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.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.

arXiv:1606.03584International Mathematics Research NoticesApr 27, 20188 citations

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.