  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.
