Bounds on Shannon distinguishability in terms of partitioned measures

TL;DR

This paper introduces bounds on quantum Shannon distinguishability using partitioned measures, relating them to trace distances and fidelities, and explores their implications for quantum cryptography and metric equivalences.

Contribution

It presents new bounds on quantum distinguishability measures based on partitioned trace distances and fidelities, generalizing existing bounds and analyzing metric equivalences in quantum cryptography.

Findings

Derived bounds on quantum Shannon distinguishability.

Showed equivalence of metrics under exponential convergence.

Connected measures to cryptographic indistinguishability.

Abstract

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.