Transition probabilities of normal states determine the Jordan structure of a quantum system

Let $\Phi:\mathfrak{S}(M_1)\to \mathfrak{S}(M_2)$ 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 Uhlmann transition probability $P_U$, the Raggio transition probability $P_B$ and an "asymmetric transition probability" $P_0$ as defined 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, i.e., for any normal states $\mu$ and $\nu$, we have $$P\big(\Phi(\mu),\Phi(\nu)\big) = 0 \quad \text{if and only if} \quad P(\mu,\nu)=0,$$ 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 $\Theta:M_2\to M_1$ with $$\Phi = \Theta^*|_{\mathfrak{S}(M_1)}$$ 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.