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 x 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 phi : M-mn -> M-mn satisfies w(phi(A circle times B)) = w (A circle times B) for all A is an element of M-m, and B is an element of M-n if and only if there is a unitary matrix U is an element of M-mn and a complex unit xi such that phi(A circle times B) = xi U(phi(1)(A) circle times phi(2)(B))U* for all A is an element of M-m and B is an element of M-n, where phi(k) is the identity map or the transposition map X bar right arrow X-t for k = 1, 2, and the maps phi(1) and phi(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. (c) 2013 Elsevier Inc. All rights reserved.