Choi matrices revisited. II

TL;DR

This paper explores all variants of Choi matrices for linear maps in finite-dimensional spaces, characterizing their structure via non-degenerate bilinear forms and analyzing their properties in matrix algebras.

Contribution

It provides a comprehensive classification of Choi matrix variants based on bilinear forms and examines their implications for positivity and entanglement criteria.

Findings

All variants of Choi matrices are determined by non-degenerate bilinear forms.

Characterization of variants that preserve key positivity correspondences.

Comparison of different bilinear form definitions by de Pillis and Choi.

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 $\le k$ as well as $k$-positivity and $k$-block-positivity. We also compare de Pillis' definition [Pacific J. Math. 23 (1967), 129--137] and Choi's definition [Linear Alg. Appl. 10 (1975), 285--290], which arise from different bilinear forms.