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Bounds on Shannon distinguishability in terms of partitioned measures

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.