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

Abstract

We prove a Schoenberg-type correspondence for non-unital semigroups which generalizes an analogous result for unital semigroup proved by Michael Schürmann. 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.

arXiv comment18 pages, v2 contains minor corrections. v3: parts of Section 2 moved to Sections 4 and 6, additional details are inserted in several proofs, and further minor corrections, v4 cibtains final minor corrections

References in corpus (8)

• Separability Criterion for Density Matrices
• Separability of Mixed States: Necessary and Sufficient Conditions
• General Entanglement Breaking Channels
• Cones of positive maps and their duality relations
• When Do Composed Maps Become Entanglement Breaking?
• On a family of a linear maps from $M_n(\mathbb{C})$ to $M_{n^2}(\mathbb{C})$
• Characterization of equivariant maps and application to entanglement detection
• Mapping cone of $k$-Entanglement Breaking Maps