Linear maps preserving numerical radius of tensor products of matrices

Let m, n≥2 be positive integers. Denote by Mmthe set of m×m complex matrices and by w(X) the numerical radius of a square matrix X. Motivated by the study of operations on bipartite systems of quantum states, we show that a linear map φ:Mmn→Mmnsatisfies w(φ(A⊗B))=w(A⊗B)forall A∈Mm and B∈Mn if and only if there is a unitary matrix U∈Mmnand a complex unit ξ such that φ(A⊗B)=ξU(φ1(A)⊗φ2(B))U*forall A∈Mm and B∈Mn, where φkis the identity map or the transposition map X{mapping}Xtfor k=1, 2, and the maps φ1and φ2will be of the same type if m, n≥3. In particular, if m, 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.