# Journal of Operator Theory

Volume 90, Issue 2, Autumn 2023 pp. 365-384.

Abstract operator systems over the cone of positive semidefinite matrices**Authors**: Martin Berger (1), Tim Netzer (2)

**Author institution:**(1) Department of Mathematics, University of Innsbruck, Innsbruck, 6020, Austria

(2) Department of Mathematics, University of Innsbruck, Innsbruck, 6020, Austria

**Summary:**The operator systems of separable matrices, of positive semidefinite matrices, and of block positive matrices are well-known operator systems over the cone of positive semidefinite matrices. Less well-studied are the operator systems of matrices with positive semidefinite partial transpose, doubly positive matrices, and decomposable matrices. We investigate which of these systems is finitely generated, and which admits a finite-dimensional realization. We show that decomposable maps form a system which does not admit a finite-dimensional realization, whereas the system of doubly completely positive maps is not finitely generated.

**DOI:**http://dx.doi.org/10.7900/jot.2021dec10.2373

**Keywords:**operator system, positive semidefinite matrix, spectrahedra, quantum information

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