  A family of quantum measures like the Shannon distinguishability is
presented. These measures are defined over the two classes of POVM measurements
and related to separate parts in the expression for mutual information. Changes
of Ky Fan's norms and the partitioned trace distances under the operation of
partial trace are discussed. Upper and lower bounds on the introduced
quantities are obtained in terms of partitioned trace distances and Uhlmann's
partial fidelities. These inequalities provide a kind of generalization of the
well-known bounds on the Shannon distinguishability. The notion of
cryptographic exponential indistinguishability for quantum states is revisited.
When exponentially fast convergence is required, all the metrics induced by
unitarily invariant norms are shown to be equivalent.
