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Linear maps preserving (p,k) norms of tensor products of matrices (2307.00377v2)
Published 1 Jul 2023 in math.FA
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 $X\to XT$ for $s=1,2$. The result is also extended to multipartite systems.