Notes on super-operator norms induced by Schatten norms

TL;DR

This paper investigates super-operator norms induced by Schatten p-norms in quantum information, proving that for completely positive super-operators the norm is achieved by a positive semidefinite input, but not always on Hermitian inputs.

Contribution

It establishes that for completely positive super-operators, the induced norm is achieved by positive semidefinite inputs, and explores properties of these norms when tensoring with identity super-operators.

Findings

Norms are achieved by positive semidefinite inputs for completely positive super-operators.

Existence of super-operators where the norm is not achieved on Hermitian inputs.

Properties of super-operator norms when tensored with identity operators.

Abstract

This paper considers basic properties of super-operator norms induced by Schatten p-norms. Such super-operator norms arise in various contexts in the study of quantum information. It is proved that for completely positive super-operators, the value of any such norm is achieved by a positive semidefinite input, answering a question recently posed by King and Ruskai. However, for any choice of p, there exists a super-operator that is the difference of two completely positive, trace-preserving super-operators such that the value of the super-operator norm is not even achieved on a Hermitian input operator. Also considered are the properties of the above norms for super-operators tensored with the identity super-operator.