Linear preservers and quantum information science

Abstract

Let $m,n\ge 2$ be positive integers, $M_m$ the set of $m\times m$ complex matrices and $M_n$ the set of $n\times n$ complex matrices. Regard $M_{mn}$ as the tensor space $M_m\otimes M_n$. Suppose $|\cdot |$ is the Ky Fan $k$-norm with $1\le k\le mn$, or the Schatten $p$-norm with $1\le p\le \infty$ ($p\ne 2$) on $M_{mn}$. It is shown that a linear map $\varphi :M_{mn}\to M_{mn}$ satisfying $$|A\otimes B|=|\varphi (A\otimes B)|$$ for all $A\in M_m$ and $B\in M_n$ if and only if there are unitary $U,V\in M_{mn}$ such that $\varphi$ has the form $A\otimes B\mapsto U({\phi }_1(A)\otimes {\phi }_2(B))V$, where ${\phi }_i(X)$ is either the identity map $X\mapsto X$ or the transposition map $X\mapsto X^t$. The results are extended to tensor space $M_{n_1}\otimes ...\otimes M_{n_m}$ of higher level. The connection of the problem to quantum information science is mentioned.

13 pages

References in corpus (1)

• Minimal and Maximal Operator Spaces and Operator Systems in Entanglement Theory