Transition probabilities of normal states determine the Jordan structure of a quantum system
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AbstractLet Phi : G(M-1) -> G(M-2) be a bijection (not assumed affine nor continuous) between the sets of normal states of two quantum systems, modelled on the self-adjoint parts of von Neumann algebras M-1 and M-2, respectively. This paper concerns with the situation when Phi preserves (or partially preserves) one of the following three notions of "transition probability" on the normal state spaces: the transition probability P-U introduced by Uhlmann [Rep. Math. Phys. 9, 273-279 (1976)], the transition probability P-R introduced by Raggio [Lett. Math. Phys. 6, 233-236 (1982)], and an "asymmetric transition probability" P-0 (as introduced in this article). It is shown that the two systems are isomorphic, i.e., M-1 and M-2 are Jordan *-isomorphic, if Phi preserves all pairs with zero Uhlmann (respectively, Raggio or asymmetric) transition probability, in the sense that for any normal states mu and nu, we have P(Phi(mu), Phi(nu)) = 0 if and only if P(mu, nu) = 0, where P stands for P-U (respectively, P-R or P-0). Furthermore, as an extension of Wigner's theorem, it is shown that there is a Jordan *-isomorphism Theta : M-2 -> M-1 satisfying Phi = Theta*vertical bar(G(M1)) if and only if Phi preserves the "asymmetric transition probability." This is also equivalent to Phi preserving the Raggio transition probability. Consequently, if Phi preserves the Raggio transition probability, it will preserve the Uhlmann transition probability as well. As another application, the sets of normal states equipped with either the usual metric, the Bures metric or "the metric induced by the self-dual cone," are complete Jordan *-invariants for the underlying von Neumann algebras. (C) 2015 AIP Publishing LLC.
All Author(s) ListLeung CW, Ng CK, Wong NC
Journal nameJournal of Mathematical Physics
Year2016
Month1
Day1
Volume Number57
Issue Number1
PublisherAMER INST PHYSICS
ISSN0022-2488
eISSN1089-7658
LanguagesEnglish-United Kingdom
Web of Science Subject CategoriesPhysics; Physics, Mathematical

Last updated on 2021-20-01 at 01:39