Wigner's theorem and its generalizations

Kan He, Hou Jin-chuan

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.

Pankaj Sharan

Andrzej Łuczak

A general form of the Wigner-Araki-Yanase theorem is proved in the framework of quantum measurement theory based on the notion of instrument.

Pankaj Sharan

R. Sala Mayato, S. Brouard, Juan Gonzalo Muga

It is found that, in general, Wigner trajectories satisfy Liouville’s theorem only locally, i.e., for restricted phase space and time domains. This fact limits their possible applications. Examples are provided to visualize the process of creation and destruction of Wigner trajectories. It is argued that Weyl transforms of Heisenberg operators are, however, viable alternatives to Wigner trajectories, even though they do not satisfy Liouville’s theorem either.