LINEAR MAPS PRESERVING TENSOR PRODUCTS OF RANK-ONE HERMITIAN MATRICES

For a positive integer n >= 2, let M-n be the set of n x n complex matrices and H-n the set of Hermitian matrices in M-n. We characterize injective linear maps phi : H-m1... ml -> H-n satisfying rank(A(1) circle times . . . circle times A(l)) = 1 double right arrow rank(phi (A(1) circle times . . . circle times A(l))) = 1 for all A(k) epsilon H-mk, k = 1, . . . , l, where l; m(1), . . . , m(l) >= 2 are positive integers. The necessity of the injectivity assumption is shown. Moreover, the connection of the problem to quantum information science is mentioned.