A Shrinking Factor for Unitarily Invariant Norms under a Completely Positive Map

TL;DR

This paper investigates how unitarily invariant norms of Hermitian operators change under completely positive maps, establishing a bound on the shrinking factor based on spectral and trace norms.

Contribution

It introduces a bound on the shrinking factor for unitarily invariant norms under completely positive maps, linking it to spectral and trace norms.

Findings

Shrinking factor is bounded by the maximum of spectral and trace norms.

The relation applies to Hermitian operators under completely positive maps.

Provides a unified bound for all unitarily invariant norms.

Abstract

A relation between values of a unitarily invariant norm of Hermitian operator before and after action of completely positive map is studied. If the norm is jointly defined on both the input and output Hilbert spaces, one defines a shrinking factor under the restriction of given map to Hermitian operators. As it is shown, for any unitarily invariant norm this shrinking factor is not larger than the maximum of two values for the spectral norm and the trace norm.