Asymptotic relative submajorization of multiple-state boxes

TL;DR

This paper extends the concept of relative submajorization to multiple quantum state boxes, characterizing error probabilities and asymptotic transformations in quantum hypothesis testing using sandwiched Rènyi divergences.

Contribution

It introduces a new preorder for multiple-state boxes and provides a characterization of asymptotic transformations and error exponents in quantum hypothesis testing.

Findings

Characterizes error probabilities in composite hypothesis testing.

Provides conditions for catalytic transformations between boxes.

Shows the strong converse exponent equals the maximum of pairwise exponents.

Abstract

Pairs of states, or "boxes" are the basic objects in the resource theory of asymmetric distinguishability (Wang and Wilde, 2019), where free operations are arbitrary quantum channels that are applied to both states. From this point of view, hypothesis testing is seen as a process by which a standard form of distinguishability is distilled. Motivated by the more general problem of quantum state discrimination, we consider boxes of a fixed finite number of states and study an extension of the relative submajorization preorder to such objects. In this relation a tuple of positive operators is greater than another if there is a completely positive trace nonincreasing map under which the image of the first tuple satisfies certain semidefinite constraints relative to the other one. This preorder characterizes error probabilities in the case of testing a composite null hypothesis against a simple alternative hypothesis, as well as certain error probabilities in state discrimination. We present a sufficient condition for the existence of catalytic transformations between boxes, and a characterization of an associated asymptotic preorder, both expressed in terms of sandwiched R\'enyi divergences. This characterization of the asymptotic preorder directly shows that the strong converse exponent for a composite null hypothesis is equal to the maximum of the corresponding exponents for the pairwise simple hypothesis testing tasks.