints, with implications for quantum resource theories and second laws.

Contribution

It generalizes relative submajorization to equivariant settings, characterizes conditions for catalytic and asymptotic transformations, and introduces monotones related to quantum divergences.

Findings

Established a sufficient condition for catalytic transformations.

Provided a characterization of asymptotic relaxation of the relation.

Identified monotones related to sandwiched quantum Rényi divergences.

Abstract

We study a generalization of relative submajorization that compares pairs of positive operators on representation spaces of some fixed group. A pair equivariantly relatively submajorizes another if there is an equivariant subnormalized channel that takes the components of the first pair to a pair satisfying similar positivity constraints as in the definition of relative submajorization. In the context of the resource theory approach to thermodynamics, this generalization allows one to study transformations by Gibbs-preserving maps that are in addition time-translation symmetric. We find a sufficient condition for the existence of catalytic transformations and a characterization of an asymptotic relaxation of the relation. For classical and certain quantum pairs the characterization is in terms of explicit monotone quantities related to the sandwiched quantum R\'enyi divergences. In the general quantum case the relevant quantities are given only implicitly. Nevertheless, we find a large collection of monotones that provide necessary conditions for asymptotic or catalytic transformations. When applied to time-translation symmetric maps, these give rise to second laws that constrain state transformations allowed by thermal operations even in the presence of catalysts.