Note on the saturation of the norm inequalities between diamond and nuclear norm

TL;DR

This paper characterizes operators that saturate the bounds between the diamond norm and the nuclear norm, which are important in quantum information and low-rank matrix recovery.

Contribution

It provides a simple characterization of all operators that achieve the bounds between the diamond norm and the nuclear norm.

Findings

Identifies operators saturating the bounds between the diamond and nuclear norms

Provides explicit conditions for norm saturation

Enhances understanding of norm inequalities in quantum information

Abstract

The diamond norm plays an important role in quantum information and operator theory. Recently, it has also been proposed as a regularizer for low-rank matrix recovery. The norm constants that bound the diamond norm in terms of the nuclear norm (also known as trace norm) are explicitly known. This note provides a simple characterization of all operators saturating the upper and the lower bound.