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Let : (M1) (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 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 preserves all pairs with zero Uhlmann (respectively, Raggio or asymmetric) transition probability, in the sense that for any normal states and , we have P(),()=0 if and only if P(,

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