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

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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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Authors and Affiliations

  1. Graduate School of Information Sciences, Tohoku University, Aoba-ku, Sendai, 980-8579, Japan

    Wataru Kumagai

  2. Graduate School of Mathematics, Nagoya University, Nagoya, Japan

    Wataru Kumagai & Masahito Hayashi

  3. Centre for Quantum Technologies, National University of Singapore, Singapore, Singapore

    Masahito Hayashi

Authors
  1. Wataru Kumagai
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  2. Masahito Hayashi
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Correspondence to Wataru Kumagai.

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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

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