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, practically not very useful. In this paper, we develop further an approach initiated in [Mosonyi, Szil\'agyi, 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.

翻译：二元复合独立同分布态判别的最优误差指数平凡地受限于区分两个假设所对应集合中单个元素的最坏情况两两指数；在有限维经典情形中，这些界实际上给出了误差指数的精确单拷贝表达式。相比之下，在非对易情形中，最优指数仅能通过正则化散度表达，所得公式虽具有概念意义，但实际用途有限。本文进一步发展了[Mosonyi, Szil\'agyi, Weiner, IEEE Trans. Inf. Th. 68(2):1032--1067, 2022]中提出的方法，通过不仅比较两个假设中的单个态，还比较与之相关的各类非归一化半正定算子，从而给出误差指数的改进单拷贝界。我们展示了给出有效界的此类算子的若干等价刻画，并证明在对易情形中考虑态的加权几何平均，以及在每个假设包含两个态的情形中考虑加权久保-安藤几何平均，是本方法的最优选择。由此，我们给出了加权久保-安藤几何平均的新刻画：它们是唯一满足块可加性、张量乘性及算术-几何平均不等式的二元算子几何平均。我们还将结果推广至复合量子信道判别，证明了两个量子信道的加权久保-安藤几何平均（这一概念似乎是新的）具有类似的最优性。通过扩展此概念，我们定义了超算子透视函数的概念并建立了其若干基本性质，这些性质可能具有独立的研究价值。