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Annals of Physics

A direct proof of Wigner's theorem on maps which preserve transition probabilities between pure states of quantum systems

F. Almeida. Mathematical Research Unit

Published Dec 1, 1989• 3 citations
Publisher
DOI
10.1016/0003-4916(89)90184-x

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Abstract Let H be either a complex inner product space of dimension at least two or a real inner product space of dimension at least three, and let us fix an $\alpha \in (0,\tfrac{\pi }{2} )$. The purpose of this paper is to characterise all bijective transformations on the projective space P(H) which preserve the quantum angle $\alpha$ (or Fubini–Study distance $\alpha$) between lines in both directions. (Let us emphasise that we do not assume anything about the preservation of other quantum angles). For real inner product spaces and when $H=\mathbb{C}^2$ we do this for every $\alpha$, and when H is a complex inner product space of dimension at least three we describe the structure of such transformations for $\alpha \leq \tfrac{\pi }{4}$. Our result immediately gives an Uhlhorn-type generalisation of Wigner’s theorem on quantum mechanical symmetry transformations, that is considered to be a cornerstone of the mathematical foundations of quantum mechanics. Namely, under the above assumptions, every bijective map on the set of pure states of a quantum mechanical system that preserves the transition probability $\cos ^2\alpha$ in both directions is a Wigner symmetry (thus automatically preserves all transition probabilities), except for the case when $H=\mathbb{C}^2$ and $\alpha = \tfrac{\pi }{4}$ where an additional possibility occurs. (Note that the classical theorem of Uhlhorn is the solution for the $\alpha = \tfrac{\pi }{2}$ case). Usually in the literature, results which are connected to Wigner’s theorem are discussed under the assumption of completeness of H; however, here we shall remove this unnecessary hypothesis in our investigation. Our main tool is a characterisation of bijective maps on unit spheres of real inner product spaces which preserve one spherical angle in both directions.

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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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Abstract Let $H$ be a Hilbert space and $P(H)$ be the projective space of all quantum pure states. Wigner’s theorem states that every bijection $\phi \colon P(H)\to P(H)$ that preserves the quantum angle between pure states is automatically induced by either a unitary or an antiunitary operator $U\colon H\to H$. Uhlhorn’s theorem generalizes this result for bijective maps $\phi $ that are only assumed to preserve the quantum angle $\frac{\pi }{2}$ (orthogonality) in both directions. Recently, two papers, written by Li–Plevnik–Šemrl and Gehér, solved the corresponding structural problem for bijections that preserve only one fixed quantum angle $\alpha $ in both directions, provided that $0 < \alpha \leq \frac{\pi }{4}$ holds. In this paper we solve the remaining structural problem for quantum angles $\alpha $ that satisfy $\frac{\pi }{4} < \alpha < \frac{\pi }{2}$, hence complete a programme started by Uhlhorn. In particular, it turns out that these maps are always induced by unitary or antiunitary operators, however, our assumption is much weaker than Wigner’s.

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Abstract Let H be either a complex inner product space of dimension at least two or a real inner product space of dimension at least three, and let us fix an α∈(0,π2)\alpha \in (0,\tfrac{\pi }{2} )α∈(0,2π​). The purpose of this paper is to characterise all bijective transformations on the projective space P(H) which preserve the quantum angle α\alphaα (or Fubini–Study distance α\alphaα) between lines in both directions. (Let us emphasise that we do not assume anything about the preservation of other quantum angles). For real inner product spaces and when H=C2H=\mathbb{C}^2H=C2 we do this for every α\alphaα, and when H is a complex inner product space of dimension at least three we describe the structure of such transformations for α≤π4\alpha \leq \tfrac{\pi }{4}α≤4π​. Our result immediately gives an Uhlhorn-type generalisation of Wigner’s theorem on quantum mechanical symmetry transformations, that is considered to be a cornerstone of the mathematical foundations of quantum mechanics. Namely, under the above assumptions, every bijective map on the set of pure states of a quantum mechanical system that preserves the transition probability cos⁡2α\cos ^2\alphacos2α in both directions is a Wigner symmetry (thus automatically preserves all transition probabilities), except for the case when H=C2H=\mathbb{C}^2H=C2 and α=π4\alpha = \tfrac{\pi }{4}α=4π​ where an additional possibility occurs. (Note that the classical theorem of Uhlhorn is the solution for the α=π2\alpha = \tfrac{\pi }{2}α=2π​ case). Usually in the literature, results which are connected to Wigner’s theorem are discussed under the assumption of completeness of H; however, here we shall remove this unnecessary hypothesis in our investigation. Our main tool is a characterisation of bijective maps on unit spheres of real inner product spaces which preserve one spherical angle in both directions.
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.
Abstract Let HHH be a Hilbert space and P(H)P(H)P(H) be the projective space of all quantum pure states. Wigner’s theorem states that every bijection ϕ ⁣:P(H)→P(H)\phi \colon P(H)\to P(H)ϕ:P(H)→P(H) that preserves the quantum angle between pure states is automatically induced by either a unitary or an antiunitary operator U ⁣:H→HU\colon H\to HU:H→H. Uhlhorn’s theorem generalizes this result for bijective maps $\phi $ that are only assumed to preserve the quantum angle π2\frac{\pi }{2}2π​ (orthogonality) in both directions. Recently, two papers, written by Li–Plevnik–Šemrl and Gehér, solved the corresponding structural problem for bijections that preserve only one fixed quantum angle $\alpha $ in both directions, provided that 0<α≤π40 < \alpha \leq \frac{\pi }{4}0<α≤4π​ holds. In this paper we solve the remaining structural problem for quantum angles $\alpha $ that satisfy π4<α<π2\frac{\pi }{4} < \alpha < \frac{\pi }{2}4π​<α<2π​, hence complete a programme started by Uhlhorn. In particular, it turns out that these maps are always induced by unitary or antiunitary operators, however, our assumption is much weaker than Wigner’s.