Quantum Hypothesis Testing for Gaussian States: Quantum Analogues of χ2, t-, and F-Tests

We consider quantum counterparts of testing problems for which the optimal tests are the χ2, t-, and F-tests. These quantum counterparts are formulated as quantum hypothesis testing problems concerning Gaussian state families, and they contain nuisance parameters, which have group symmetry. The quantum Hunt-Stein theorem removes some of these nuisance parameters, but other difficulties remain. In order to remove them, we combine the quantum Hunt-Stein theorem and other reduction methods to establish a general reduction theorem that reduces a complicated quantum hypothesis testing problem to a fundamental quantum hypothesis testing problem. Using these methods, we derive quantum counterparts of the χ2, t-, and F-tests as optimal tests in the respective settings.