  We treat quantum counterparts of testing problems whose optimal tests are
given by chi-square, t and F tests. These quantum counterparts are formulated
as quantum hypothesis testing problems concerning quantum Gaussian states
families, and contain disturbance parameters, which have group symmetry.
Quantum Hunt-Stein Theorem removes a part of these disturbance parameters, but
other types of difficulty still remain. In order to remove them, combining
quantum Hunt-Stein theorem and other reduction methods, we 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 chi-square, t and F tests as optimal tests in
the respective settings.
