A class of permutation-invariant measurements and their relation to quantum relative entropies

Abstract

We characterize the asymptotic performance of a class of positive operator valued measurements (POVMs) where the only task is to make measurements on independent and identically distributed quantum states on finite-dimensional systems. The POVMs we utilize here can be efficiently described in terms of a reasonably small set of parameters. Their analysis furthers the development of a quantum method of types. They deliver provably optimal performance in asymmetric hypothesis testing and in the transmission of classical messages over quantum channels. We now relate them to the recently developed $\alpha-z$ divergences $D_{\alpha,z}$ by giving an operational interpretation for the limiting case $\lim_{\alpha\to1}D_{\alpha,1-\alpha}$ in terms of probabilities for certain measurement outcomes. This explains one of the more surprising findings of [1] in terms of the theory of group representations. In addition, we provide a Cauchy-Binet type formula for unitary matrices which connects the underlying representation theoretic objects to partial sums of the entries of unitary matrices. At last, we concentrate on the special case of qubits. We are able to give a complete description of the asymptotic detection probabilities for all POVM elements described here. We take the opportunity to define a family of functions on pairs of semi-definite matrices which obeys the quantum generalizations of R\'enyi's axioms except from the generalized mean value axiom. This family is described by limiting values of $\alpha-z$ divergences for the extremal values of the parameter.