Complete positivity of the map from a basis to its dual basis

Vern I. Paulsen
Fred Shultz

Copyright (2013) American Institute of Physics. The published article may be found at http://dx.doi.org/10.1063/1.4812329)

Abstract

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 M n to its dual basis is a complete order isomorphism. We exhibit “natural” orthonormal bases for M n 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 {E ij } can be replaced by other bases.