Error bounds for composite quantum hypothesis testing and a new characterization of the weighted Kubo-Ando geometric means
Péter E. Frenkel, Milán Mosonyi, Péter Vrana, Mihály Weiner
TL;DR
The paper tackles tightening the single-copy bounds on error exponents for binary composite quantum hypothesis testing by leveraging maximal operator-mean bounds, rather than relying solely on trivial worst-case pairwise divergences.A general framework of Y-variable positive operator functions is developed, focusing on tensor multiplicativity and AM–GM-type properties, with a central role for weighted Kubo–Ando geometric means.A complete characterization is achieved in two key cases: (i) commuting (classical) hypotheses, where maximal bounds correspond to weighted geometric means; (ii) two-variable non-commutative cases, where maximal bounds are the $t$-weighted KA means, yielding exact single-copy bounds and new KA-characterizations.These results extend to composite quantum channel discrimination via a channel analogue of KA means and introduce a superoperator perspective that may be of independent interest, offering a unified view across state and channel discrimination.Open questions remain for extending exact maximality to more than two non-commuting operators and for fully explicit multi-variable geometric means beyond the two-variable KA framework.
Abstract
The optimal error exponents of binary composite i.i.d. state discrimination are trivially bounded by the worst-case pairwise exponents of discriminating individual elements of the sets representing the two hypotheses, and in the finite-dimensional classical case, these bounds in fact give exact single-copy expressions for the error exponents. In contrast, in the non-commutative case, the optimal exponents are only known to be expressible in terms of regularized divergences, resulting in formulas that, while conceptually relevant, are practically not very useful. In this paper, we develop further an approach initiated in [Mosonyi, Szilágyi, Weiner, IEEE Trans. Inf. Th. 68(2):1032--1067, 2022] to give improved single-copy bounds on the error exponents by comparing not only individual states from the two hypotheses, but also various unnormalized positive semi-definite operators associated to them. Here, we show a number of equivalent characterizations of such operators giving valid bounds, and show that in the commutative case, considering weighted geometric means of the states, and in the case of two states per hypothesis, considering weighted Kubo-Ando geometric means, are optimal for this approach. As a result, we give a new characterization of the weighted Kubo-Ando geometric means as the only $2$-variable operator geometric means that are block additive, tensor multiplicative, and satisfy the arithmetic-geometric mean inequality. We also extend our results to composite quantum channel discrimination, and show an analogous optimality property of the weighted Kubo-Ando geometric means of two quantum channels, a notion that seems to be new. We extend this concept to defining the notion of superoperator perspective function and establish some of its basic properties, which may be of independent interest.