Sample Complexity of Composite Quantum Hypothesis Testing

TL;DR

This paper characterizes the minimum number of quantum state copies needed for reliable composite hypothesis testing, providing tight bounds and extending results to privacy-preserving scenarios.

Contribution

It derives tight bounds on sample complexity for finite and infinite uncertainty sets in quantum hypothesis testing, including privacy considerations.

Findings

Derived lower bounds generalizing simple QHT sample complexity

Introduced new upper bounds matching lower bounds up to constants

Extended analysis to differentially private quantum hypothesis testing

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.