A Converse Bound via the Nussbaum-Szko{\l}a Mapping for Quantum Hypothesis Testing

TL;DR

This paper introduces a new lower bound for quantum hypothesis testing using the Nussbaum-Szko{ }la mapping, unifying converse results across asymptotic regimes and improving tightness over existing bounds.

Contribution

It presents a novel, unified lower bound based on a single expression that enhances non-asymptotic analysis and classical approximation in quantum hypothesis testing.

Findings

Provides accurate approximations at small blocklengths.

Demonstrates improved tightness over existing bounds.

Unifies converse results across all major asymptotic regimes.

Abstract

Quantum hypothesis testing concerns the discrimination between quantum states. This paper introduces a novel lower bound for asymmetric quantum hypothesis testing that is based on the Nussbaum-Szko{\l}a mapping. The lower bound provides a unified recovery of converse results across all major asymptotic regimes, including large-, moderate-, and small-deviations. Unlike existing bounds, which either rely on technically involved information-spectrum arguments or suffer from fixed prefactors and limited applicability in the non-asymptotic regime, the proposed bound arises from a single expression and enables, in some cases, the direct use of classical results. It is further demonstrated that the proposed bound provides accurate approximations to the optimal quantum error trade-off function at small blocklengths. Numerical comparisons with existing bounds, including those based on fidelity and information spectrum methods, highlight its improved tightness.