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arXiv:2410.13120

Choi matrices revisited. III

Kyung Hoon Han, Seung-Hyeok Kye

Published Oct 17, 2024• 0 citations
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Abstract

We look for all linear isomorphisms from the mapping spaces onto the tensor products of matrices which send $k$-superpositive maps onto unnormalized bi-partite states of Schmidt numbers less than or equal to $k$. They also send $k$-positive maps onto $k$-block-positive matrices. We also look for all the bilinear pairings between the mapping spaces and tensor products of matrices which retain the usual duality between $k$-positivity and Schmidt numbers $\le k$. They also retain the duality between $k$-superpositivity and $k$-block-positivity.

DOI
10.48550/arxiv.2410.13120

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We look for all linear isomorphisms from the mapping spaces onto the tensor products of matrices which send kkk-superpositive maps onto unnormalized bi-partite states of Schmidt numbers less than or equal to kkk. They also send kkk-positive maps onto kkk-block-positive matrices. We also look for all the bilinear pairings between the mapping spaces and tensor products of matrices which retain the usual duality between kkk-positivity and Schmidt numbers ≤k\le k≤k. They also retain the duality between kkk-superpositivity and kkk-block-positivity.
Building on recent works that investigate positivity preserving matrix products, we {examine} the class of \JCP (\jcp) matrix products. A bilinear map on the Cartesian product of the space of n by n matrices with itself into m by m matrices is a \jcp matrix product if the natural linear map it induces on the tensor product of the space of n by n matrices with itself into m by m matrices is completely positive. In particular, a matrix product is \jcp if and only if its naturally associated Choi matrix is positive semidefinite. Similarly, a matrix product is \jcp if and only if it admits a Choi-Kraus representation. We use the Choi-Kraus representation of \jcp matrix products to study various basic properties, including positivity lower bounds, commutativity, units, causality, and separability. As examples, we apply our results to the Schur (Hadamard) product and the convolution product.
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