1211.0396 (Ajda Fosner et al.)

Linear maps preserving Ky Fan norms and Schatten norms of tensor product
of matrices    [PDF]

Ajda Fosner, Zejun Huang, Chi-Kwong Li, Nung-Sing Sze

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)| for all $A \in M_m$ and $ \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 $M_{n_1} \otimes ... \otimes M_{n_m}$ of higher level. The connection of the problem to quantum information science is mentioned.

View original: http://arxiv.org/abs/1211.0396