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arXiv:1510.01487 · Journal of Mathematical Physics

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

Chi-Wai Leung, Chi–Keung Ng, Ngai‐Ching Wong

Published Dec 1, 2015• 13 citations
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Abstract

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 PΦ(μ),Φ(ν)=0 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.

DOI
10.1063/1.4936404

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Let n>1 be a positive integer, {H1,…,Hn} be a finite collection of complex Hilbert spaces with dim⁡(Hk)≥2, and P1(Hk) be the set of all rank-1 self-adjoint projections on Hk, k=1,…,n. Set DP1⨂k=1nHk={A1⊗⋯⊗An:Ak∈P1(Hk),k=1,…,n}.We characterize the maps ϕ from D(P1(⨂k=1nHk)) to P1(⨂k=1nHk) preserving transition probability, i.e. tr(AB)=tr(ϕ(A)ϕ(B)),for all A,B∈DP1⨂k=1nHk.A particular case corresponding to n = 1 is well known as (non-surjective version) Wigner's theorem. Our result may be considered as a generalization of Wigner's theorem.

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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 PΦ(μ),Φ(ν)=0 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.
In the algebraic approach to quantum theory, a quantum observable is given by an element of a Jordan algebra and a state of the system is modelled by a normalized positive functional on the underlying algebra. Maximal deviation of a quantum observable is the largest statistical deviation one can obtain in a particular state of the system. The main result of the paper shows that each linear bijective transformation between JBW algebras preserving maximal deviations is formed by a Jordan isomorphism or a minus Jordan isomorphism perturbed by a linear functional multiple of an identity. It shows that only one numerical statistical characteristic has the power to determine the Jordan algebraic structure completely. As a consequence, we obtain that only very special maps can preserve the diameter of the spectra of elements. Nonlinear maps preserving the pseudometric given by maximal deviation are also described. The results generalize hitherto known theorems on preservers of maximal deviation in the case of self-adjoint parts of von Neumann algebras proved by Molnár.
Five conceptually distinct notions of symmetry in quantum theory are studied in the algebraic setting where a quantum system is characterized by a von Neumann algebra of observables and the set of normal states on the algebra. It is shown that all five symmetry notions are closely related and that the glue binding them together is the concept of a Jordan ∗-automorphism. For factor algebras a Jordan ∗-automorphism reduces either to an ∗-automorphism or a ∗-anti-automorphism. If the algebra is put in standard form then a ∗-automorphism is always unitarily implementable, whereas a ∗-anti-automorphism is always anti-unitarily implementable. However, there is no guarantee that a general von Neumann algebra admits ∗-anti-automorphisms or, if it does, that it admits order two (or involutory) ∗-anti-automorphisms). For non-factor algebras there can be genuine Jordan ∗-automorphisms that are neither ∗-automorphisms nor ∗-anti-automorphisms, and implementation is possible only through partial isometries. These developments enable generalized versions of Wigner's theorem on the implementation of transition probability preserving symmetries for von Neumann algebras. This review is largely an exercise in connecting the dots in existing mathematics and physics literature. But in the service of the philosophy of physics it is an exercise worth doing since the practitioners in this field seem largely unaware of or unappreciative of this literature and how it fits together to yield a multifaceted but unified picture of quantum symmetries. Along the way various interpretations issues worthy of further discussion are flagged.
The empirical content of quantum theory is entirely encoded in the Born probability pairing between states and effects. In finite dimension, we study the following structural problem: which reparametrizations of states and effects preserve all Born probabilities? We consider bijective affine maps on the state space and effect space that leave the Born pairing invariant for all state–effect pairs, and provide a complete classification of all such transformations. We prove that every probability-preserving reparametrization is necessarily induced by a Jordan *-automorphism of the matrix algebra B(H). In finite dimension, this restricts the transformation to the form A→UAU† or A→UA⊤U†, for some unitary operator U. In particular, equivalence classes of states under probability-preserving reparametrizations coincide with unitary orbits and are completely characterized by the spectrum of the density operator. The result shows that unitary and antiunitary conjugations exhaust all probability-preserving affine reparametrizations of the state–effect framework.
The interplay between the algebraic structure (operator algebras) for the quantum observables and the convex structure of the state space has been explored for a long time, and most advanced results are due to Alfsen and Shultz. Here we present a more elementary approach with a more general structure for the observables, which focuses on the transition probability of the quantum logical atoms. The binary case gives rise to the generalized qubit models and was fully developed in a preceding paper. Here we consider any case with finite information capacity (binary means that the information capacity is 2). A novel geometric property that makes any compact convex set a matching state space is presented. Generally, the transition probability is not symmetric; if it is symmetric, we get an inner product and a self-dual cone. The emerging mathematical structure comes close to the Euclidean Jordan algebras and becomes a new mathematical model for a potential extension of quantum theory.
Let n>1 be a positive integer, {H1,…,Hn} be a finite collection of complex Hilbert spaces with dim⁡(Hk)≥2, and P1(Hk) be the set of all rank-1 self-adjoint projections on Hk, k=1,…,n. Set DP1⨂k=1nHk={A1⊗⋯⊗An:Ak∈P1(Hk),k=1,…,n}.We characterize the maps ϕ from D(P1(⨂k=1nHk)) to P1(⨂k=1nHk) preserving transition probability, i.e. tr(AB)=tr(ϕ(A)ϕ(B)),for all A,B∈DP1⨂k=1nHk.A particular case corresponding to n = 1 is well known as (non-surjective version) Wigner's theorem. Our result may be considered as a generalization of Wigner's theorem.