Linear Maps Preserving Numerical Radius of Tensor Products of Matrices

Let m,n ≥ 2 be positive integers. Denote by Mm 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 : 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 ∈Mmn and a complex unit ξ such that φ(A⊗B) = ξU(φ1(A)⊗ φ2(B))U for all A ∈Mm and B ∈Mn, where φk is the identity map or the transposition map X 7→ X for k = 1, 2, and the maps φ1 and φ2 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. 2010 Math. Subj. Class.: 15A69, 15A86, 15A60, 47A12.