Complete positivity of the map from a basis to its dual basis
Vern I. Paulsen, Fred Shultz · Journal of Mathematical Physics · 2013
The dual of a matrix ordered space has a natural matrix ordering that makes the dual space matrix ordered as well. The purpose of these notes is to give a condition that describes when the linear map taking a basis of Mn to its dual basis is a complete order isomorphism. We exhibit “natural” orthonormal bases for Mn such that this map is an order isomorphism, but not a complete order isomorphism. Included among such bases is the Pauli basis. Our results generalize the Choi matrix by giving conditions under which the role of the standard basis {Eij} can be replaced by other bases.