Sample Complexity of Composite Quantum Hypothesis Testing

TL;DR

This work studies symmetric binary composite quantum hypothesis testing in the finite-sample regime, introducing the delta-sample complexity $n^*(\delta)$ and providing tight lower and upper bounds across regimes with finite and infinite uncertainty-set cardinalities. It derives a min-max reformulation of the error probability, connects the bounds to the maximum fidelity between sets and to the Bures distance, and applies quantum Chernoff-type arguments to establish nontrivial finite-sample guarantees. The results show that, up to universal constants, $n^*(\delta) = \Theta( \frac{\ln(1/\delta)}{-\ln F_{max}} )$ under broad conditions and provide specialized bounds for quantum state verification and finite/infinite cardinality scenarios; these bounds extend to locally differentially private QHT, quantifying privacy-induced sample-size costs. The work thus offers a comprehensive finite-sample framework for composite quantum state discrimination with practical implications for quantum sensing and secure quantum information processing.

Abstract

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.

Theorems & Definitions (18)

• definition 1: Sample Complexity of Simple QHT
• theorem 2: cheng2025invitation
• definition 3: Sample Complexity of Composite QHT
• remark 4
• proposition 5
• lemma 6
• proposition 7
• theorem 8
• theorem 9
• lemma 10