Quantum sequential parameter testing

Abstract

Sequential strategies in hypothesis testing use a variable number of measurement rounds, allowing a decision to be made as soon as the observed data provide a prescribed level of error tolerance. Although sequential testing is well established for a discrete set of hypotheses, extending this framework to a continuous parameter space poses additional challenges and has remained largely unexplored. In this article, we introduce sequential \textit{parameter testing}, a framework for determining an unknown parameter up to a prescribed tolerance by ruling out sufficiently distant competing values with a target error tolerance. We clarify its operational distinction from conventional parameter estimation and develop sequential tests for continuous families of hypotheses, introducing the \textit{twin-peaks test} as a natural and computationally efficient analog of sequential likelihood-ratio testing. We apply the framework to two paradigmatic quantum tasks: testing the phase and the purity of a qubit. For phase testing, we show numerically that adaptive projective measurements achieve the same average sample cost as collective covariant measurements with a fixed number of copies. For purity testing, local measurements are optimal, and sequential parameter testing yields significant average sample savings over fixed sample-size protocols. Our results establish parameter testing as an operationally meaningful framework for resource-efficient certification tasks involving continuous parameters.