Partitioned trace distances

TL;DR

The paper introduces partitioned trace distances, a new quantum metric based on Ky Fan norms, which retain key properties of standard trace distance and have potential applications in quantum information processing.

Contribution

It defines and analyzes partitioned trace distances, extending the standard trace distance with properties like unitary invariance and convexity, and reformulates these properties through majorization relations.

Findings

Partitioned trace distances are unitary invariant.

They are strongly convex.

They do not increase under certain quantum operations.

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.