Let K be a field of characteristic not two or three with an involution and F be its fixed field. Let Hm be the F-vector space of all m-square Hermitian matrices over K. Let �m denote the set of all rank-one matrices in Hm. In the tensor product space N k=1 Hmi, let N k i=1 �mi denote the set of all decomposable elements N k i=1 Ai such that Ai 2 �mi , i = 1,...,k. In this paper, additive maps T from Hm Hn to Hs Ht such that T(�mn) � (�st) ( {0} are characterized. From this, a characterization of linear maps is found between tensor products of two real vector spaces of complex Hermitian matrices that send separable pure states to separable pure states. Also classified in this paper are almost surjective additive maps L from N k=1 Hmi to Nl i=1 Hni such that L k i=1�mi � � Nl i=1 �ni where 2 � kl. When K is algebraically closed and K = F, it is shown that every linear map on Nk=1 Hmi that preserves Nk=1 �mi is induced by k bijective linear rank-one preservers on Hmi , i = 1,...,k.