Generalizations of Wigner's theorem from rank-$ 1 $ projections to rank-$ n $ projections

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.