We treat quantum counterparts of testing problems whose optimal tests aregiven by chi-square, t and F tests. These quantum counterparts are formulatedas quantum hypothesis testing problems concerning quantum Gaussian statesfamilies, and contain disturbance parameters, which have group symmetry.Quantum Hunt-Stein Theorem removes a part of these disturbance parameters, butother types of difficulty still remain. In order to remove them, combiningquantum Hunt-Stein theorem and other reduction methods, we establish a generalreduction theorem that reduces a complicated quantum hypothesis testing problemto a fundamental quantum hypothesis testing problem. Using these methods, wederive quantum counterparts of chi-square, t and F tests as optimal tests inthe respective settings.
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