Converse bounds for quantum hypothesis exclusion: A divergence-radius approach

TL;DR

This paper presents alternative proofs for upper bounds on the error exponents in quantum hypothesis exclusion tasks, using divergence radii and strong converse results for binary hypothesis testing.

Contribution

It introduces a divergence-radius approach and strong converse techniques as a new method to derive bounds in quantum hypothesis exclusion.

Findings

Upper bounds on asymptotic error exponents established

Divergence radii characterize the bounds

Alternative proof techniques demonstrated

Abstract

Hypothesis exclusion is an information-theoretic task in which an experimenter aims at ruling out a false hypothesis from a finite set of known candidates, and an error occurs if and only if the hypothesis being ruled out is the ground truth. For the tasks of quantum state exclusion and quantum channel exclusion -- where hypotheses are represented by quantum states and quantum channels, respectively -- efficiently computable upper bounds on the asymptotic error exponents were established in a recent work of the current authors [Ji et al., arXiv:2407.13728 (2024)], where the derivation was based on nonasymptotic analysis. In this companion paper of our previous work, we provide alternative proofs for the same upper bounds on the asymptotic error exponents of quantum state and channel exclusion, but using a conceptually different approach from the one adopted in the previous work. Specifically, we apply strong converse results for asymmetric binary hypothesis testing to distinguishing an arbitrary ``dummy'' hypothesis from each of the concerned candidates. This leads to the desired upper bounds in terms of divergence radii via a geometrically inspired argument.