The structure of projection-valued states: A generalization of Wigner's theorem

Georges Chevalier

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.

Filippus S. Roux

Dorje C. Brody

Wigner’s theorem asserts that an isometric (probability conserving) transformation on a quantum state space must be generated by a Hamiltonian that is Hermitian. It is shown that when the hermiticity condition on the Hamiltonian is relaxed, we obtain the following complex generalization of Wigner’s theorem: a holomorphically projective (complex geodesic curves preserving) transformation on a quantum state space must be generated by a Hamiltonian that is not necessarily Hermitian.

William K. Wootters

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.