Complete positivity of the map from a basis to its dual basis
| Autor: | Vern I. Paulsen, Fred Shultz |
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| Rok vydání: | 2012 |
| Předmět: |
Pure mathematics
Quantum Physics Order isomorphism Basis (linear algebra) Dual space 010102 general mathematics Mathematics - Operator Algebras FOS: Physical sciences Statistical and Nonlinear Physics 01 natural sciences Linear map Matrix (mathematics) 46N50 47L07 0103 physical sciences Dual basis FOS: Mathematics Orthonormal basis 010307 mathematical physics Isomorphism 0101 mathematics Quantum Physics (quant-ph) Operator Algebras (math.OA) Mathematical Physics Mathematics |
| DOI: | 10.48550/arxiv.1212.4787 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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