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

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.

Simplified proof of the main theorem in [arXiv:1511.01513] and a converse statement. 3+1 pages. V2: small changes and incomplete funding information corrected. V3: minor polishing, published version

References in corpus (1)

• Improving compressed sensing with the diamond norm

Cited by in corpus (5)

• Models of quantum complexity growth
• Almost all quantum channels are equidistant
• Operational applications of the diamond norm and related measures in quantifying the non-physicality of quantum maps
• Guaranteed recovery of quantum processes from few measurements
• Improving compressed sensing with the diamond norm