Towards Quantum Universal Hypothesis Testing

Abstract: Hoeffding's formulation and solution to the universal hypothesis testing (UHT) problem had a profound impact on many subsequent works dealing with asymmetric hypotheses. In this work, we introduce a quantum universal hypothesis testing framework that serves as a quantum analog to Hoeffding's UHT. Motivated by Hoeffding's approach, which estimates the empirical distribution and uses it to construct the test statistic, we employ quantum state tomography to reconstruct the unknown state prior to forming the test statistic. Leveraging the concentration properties of quantum state tomography, we establish the exponential consistency of the proposed test: the type II error probability decays exponentially quickly, with the exponent determined by the trace distance between the true state and the nominal state.