Linear maps preserving determinant of tensor products of Hermitian matrices

Let m,n≥2m,n≥2 be positive integers and let HnHn be the set of n×nn×n complex Hermitian matrices. We study linear maps ϕ:Hmn→Hmnϕ:Hmn→Hmn satisfying det⁡(A⊗B)=det⁡(ϕ(A⊗B))det⁡(A⊗B)=det⁡(ϕ(A⊗B)) for all A∈HmA∈Hm, B∈HnB∈Hn. The connection of the problem to quantum information science is mentioned.