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

TL;DR

This paper characterizes linear maps on tensor product matrices that preserve the (p,k) norms, showing they are essentially unitary conjugations combined with identity or transposition operations, extending to multipartite systems.

Contribution

It provides a complete characterization of linear maps preserving (p,k) norms of tensor products, identifying their structure as unitary conjugations with identity or transposition.

Findings

Preservers are unitary conjugations with identity or transposition

Characterization applies to all (p,k) norms with 2<p<∞

Results extend to multipartite tensor systems

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.