Sample Complexity of Composite Quantum Hypothesis Testing

This paper investigates symmetric composite binary quantum hypothesis testing (QHT), where the goal is to determine which of two uncertainty sets contains an unknown quantum state. While asymptotic error exponents for this problem are well-studied, the finite-sample regime remains poorly understood. We bridge this gap by characterizing the sample complexity – the minimum number of state copies required to achieve a target error level. Specifically, we derive lower bounds that generalize the sample complexity of simple QHT and introduce new upper bounds for various uncertainty sets, including of both finite and infinite cardinalities. Notably, our upper and lower bounds match up to universal constants, providing a tight characterization of the sample complexity. Finally, we extend our analysis to the differentially private setting, establishing the sample complexity for privacy-preserving composite QHT.

💡 Research Summary

This paper addresses a fundamental gap in quantum information theory: the finite‑sample behavior of symmetric composite binary quantum hypothesis testing (QHT). In the composite setting, an unknown quantum state is promised to belong to one of two uncertainty sets D₁ or D₂, and the task is to decide which set contains the state. While asymptotic error exponents (quantum Chernoff, Stein) for this problem are well understood, practical quantum devices operate with a limited number of copies, making it essential to know how many copies n are needed to achieve a target average error δ. The authors define the δ‑sample complexity n*(δ) as the smallest n such that the minimax error probability does not exceed δ, and they provide matching lower and upper bounds that are tight up to universal constants.

The technical development proceeds as follows. First, the authors rewrite the minimax error probability in terms of the trace distance between convex hulls of the n‑copy product states generated by the two sets (Proposition 5). This representation enables the use of Sion’s minimax theorem and shows that the error is non‑increasing with n (Lemma 6). Next, they apply the quantum Chernoff bound and the relationship between trace distance and Uhlmann fidelity to obtain a hierarchy of upper bounds (Equations 10‑12). The key quantity that appears is the maximum fidelity between any pair of states from the two sets, \