Playing with fidelities. (English) Zbl 1045.46046
Summary: The notion of partial fidelities as invented by A. Uhlmann [Rep. Math. Phys. 45, 407-418 (2000; Zbl 0962.47003)] for pairs of finite-dimensional density matrices is extended to the \(vN\)-algebraic context and is considered and thoroughly discussed in detail from a mathematical point of view. Especially, in the case of semifinite \(vN\)-algebras, formulae and estimates for the partial fidelity between the functionals of a dense cone of inner derived normal positive linear forms are obtained. Also, some generalities on the notion of fidelity in quantum physics are collected in the Appendix, and another system of mathematical axioms for fidelity over density operators, which is based on the concept of relative majorization and which is intimately related to complete positivity, is proposed.
MSC:
| 46L89 | Other “noncommutative” mathematics based on \(C^*\)-algebra theory |
| 58B10 | Differentiability questions for infinite-dimensional manifolds |
| 46L10 | General theory of von Neumann algebras |
| 58B20 | Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds |
| 81R15 | Operator algebra methods applied to problems in quantum theory |
| 81T05 | Axiomatic quantum field theory; operator algebras |
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