Partitioned trace distances

Abstract

New quantum distance is introduced as a half-sum of several singular values of difference between two density operators. This is, up to factor, the metric induced by so-called Ky Fan norm. The partitioned trace distances enjoy similar properties to the standard trace distance, including the unitary invariance, the strong convexity and the close relations to the classical distances. The partitioned distances cannot increase under quantum operations of certain kind including bistochastic maps. All the basic properties are re-formulated as majorization relations. Possible applications to quantum information processing are briefly discussed.

8 pages, no figures. Significant changes are made. New section on majorization is added. Theorem 4.1 is extended. The bibliography is enlarged.

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• Fano type quantum inequalities in terms of $q$-entropies
• Disguising quantum channels by mixing and channel distance trade-off
• A Shrinking Factor for Unitarily Invariant Norms under a Completely Positive Map
• Optimal cloning with respect to the relative error