Linear maps preserving Ky Fan norms and Schatten norms of tensor products of matrices

Abstract

For a positive integer $n$, let $M_n$ be the set of $n\times n$ complex matrices. Suppose $\Vert \cdot \Vert$ 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 $M_{mn}$, where $m,n\ge 2$ are positive integers. It is shown that a linear map $\varphi :M_{mn}\to M_{mn}$ satisfying $$\Vert A\otimes B\Vert =\Vert \varphi (A\otimes B)\Vert for\; allA\in M_{m}andB\in M_n$$ 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 ${\phi }_s(X)$ is either the identity map $X\mapsto X$ or the transposition map $X\mapsto X^t$. The results are extended to tensor space $M_{n_1}\otimes \cdots \otimes M_{n_m}$ of higher level. The connection of the problem to quantum information science is mentioned.

14 pages, to appear in SIAM J. Matrix Analysis Appl