Linear maps preserving Ky Fan norms and Schatten norms of tensor products of matrices

For a positive integer n, let Mn be the set of n × n complex matrices. Suppose || · || is the Ky Fan k-norm with 1 ≤ k ≤ mn or the Schatten p-norm with 1 ≤ p ≤ ∞ (p ≠ 2) on M mn, where m, n ≥ 2 are positive integers. It is shown that a linear map φ : Mmn → Mmn satisfying ||A⊗B|| = ||φ(A⊗B)|| for all A ∈ Mm and B ∈ Mn if and only if there are unitary U, V ∈ Mmn such that φ has the form A ⊗ B → U(φ1(A) ⊗ φ2(B))V , where φs(X) is either the identity map X → X or the transposition map X → Xt. The results are extended to tensor space Mn1 ⊗ ⋯ ⊗ Mnm of higher level. The connection of the problem to quantum information science is mentioned.