Transition probabilities of normal states determine the Jordan structure of a quantum system
Publication in refereed journal

香港中文大學研究人員

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摘要Let 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.
著者Leung CW, Ng CK, Wong NC
期刊名稱Journal of Mathematical Physics
出版年份2016
月份1
日期1
卷號57
期次1
出版社AMER INST PHYSICS
國際標準期刊號0022-2488
電子國際標準期刊號1089-7658
語言英式英語
Web of Science 學科類別Physics; Physics, Mathematical

上次更新時間 2021-21-02 於 01:12