Matrix Majorization in Large Samples with Varying Support Restrictions

TL;DR

This paper investigates matrix majorization in large samples with varying support restrictions, providing conditions based on divergences that influence quantum thermodynamics transformations.

Contribution

It extends matrix majorization theory to cases with support restrictions and introduces divergence-based conditions for large samples and catalytic regimes.

Findings

Identifies sufficient and almost necessary conditions for majorization.

Shows support restrictions significantly impact divergence measures.

Applies results to catalytic state transformations in quantum thermodynamics.

Abstract

We say that a matrix $P$ with non-negative entries majorizes another such matrix $Q$ if there is a stochastic matrix $T$ such that $Q=TP$. We study matrix majorization in large samples and in the catalytic regime in the case where the columns of the matrices need not have equal support, as has been assumed in earlier works. We focus on two cases: either there are no support restrictions (except for requiring a non-empty intersection for the supports) or the final column dominates the others. Using real-algebraic methods, we identify sufficient and almost necessary conditions for majorization in large samples or when using catalytic states under these support conditions. These conditions are given in terms of multivariate divergences that generalize the R\'enyi divergences. We notice that varying support conditions dramatically affect the relevant set of divergences. Our results find an application in the theory of catalytic state transformation in quantum thermodynamics.