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Let  n &gt;1 be a positive integer, { H1 , …,  Hn} be a finite collection of complex Hilbert spaces with, and  P1 ( Hk) be the set of all rank-1 self-adjoint projections on  Hk,  k  = 1, …,  n . SetWe characterize the maps  ϕ  fromtopreserving transition probability, i.e.A particular case corresponding to  n  = 1 is well known as (non-surjective version) Wigner's theorem. Our result may be considered as a generalization of Wigner's theorem.

Maps preserving transition probability from pure product states to pure states