Abstract
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.
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Communicated by M. B. Ruskai
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Kumagai, W., Hayashi, M. Quantum Hypothesis Testing for Gaussian States: Quantum Analogues of χ2, t-, and F-Tests. Commun. Math. Phys. 318, 535–574 (2013). https://doi.org/10.1007/s00220-013-1678-1
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DOI: https://doi.org/10.1007/s00220-013-1678-1