In this paper, we study a matrix equation involving the matrix geometric mean
A♮tB=A1/2(A−1/2BA−1/2)tA1/2, t∈(1,2) . We study several properties and inequalities for the unique solution of such an equation. We also use the weighted geometric mean
A♮tB to define a new quantum Hellinger divergence and show that the new quantum divergence satisfies the Data Processing Inequality.
In this paper, we introduce new quantum divergences of the form Φ(A,B)=Tr(AσB−AτB),A,B>0,where σ and τ are Kubo–Ando operator means such that σ≥τ. More precisely, we show that Φ(A,B) is a quantum divergence when σ is the weighted Kubo–Ando matrix power mean and τ is the weighted geometric mean. In addition, we construct a new quantum Hellinger-type divergence using the linear approximation of the function x. We also study the least-squares problem, the data processing inequality, and in-betweenness property of the matrix means with respect to the quantum Hellinger-type divergence.
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, continuous on monotone decreasing sequences, 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.