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

TL;DR

This work addresses the problem of establishing tight, efficiently computable upper bounds on the asymptotic error exponents for quantum state and channel exclusion. It introduces a divergence-radius (divergence-sphere) methodology based on the strong converse of asymmetric binary hypothesis testing, distilling the one-shot exclusion problem into a bound expressed via a divergence radius. The main contributions are alternative proofs of the log-Euclidean upper bound for state exclusion and the corresponding bound for channel exclusion, with a clear geometric interpretation of divergence radii as converse limits. The results enhance understanding of antidistinguishability in quantum information and yield computable, principled bounds applicable to both states and channels.

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.