1 results
Linear maps preserving
$(p,k)$-norms of tensor products of matrices
- Part of
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- Journal:
- Canadian Journal of Mathematics , First View
- Published online by Cambridge University Press:
- 15 December 2023, pp. 1-21
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- Article
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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.
