Related papers: Infinite dimensional analogues of Choi matrices

Infinite dimensional analogues of Choi matrices

URL: http://arxiv.org/abs/2311.18240v2
Date: Sat, 6 Jul 2024 06:55:43 GMT
Title: Infinite dimensional analogues of Choi matrices
Authors: Kyung Hoon Han, Seung-Hyeok Kye, Erling Størmer,
Abstract summary: Choi matrices are useful to characterize positivity of maps as well as complete positivity. It turns out that such correspondences are possible for every normal completely bounded map if and only if the factor is of type I. We also define the notion of $k$-superpositive maps, which turns out to be equivalent to the property of $k$-partially entanglement breaking.
Score: 0.0
License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
Abstract: For a class of linear maps on a von Neumann factor, we associate two objects, bounded operators and trace class operators, both of which play the roles of Choi matrices. Each of them is positive if and only if the original map on the factor is completely positive. They are also useful to characterize positivity of maps as well as complete positivity. It turns out that such correspondences are possible for every normal completely bounded map if and only if the factor is of type I. As an application, we provide criteria for Schmidt numbers of normal positive functionals in terms of Choi matrices of $k$-positive maps, in infinite dimensional cases. We also define the notion of $k$-superpositive maps, which turns out to be equivalent to the property of $k$-partially entanglement breaking.

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