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

TL;DR

This paper characterizes linear maps on tensor product matrices that preserve Ky Fan and Schatten norms, revealing they are essentially unitary conjugations combined with identity or transposition operations, with implications for quantum information.

Contribution

It provides a complete characterization of linear maps preserving these norms on tensor products, extending previous results and connecting to quantum information science.

Findings

Preservers are unitary conjugations with identity or transpose operations.

Results extend to higher tensor levels beyond bipartite systems.

Connections to quantum information science are discussed.

Abstract

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.