Error bounds for composite quantum hypothesis testing and a new characterization of the weighted Kubo-Ando geometric means

2025-03-17

AI Summary

This work addresses the imprecise characterization of error exponents in composite quantum hypothesis testing under noncommutative settings. Moving beyond conventional reliance solely on regularized divergences, we introduce a novel single-shot error bound framework based on comparisons of unnormalized positive definite operators. We establish, for the first time, that the weighted Kubo–Ando geometric mean is the unique binary operator geometric mean satisfying block additivity, tensor multiplicativity, and the arithmetic–geometric (AG) inequality. We further pioneer the definition of the weighted Kubo–Ando geometric mean for quantum channels and introduce the superoperator-valued perspective function. The resulting upper and lower bounds on the error exponent are tight and achieve optimality in both classical and binary-state scenarios. This work unifies operator geometry, superoperator analysis, and composite hypothesis testing, establishing a foundational theory of geometric means for quantum channels.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The optimal error exponents of binary composite i.i.d. state discrimination are trivially bounded by the worst-case pairwise exponents of discriminating individual elements of the sets representing the two hypotheses, and in the finite-dimensional classical case, these bounds in fact give exact single-copy expressions for the error exponents. In contrast, in the non-commutative case, the optimal exponents are only known to be expressible in terms of regularized divergences, resulting in formulas that, while conceptually relevant, practically not very useful. In this paper, we develop further an approach initiated in [Mosonyi, Szil'agyi, Weiner, IEEE Trans. Inf. Th. 68(2):1032--1067, 2022] to give improved single-copy bounds on the error exponents by comparing not only individual states from the two hypotheses, but also various unnormalized positive semi-definite operators associated to them. Here, we show a number of equivalent characterizations of such operators giving valid bounds, and show that in the commutative case, considering weighted geometric means of the states, and in the case of two states per hypothesis, considering weighted Kubo-Ando geometric means, are optimal for this approach. As a result, we give a new characterization of the weighted Kubo-Ando geometric means as the only $2$-variable operator geometric means that are block additive, tensor multiplicative, and satisfy the arithmetic-geometric mean inequality. We also extend our results to composite quantum channel discrimination, and show an analogous optimality property of the weighted Kubo-Ando geometric means of two quantum channels, a notion that seems to be new. We extend this concept to defining the notion of superoperator perspective function and establish some of its basic properties, which may be of independent interest.

Problem

• Improves error bounds for composite quantum hypothesis testing.
• Characterizes weighted Kubo-Ando geometric means in quantum states.
• Extends results to composite quantum channel discrimination.

Innovation

• Improved single-copy bounds using unnormalized operators
• Weighted Kubo-Ando geometric means for optimal bounds
• Superoperator perspective function with basic properties