Choi matrices revisited

TL;DR

This paper investigates the conditions under which the Choi matrix representation of linear maps between matrix algebras remains valid when using bases other than matrix units, establishing necessary and sufficient criteria.

Contribution

It extends previous work by providing a complete characterization of bases that preserve the Choi matrix correspondence for complete positivity.

Findings

Paulsen and Shultz's condition is necessary and sufficient.

Necessary and sufficient conditions for basis independence are established.

The results clarify the basis dependence of Choi matrices in quantum information theory.

Abstract

A linear map between matrix algebras corresponds to the Choi matrix in the tensor product of two matrix algebras, whose definition depends on the matrix units. Paulsen and Shultz [J. Math. Phys. {\bf 54} (2013), 072201] considered the question if one can replace matrix units by another basis of matrix algebras in the definition of Choi matrix to retain the correspondence between complete positivity of maps and 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.