Inequalities that Collectively Completely Characterize the Catalytic Majorization Relation

TL;DR

This paper characterizes the catalytic majorization relation for probability vectors using an infinite family of functions, providing a complete set of inequalities that determine when one vector can be transformed into another with the help of a catalyst.

Contribution

It introduces an infinite family of functions that fully characterize catalytic majorization, advancing the understanding of quantum state transformations.

Findings

Provides an if-and-only-if condition for catalytic majorization.

Introduces an infinite family of functions to characterize the relation.

Connects the mathematical framework to quantum state transformation possibilities.

Abstract

For probability vectors x and y, the catalytic majorization relation x prec_T y is defined to hold when there exists a probability vector z such that x otimes z is majorized by y otimes z. In this paper, an infinite family of functions is given such that, subject to some trivial restrictions, x prec_T y if and only if f_r(x) < f_r(y) for all functions f_r in the family. An outline of a proof of this result is provided. The catalytic majorization relation is known to provide a determination of which transformations of jointly held pure quantum states are possible using local operations and classical communication when an additional jointly held state may be specified to facilitate the transformation without being consumed.