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arXiv:2307.09247 · Journal of Mathematical Physics

Choi matrices revisited. II

Kyung Hoon Han, Seung-Hyeok Kye

Published Oct 1, 2023• 2 citations
arXivHTMLPDF

Abstract

In this paper, we consider all possible variants of Choi matrices of linear maps, and show that they are determined by non-degenerate bilinear forms on the domain space. We will do this in the setting of finite dimensional vector spaces. In case of matrix algebras, we characterize all variants of Choi matrices which retain the usual correspondences between k-superpositivity and Schmidt number ≤k as well as k-positivity and k-block-positivity. We also compare de Pillis’ definition [Pac. J. Math. 23, 129–137 (1967)] and Choi’s definition [Linear Algebra Appl. 10, 285–290 (1975)], which arise from different bilinear forms.

DOI
10.1063/5.0169483

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In this paper, we consider all possible variants of Choi matrices of linear maps, and show that they are determined by non-degenerate bilinear forms on the domain space. We will do this in the setting of finite dimensional vector spaces. In case of matrix algebras, we characterize all variants of Choi matrices which retain the usual correspondences between k-superpositivity and Schmidt number ≤k as well as k-positivity and k-block-positivity. We also compare de Pillis’ definition [Pac. J. Math. 23, 129–137 (1967)] and Choi’s definition [Linear Algebra Appl. 10, 285–290 (1975)], which arise from different bilinear forms.
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.
A linear map between matrix algebras corresponds to the Choi matrix in the tensor product of two matrix algebras whose definition depends on matrix units. Paulsen and Shultz [J. Math. Phys. 54, 072201 (2013)] considered the question if one can replace matrix units by another basis of matrix algebras in the definition of the Choi matrix to retain the correspondence between the complete positivity of maps and the positivity of Choi matrices and gave a sufficient condition on basis under which this is true. In this note, we provide necessary and sufficient conditions to see that the Paulsen–Shultz condition is also necessary.
Several investigations have been done on positive maps on their algebraic structures with more emphasis on completely positive maps. In this study, we have described the structure of the Choi matrices for 2-positive maps on positive semidefinite matrices and the conditions for complete positivity of positive linear maps from n to n+1. The motivation behind these objectives is work done by Majewski and Marciniak on the structure of positive maps ф from Mn to Mn + 1(2 ≥ 2) between matrix algebras.
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.