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

Abstract

Let $m,n\ge 2$ be integers. Denote by $M_n$ the set of $n\times n$ complex matrices. Let $\|\cdot\|_{(p,k)}$ be the $(p,k)$ norm on $M_{mn}$ with $1\leq k\leq mn$ and $2<p<\infty$. We show that a linear map $\phi:M_{mn}\rightarrow M_{mn}$ satisfies $$\|\phi(A\otimes B)\|_{(p,k)}=\|A\otimes B\|_{(p,k)} {\rm\quad for~ all\quad}A\in M_m {\rm ~and ~}B\in M_n$$ if and only if there exist unitary matrices $U,V\in M_{mn}$ such that $$\phi(A\otimes B)=U(\varphi_1(A)\otimes \varphi_2(B))V {\rm\quad for~ all\quad}A\in M_m {\rm~ and~ }B\in M_n,$$ where $\varphi_s$ is the identity map or the transposition map $X\to X^T$ for $s=1,2$. The result is also extended to multipartite systems.