Hermitian Maps: Approximations and Completely Positive Extensions

TL;DR

This paper analyzes Hermitian linear maps, establishing bounds on their CP decompositions, optimal approximations, and minimal auxiliary space dimensions for extensions, with practical examples.

Contribution

It provides a precise lower bound on the negative part in CP decompositions, shows the positive part yields optimal CP approximations, and determines minimal auxiliary space dimensions for extensions.

Findings

Lower bound on Hilbert-Schmidt norm of negative part in CP decomposition

Positive part of decomposition yields optimal CP approximation

Explicit constructions for minimal auxiliary space dimensions

Abstract

This study investigates Hermitian linear maps, focusing on their decomposition into completely positive (CP) maps and their extensions to CP maps using auxiliary spaces. We derive a precise lower bound on the Hilbert-Schmidt norm of the negative component in any CP decomposition, proving its attainability through the Jordan decomposition. Additionally, we demonstrate that the positive part of this decomposition provides the optimal CP approximation in the Hilbert-Schmidt norm. We also determine the minimal dimension of an auxiliary space required to extend a Hermitian map to a CP map, with explicit constructions provided. Practical examples illustrate the application of our results.