保持平行对的矩阵上的线性映射

A Characterization of Completely-Positive Trace-Preserving Maps on 2x2 Matrices

This paper has been superseded by quant-ph/0101003.

Hua's fundamental theorem of geometry of rectangular matrices over EAS division rings

The fundamental theorem of geometry of rectangular matrices describes the general form of bijective maps on the space of all $m\times n$ matrices over a division ring $\mathbb{D}$ which preserve adjacency in both directions. This result proved by Hua in the nineteen forties has been recently improved in several directions. One can study such maps without the bijectivity assumption or one can try to get the same conclusion under the weaker assumption that adjacency is preserved in one direction only. And the last possibility is to study maps acting between matrix spaces of different sizes. The …

矩阵几何中的保直径满射

Diameter preserving surjections in the geometry of matrices

本研究针对一类直径有限的图，证明保直径对的满射必为图同构，并将该结果推广应用到矩形矩阵空间、埃尔米特矩阵空间及Grassmann空间对应的邻接图中。

Strong $k$-commutativity preserving maps on 2$\times$2 matrices

Let ${\mathcal M}_2(\mathbb F)$ be the algebra of 2$\times$2 matrices over the real or complex field $\mathbb F$. For a given positive integer $k\geq 1$, the $k$-commutator of $A$ and $B$ is defined by $[A,B]_k=[[A,B]_{k-1},B]$ with $[A,B]_0=A$ and $[A,B]_1=[A,B]=AB-BA$. The main result is shown that a map $Φ: {\mathcal M}_2(\mathbb F)\to {\mathcal M}_2(\mathbb F)$ with range containing all rank one matrices satisfies that $[Φ(A),Φ(B)]_k = [A,B]_k $ for all $A, B\in{\mathcal M}_2(\mathbb F)$ if and only if there exist a functional $h :{\mathcal M}_2(\mathbb F) \rightarrow {\mathbb F}$ and a sc…

The Lie algebra preserving a degenerate bilinear form

Let $k$ be an arbitrary field and $d$ a positive integer. For each degenerate symmetric or antisymmetric bilinear form $M$ on $k^{d}$ we determine the structure of the Lie algebra of matrices that preserve $M$, and of the Lie algebra of matrices that preserve the subspace spanned by $M$. We show that these Lie algebras are semidirect products of classical Lie algebras and certain representations, and determine their radicals, derived series and semisimple quotients. Our main motivation and application is to determine the structure of the graded Lie algebra of derivations of each commutative or…

An Analysis of Completely-Positive Trace-Preserving Maps on 2x2 Matrices

We give a useful new characterization of the set of all completely positive, trace-preserving (i.e., stochastic) maps from 2x2 matrices to 2x2 matrices. These conditions allow one to easily check any trace-preserving map for complete positivity. We also determine explicitly all extreme points of this set, and give a useful parameterization after reduction to a certain canonical form. This allows a detailed examination of an important class of non-unital extreme points which can be characterized as having exactly two images on the Bloch sphere. We also discuss a number of related issues about t…

保持 $\ell_p$-范数平行向量的线性映射

Linear maps preserving $\ell_p$-norm parallel vectors

本文刻画了赋范空间中保持平行对与三角形等式达到（TEA）对的线性映射，给出了 $\ell_p$-范数下各类情形的具体刻画，并将其推广至无限维空间。

Nonsurjective maps between rectangular matrix spaces preserving disjointness, triple products, or norms

Let $M_{m,n}$ be the space of $m\times n$ real or complex rectangular matrices. Two matrices $A, B \in M_{m,n}$ are disjoint if $A^*B = 0_n$ and $AB^* = 0_m$. In this paper, a characterization is given for linear maps $Φ: M_{m,n} \rightarrow M_{r,s}$ sending disjoint matrix pairs to disjoint matrix pairs, i.e., $A, B \in M_{m,n}$ are disjoint ensures that $Φ(A), Φ(B) \in M_{r,s}$ are disjoint. More precisely, it is shown that $Φ$ preserves disjointness if and only if $Φ$ is of the form $$Φ(A) = U\begin{pmatrix} A \otimes Q_1 & 0 & 0 \cr 0 & A^t \otimes Q_2 & 0 \cr 0 & 0 & 0 \cr\end{pmatrix}V$$…