Speaker
Description
Hypothesis testing is a fundamental framework for making decisions between competing explanations of a system on the basis of observed data. In quantum information science, it underlies tasks such as signal detection, quantum sensing, communication, and the verification of quantum devices, where one must infer which quantum state or physical process generated the observed measurement outcomes. Standard quantum hypothesis testing assumes that each hypothesis specifies a single quantum state. In many realistic situations, however, unknown parameters, calibration errors, or incomplete prior information mean that the state under each hypothesis is only known to belong to a prescribed set. Composite quantum hypothesis testing addresses this uncertainty by seeking a state-independent measurement that performs optimally over all states consistent with the competing hypotheses.
We study single-copy binary composite quantum hypothesis testing for qubit states under a minimax Bayesian error criterion. The null hypothesis consists of a fixed pure state, while the alternative hypothesis is a continuous family of pure states whose Bloch vectors form an arc on a great circle. By exploiting the linearity of the error probability in the space of quantum states, it follows from Fang and Hayashi (2025) that replacing the original nonconvex alternative set by its convex hull leaves not only the minimum worst-case error probability but also the set of minimax-optimal POVMs unchanged. The optimization is then reduced to a geometric problem involving the convex hull of an affine image of the Bloch vector arc.
We derive an explicit characterization of the least favorable state and the optimal binary measurement for arbitrary prior probabilities and arc endpoints. Depending on the geometry and the prior imbalance, the least favorable state is either an endpoint pure state or a mixed state formed from the two endpoint states. Accordingly, the minimax-optimal measurement is either the Helstrom measurement for the null state and an endpoint state, the Helstrom measurement for the null state and a least favorable endpoint mixture, or a trivial decision rule that always selects the more probable hypothesis. We also identify the boundary cases in which the weighted state difference becomes singular and the optimal effect operator is nonunique. These results provide a complete Bloch-sphere interpretation of single-copy composite qubit discrimination and explicitly connect geometric convex optimization with general minimax results for composite quantum hypothesis testing.