Linear maps preserving numerical radius of tensor products of matrices

Let m,n≥2m,n≥2 be positive integers. Denote by MmMm the set of m×mm×m complex matrices and by w(X)w(X) the numerical radius of a square matrix XX. Motivated by the study of operations on bipartite systems of quantum states, we show that a linear map ϕ:Mmn→Mmnϕ:Mmn→Mmn satisfies w(ϕ(A⊗B))=w(A⊗B)for all  A∈Mm  and  B∈Mn if and only if there is a unitary matrix U∈MmnU∈Mmn and a complex unit ξξ such that ϕ(A⊗B)=ξU(φ1(A)⊗φ2(B))U∗for all  A∈Mm  and  B∈Mn, where φkφk is the identity map or the transposition map X↦XtX↦Xt for k=1,2k=1,2, and the maps φ1φ1 and φ2φ2 will be of the same type if m,n≥3m,n≥3. In particular, if m,n≥3m,n≥3, the map corresponds to an evolution of a closed quantum system (under a fixed unitary operator), possibly followed by a transposition. The results are extended to multipartite systems.