Quantum Sequential Hypothesis Testing

TL;DR

This paper develops a quantum sequential hypothesis testing framework, establishing optimal bounds on the number of quantum state copies needed for discrimination with error thresholds, and demonstrates advantages over fixed measurement strategies.

Contribution

It introduces a quantum sequential analysis method for hypothesis testing, providing optimal bounds and practical strategies that outperform fixed measurement approaches.

Findings

Optimal lower bounds on the average number of copies for quantum hypothesis testing.

Sequential strategies can outperform fixed measurement methods.

For pure states, finite samples suffice even for perfect discrimination.

Abstract

We introduce sequential analysis in quantum information processing, by focusing on the fundamental task of quantum hypothesis testing. In particular our goal is to discriminate between two arbitrary quantum states with a prescribed error threshold, $\epsilon$, when copies of the states can be required on demand. We obtain ultimate lower bounds on the average number of copies needed to accomplish the task. We give a block-sampling strategy that allows to achieve the lower bound for some classes of states. The bound is optimal in both the symmetric as well as the asymmetric setting in the sense that it requires the least mean number of copies out of all other procedures, including the ones that fix the number of copies ahead of time. For qubit states we derive explicit expressions for the minimum average number of copies and show that a sequential strategy based on fixed local measurements outperforms the best collective measurement on a predetermined number of copies. Whereas for general states the number of copies increases as $\log 1/\epsilon$, for pure states sequential strategies require a finite average number of samples even in the case of perfect discrimination, i.e., $\epsilon=0$.