Linear maps preserving (p,k)-norms of tensor products of matrices

Abstract

Let m,n ≥ 2 be integers. Denote by Mn the set of n × n complex matrices and ∥⋅∥(p,k) the (p, k) norm on Mmn with a positive integer k ≤ mn and a real number p > 2. We show that a linear map ϕ ∶ Mmn → Mmn satisfies ∥ϕ(A⊗B)∥(p,k) =∥A⊗B∥(p,k) for all A∈ Mm and B ∈ Mn if and only if there exist unitary matrices U,V ∈ Mmn such that ϕ(A⊗B)=U(φ1(A)⊗φ2(B))V forall A∈ Mm andB ∈ Mn, whereφs istheidentitymaporthetranspositionmap X ↦ XT fors = 1,2.Theresultisalsoextended to multipartite systems.