  For a positive integer $n$, let $M_n$ be the set of $n\times n$ complex
matrices. 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 $M_{mn}$, where
$m,n\ge 2$ are positive integers. It is shown that a linear map $\phi: M_{mn}
\rightarrow M_{mn}$ satisfying $$\|A\otimes B\| = \|\phi(A\otimes B)\| \quad
\hbox{for all} A \in M_m \hbox{and} B \in M_n$$ if and only if there are
unitary $U, V \in M_{mn}$ such that $\phi$ has the form $A\otimes B \mapsto
U(\varphi_1(A) \otimes \varphi_2(B))V$, where $\varphi_s(X)$ 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.
