Linear maps preserving numerical radius of tensor products of matrices

Let m , n ≥ 2 be positive integers. Denote by M m the 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 : M m n → M m n satisfies w ( ϕ ( A ⊗ B ) ) = w ( A ⊗ B ) for all  A ∈ M m  and  B ∈ M n if and only if there is a unitary matrix U ∈ M m n and a complex unit ξ such that ϕ ( A ⊗ B ) = ξ U ( φ 1 ( A ) ⊗ φ 2 ( B ) ) U ∗ for all  A ∈ M m  and  B ∈ M n , where φ k is the identity map or the transposition map X ↦ X t for k = 1 , 2 , and the maps φ 1 and φ 2 will 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.