Schoenberg correspondence for k-(super)positive maps on matrix algebras.

We prove a Schoenberg-type correspondence for non-unital semigroups which generalizes an analogous result for unital semigroup proved by Schürmann (in: Quantum probability and applications II, proceedings of a 2nd workshop, Heidelberg/Germany 1984, lecture notes in mathematics, vol 1136, pp 475–492, 1985). 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 (J Math Phys 17:821, 1976; Commun Math Phys 48:119-130, 1976). We present some concrete examples of semigroups of operators and study how their positivity properties can improve with time. [ABSTRACT FROM AUTHOR]