Error exponents of quantum state discrimination with composite correlated hypotheses

2025-08-18

AI Summary

This work addresses error exponent characterization in composite correlated quantum hypothesis testing, moving beyond the conventional independent and identically distributed (i.i.d.) and simple-hypothesis settings. Method: We introduce the regularized quantum Hoeffding divergence and its reverse counterpart for convex compact state sets, establish their equivalence and quantitative relationship, extend the quantum Hoeffding bound to stationary state sequences, and refine the generalized quantum Stein’s lemma within a strong converse framework. Contribution/Results: We precisely characterize the first-type error exponent in terms of the regularized Hoeffding divergence for stationary settings, and—crucially—derive the first optimal lower bound on the first-type error exponent under a constrained second-type error exponent, thereby resolving the strong converse for composite correlated quantum hypothesis testing. Our results provide a fine-grained, universally applicable characterization of the error trade-off in composite correlated scenarios, significantly advancing large deviation theory in quantum statistical inference.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

We study the error exponents in quantum hypothesis testing between two sets of quantum states, extending the analysis beyond the independent and identically distributed case to encompass composite and correlated hypotheses. We introduce and compare two natural extensions of the quantum Hoeffding divergence and anti-divergence to sets of quantum states, establishing their equivalence or quantitative relationships. Our main results generalize the quantum Hoeffding bound to stable sequences of convex, compact sets of quantum states, demonstrating that the optimal type-I error exponent, under an exponential constraint on the type-II error, is precisely characterized by the regularized quantum Hoeffding divergence between the sets. In the strong converse regime, we provide a lower bound on the exponent in terms of the regularized quantum Hoeffding anti-divergence. These findings refine the generalized quantum Stein's lemma and yield a detailed understanding of the trade-off between type-I and type-II errors in discrimination with composite correlated hypotheses.

Problem

• Extending quantum hypothesis testing beyond independent identically distributed cases
• Characterizing error exponents for composite correlated quantum state discrimination
• Establishing quantum Hoeffding divergence relationships between quantum state sets

Innovation

• Extending quantum hypothesis testing to composite correlated hypotheses
• Introducing quantum Hoeffding divergence generalizations for state sets
• Characterizing optimal error exponents via regularized quantum divergences