Optimal Adaptive Strategies for Sequential Quantum Hypothesis Testing

TL;DR

This paper develops optimal adaptive and non-adaptive sequential strategies for quantum hypothesis testing, demonstrating how adaptive measurements can significantly improve error decay rates and characterizing the limits of these strategies.

Contribution

It introduces new adaptive strategies that minimize errors in quantum hypothesis testing under sample constraints and characterizes the achievable error exponents for both adaptive and non-adaptive methods.

Findings

Adaptive strategies achieve exponential error decay rates based on measured quantum relative entropies.

Joint measurements on multiple samples increase error decay rates to quantum relative entropies.

Adaptive measurements are necessary to reach optimal error bounds under certain conditions.

Abstract

We consider sequential hypothesis testing between two quantum states using adaptive and non-adaptive strategies. In this setting, samples of an unknown state are requested sequentially and a decision to either continue or to accept one of the two hypotheses is made after each test. Under the constraint that the number of samples is bounded, either in expectation or with high probability, we exhibit adaptive strategies that minimize both types of misidentification errors. Namely, we show that these errors decrease exponentially (in the stopping time) with decay rates given by the measured relative entropies between the two states. Moreover, if we allow joint measurements on multiple samples, the rates are increased to the respective quantum relative entropies. We also fully characterize the achievable error exponents for non-adaptive strategies and provide numerical evidence showing that adaptive measurements are necessary to achieve our bounds under some additional assumptions.