Sanov theorem revisited

 

Generalised quantum Sanov theorem revisited

Ludovico Lami

Given two families of quantum states  and , called the null and the alternative hypotheses, quantum hypothesis testing is the task of determining whether an unknown quantum state belongs to  or . Mistaking  for  is a type I error, and vice versa for the type II error. In quantum Shannon theory, a fundamental role is played by the Stein exponent, i.e. the asymptotic rate of decay of the type II error probability for a given threshold on the type I error probability. Stein exponents have been thoroughly investigated -- and, sometimes, calculated. However, most currently available solutions apply to settings where the hypotheses simple (i.e. composed of a single state), or else the families  and  need to satisfy stringent constraints that exclude physically important sets of states, such as separable states or stabiliser states. In this work, we establish a general formula for the Stein exponent where both hypotheses are allowed to be composite: the alternative hypothesis  is assumed to be either composite i.i.d. or arbitrarily varying, with components taken from a known base set, while the null hypothesis  is fully general, and required to satisfy only weak compatibility assumptions that are met in most physically relevant cases -- for instance, by the sets of separable or stabiliser states. Our result extends and subsumes the findings of [BBH, CMP 385:55, 2021] (that we also simplify), as well as the 'generalised quantum Sanov theorem' of [LBR, arXiv:2408.07067]. The proof relies on a careful quantum-to-classical reduction via measurements, followed by an application of the results on classical Stein exponents obtained in [Lami, arXiv:today]. We also devise new purely quantum techniques to analyse the resulting asymptotic expressions.

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