Linear maps preserving -norms of tensor products of matrices

Abstract

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