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

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 the n by n matrices to its dual basis is a complete order isomorphism and complete co-order isomorphism. In the case of the standard matrix units this map is a complete order isomorphism and this is a restatement of the correspondence between completely positive maps and the Choi matrix. However, we exhibit natural orthonormal bases for the matrices such that this map is an order isomorphism, but not a complete order isomorphism. Some bases yield complete co-order isomorphisms. Included among such bases is the Pauli basis and tensor products of the Pauli basis. Consequently, when the Pauli basis is used in place of the the matrix unit basis, the analogue of Choi's theorem is a characterization of completely co-positive maps.