Sharp pairwise reduction for quantum hypothesis testing

Kuan-Yi Lee; Ludovico Lami

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arXiv · 2609.28440

Sharp pairwise reduction for quantum hypothesis testing

Kuan-Yi Lee·Ludovico Lami

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Abstract

We determine the optimal universal coefficient in a reduction of multiple to binary quantum hypothesis testing. Specifically, for every finite ensemble in any Hilbert space dimension (finite or infinite), we prove that the error probability of the global pretty good measurement (PGM) is at most four times the sum of the optimal binary error probabilities. By constructing a family of regular-simplex ensembles, we further show that the coefficient four is optimal, even for arbitrary global measurements. This result improves on the pairwise bounds established by Cheng and Liu [arXiv:2606.06246 (2026)] and entails an explicit, sharp guarantee for the standard PGM itself. Our proof is also simpler: rather than constructing sequential measurements and applying a union bound, our proof relies on purely matrix-analytic techniques, combining a direct block Gram matrix analysis of the PGM error and a refined inequality between quantum Hellinger distance and Bures $χ^2$-divergence. Our analysis also yields a refined, one-shot pairwise Chernoff bound and explicit sufficient number of copies for a desired discrimination error.

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Kuan-Yi Lee, Ludovico Lami. 2026-09-23. Sharp pairwise reduction for quantum hypothesis testing. https://arxiv.org/abs/2609.28440

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