Complément BIV. Théorème de Uhlhorn-Wigner

Peter Šemrl

Abstract We prove an optimal version of Uhlhorn’s generalization of Wigner’s unitary–antiunitary theorem. The main tool in our proof is Gleason’s theorem.

Georges Chevalier

Mark Pankov

Classical Wigner’s theorem characterizes unitary and anti-unitary operators as symmetries of pure states of quantum mechanical systems, i.e. rank one projections. We consider a non-injective version of Wigner’s theorem as well as Uhlhorn’s version concerning orthogonality preserving transformations and describe various extensions of these results onto other Grassmannians.

Kan He, Hou Jin-chuan

Pankaj Sharan

Yulong Tian, Jinli Xu

We provide generalizations of the classical Wigner's theorem as well as Uhlhorn's version of Wigner's theorem by considering maps that send rank-$ 1 $ projections to rank-$ n $ projections. Namely, we describe the general form of maps $ \phi:P_{1}\left (H\right) \to P_{n}\left (K\right) $ multiplying $ n $ times the transition probability and maps $ \phi:P_{1}\left (H\right) \to P_{n}\left (K\right) $ sending each complete orthogonal system of rank-$ 1 $ projections to some complete orthogonal system of rank-$ n $ projections.