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

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Abstract

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 \(M_n(\mathbb {C})\) 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.

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Acknowledgements

P.C. and U.F. were supported by the ANR Project No. ANR-19-CE40-0002. Bhat is suppported by the J C Bose Fellowship JBR/2021/000024 of SERB(India).

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The authors received no funding for the preparation of this manuscript besides the grants mentioned above.

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Authors and Affiliations

  1. Stat-Math. Unit, Indian Statistical Institute, R V College Post, Bengaluru, 560059, India

    B. V. Rajarama Bhat

  2. Université de Franche-Comté, CNRS, UMR 6623, LmB, 25000, Besançon, France

    Purbayan Chakraborty & Uwe Franz

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  1. B. V. Rajarama Bhat
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  2. Purbayan Chakraborty
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  3. Uwe Franz
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Bhat, B.V.R., Chakraborty, P. & Franz, U. Schoenberg correspondence for k-(super)positive maps on matrix algebras. Positivity 27, 51 (2023). https://doi.org/10.1007/s11117-023-01003-6

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