Let Φ : 𝔖(M1) → 𝔖(M2) 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 M1 and M2, respectively. This paper concerns with the situation when Φ preserves (or partially preserves) one of the following three notions of “transition probability” on the normal state spaces: the transition probability PU introduced by Uhlmann [Rep. Math. Phys. 9, 273-279 (1976)], the transition probability PR introduced by Raggio [Lett. Math. Phys. 6, 233-236 (1982)], and an “asymmetric transition probability” P0 (as introduced in this article). It is shown that the two systems are isomorphic, i.e., M1 and M2 are Jordan ∗-isomorphic, if Φ preserves all pairs with zero Uhlmann (respectively, Raggio or asymmetric) transition probability, in the sense that for any normal states μ and ν, we have if and only if P(μ, ν) = 0, where P stands for PU (respectively, PR or P0). Furthermore, as an extension of Wigner’s theorem, it is shown that there is a Jordan ∗-isomorphism Θ : M2 → M1 satisfying Φ = Θ∗|𝔖(M1) if and only if Φ preserves the “asymmetric transition probability.” This is also equivalent to Φ preserving the Raggio transition probability. Consequently, if Φ 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.
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January 2016
Research Article|
December 01 2015
Transition probabilities of normal states determine the Jordan structure of a quantum system
Chi-Wai Leung;
Chi-Wai Leung
a)
1Department of Mathematics,
The Chinese University of Hong Kong
, Shatin, Hong Kong
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Chi-Keung Ng;
Chi-Keung Ng
b)
2Chern Institute of Mathematics and LPMC,
Nankai University
, Tianjin 300071, China
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Ngai-Ching Wong
Ngai-Ching Wong
c)
3Department of Applied Mathematics,
National Sun Yat-sen University
, Kaohsiung 80424, Taiwan
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a)
E-mail address: [email protected]
b)
E-mail address: [email protected]
c)
E-mail address: [email protected]
J. Math. Phys. 57, 015212 (2016)
Article history
Received:
July 15 2015
Accepted:
November 11 2015
Citation
Chi-Wai Leung, Chi-Keung Ng, Ngai-Ching Wong; Transition probabilities of normal states determine the Jordan structure of a quantum system. J. Math. Phys. 1 January 2016; 57 (1): 015212. https://doi.org/10.1063/1.4936404
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