For composite quantum hypothesis testing with a mixed IID null hypothesis, the optimal type-II error exponent is the worst-case component when type-I error vanishes, but not for fixed nonzero type-I error.
von Neumann, Zur theorie der gesellschaftsspiele, Mathematische Annalen100, 295 (1928)
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Generalized quantum Stein's lemma for mixed sources
For composite quantum hypothesis testing with a mixed IID null hypothesis, the optimal type-II error exponent is the worst-case component when type-I error vanishes, but not for fixed nonzero type-I error.