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 $\varphi :P_1\left(H\right)\to P_n\left(K\right)$ multiplying $n$ times the transition probability and maps $\varphi :P_1\left(H\right)\to P_n\left(K\right)$ sending each complete orthogonal system of rank- $1$ projections to some complete orthogonal system of rank- $n$ projections.