Computing mixed Schatten norm of completely positive maps

TL;DR

This paper introduces an iterative method for computing the Schatten $p ightarrow q$ norm of completely positive maps, extending classical matrix norm computation techniques to a broader operator setting.

Contribution

The authors generalize matrix norm concepts and properties to completely positive maps and establish an iterative method for computing their Schatten norms.

Findings

Defined an iteration method for Schatten $p ightarrow q$ norm of completely positive maps.

Generalized matrix norm properties to the setting of completely positive maps.

Proved a key theorem extending classical results to this new context.

Abstract

Computing $p \rightarrow q$ norm for matrices is a classical problem in computational mathematics and power iteration is a well-known method for computing $p \rightarrow q $ norm for a matrix with nonnegative entries. Here we define an equivalent iteration method for computing $ S_p \rightarrow S_q $ norm for completely positive maps where $S_p$ is the Schatten $p$ norm. We generalize almost all of the definitions, properties, lemmas, etc. in the matrix setting to completely positive maps and prove an important theorem in this setting.