Schoenberg Correspondence for $k$-(Super)Positive Maps on Matrix Algebras

TL;DR

This paper extends the Schoenberg correspondence to non-unital semigroups of linear maps on matrix algebras, characterizing their generators based on positivity properties and providing new insights into their evolution.

Contribution

It generalizes Schoenberg's correspondence to non-unital semigroups and characterizes generators of $k$-positive, $k$-superpositive, or $k$-entanglement breaking maps.

Findings

Reproves Lindblad, Gorini, Kossakowski, Sudarshan's theorem

Provides concrete examples of semigroups with positivity properties

Studies how positivity improves over time

Abstract

We prove a Schoenberg-type correspondence for non-unital semigroups which generalizes an analogous result for unital semigroup proved by Michael Sch\"urmann. It characterizes the generators of semigroups of linear maps on $M_n(C)$ which are $k$-positive, $k$-superpositive, or $k$-entanglement breaking. As a corollary we reprove Lindblad, Gorini, Kossakowski, Sudarshan's theorem. We present some concrete examples of semigroups of operators and study how their positivity properties can improve with time.