Generalized quantum Chernoff bound

2025-08-18

AI Summary

This work addresses the optimal discrimination of composite and correlated quantum state families, generalizing both classical and quantum Chernoff bounds to generic quantum hypothesis testing. We introduce the generalized regularized quantum Chernoff divergence as the fundamental measure characterizing the optimal error exponent between state sets. Through tensor-product stability analysis, the minimax theorem, and techniques from quantum hypothesis testing on convex compact sets, we rigorously establish that this divergence fully determines the worst-case asymptotic error exponent. Furthermore, for binary composite hypotheses, we construct a universally optimal test and elucidate the operational significance of the maximum overlap in symmetric hypothesis testing. Our results unify ensemble-level discriminability with worst-case performance analysis, thereby extending the foundational interface between quantum resource theory and statistical decision theory.

Technology Category

• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

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

Abstract

We establish a generalized quantum Chernoff bound for the discrimination of multiple sets of quantum states, thereby extending the classical and quantum Chernoff bounds to the general setting of composite and correlated quantum hypotheses. Specifically, we consider the task of distinguishing whether a quantum system is prepared in a state from one of several convex, compact sets of quantum states, each of which may exhibit arbitrary correlations. Assuming their stability under tensor product, we prove that the optimal error exponent for discrimination is precisely given by the regularized quantum Chernoff divergence between the sets. Furthermore, leveraging minimax theorems, we show that discriminating between sets of quantum states is no harder than discriminating between their worst-case elements in terms of error probability. This implies the existence of a universal optimal test that achieves the minimum error probability for all states in the sets, matching the performance of the optimal test for the most challenging states. We provide explicit characterizations of the universal optimal test in the binary composite case. Finally, we show that the maximum overlap between a pure state and a set of free states, a quantity that frequently arises in quantum resource theories, is equal to the quantum Chernoff divergence between the sets, thereby providing an operational interpretation of this quantity in the context of symmetric hypothesis testing.

Problem

• Extends quantum Chernoff bound to multiple correlated quantum states
• Determines optimal error exponent for discriminating convex quantum sets
• Links maximum overlap in resource theories to Chernoff divergence

Innovation

• Generalized quantum Chernoff bound for multiple state sets
• Optimal error exponent via regularized quantum divergence
• Universal optimal test matches worst-case performance