Ela Additive Preservers of Tensor Product of Rank One Hermitian Matrices

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 ρm denote the set of all rank-one matrices in Hm. In the tensor product space ⊗ k i=1 Hmi , let ⊗ k i=1 ρmi denote the set of all decomposable elements ⊗ k i=1 Ai such that Ai ∈ ρmi , i = 1, . . . , k. In this paper, additive maps T from Hm ⊗Hn to Hs ⊗Ht such that T (ρm ⊗ ρn) ⊆ (ρs ⊗ ρt) ∪ {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 ⊗ k i=1 Hmi to ⊗ l i=1 Hni such that L ( ⊗ i=1 ρmi ) ⊆ ⊗ l i=1 ρni where 2 ≤ k ≤ l. When K is algebraically closed and K = F , it is shown that every linear map on ⊗ k i=1 Hmi that preserves ⊗ k i=1 ρmi is induced by k bijective linear rank-one preservers on Hmi , i = 1, . . . , k.