非负矩阵张量积保持问题的研究

In the context of quantum information science, the purpose of this project is to study spectrum preserver and numerical range preserver of tensor products of nonnegative matrices and give the forms of linear maps preserving spectrum, spectral radius of tensor products of nonnegative matrices and the forms of linear maps preserving numerical range, numerical radius of tensor products of nonnegative matrices, which mainly combines with spectrum theory and numerical range theory that have important applications. This project will further enrich the spectrum preservers and the numerical range preservers. On this basis, the practical value will be explored in the information transmission of quantum entangled states. The main task of this project is to generalize the spectrum,spectral radius preservers of tensor products of Hermitian matrices and the numerical range, numerical radius preservers of matrices to corresponding preservers of tensor products of nonnegative matrices.

本项目以量子信息科学为背景，结合具有重要应用价值的谱理论与数值域理论，拟开展非负矩阵张量积的谱及谱半径保持与数值域及数值半径保持问题的研究，具体给出保持非负矩阵张量积的谱、谱半径、数值域及数值半径线性映射的形式，旨在进一步丰富谱保持及数值域保持理论。在此基础上，探索其在量子纠缠态信息传输中的实际应用价值。本项目的主要工作在于将埃尔米特矩阵张量积的谱及谱半径保持与一般矩阵张量积的数值域及数值半径保持理论推广到非负矩阵张量积的谱、谱半径、数值域及数值半径保持理论。

基于量子信息传输中的实际应用背景，本项目主要研究张量的基本运算、张量特征值、奇异值等一些基本性质，主要定义了一种新的张量乘积；给出长方张量奇异值的Perron-Frobenius定理；给出了张量特征值的包含域，及判定张量正定性与非奇异性的方法；给出了最小秩为1的符号模式张量的完整刻画，证明符号模式张量的最大秩大于等于项秩，并且刻画了具有符号左逆和符号右逆的张量。此研究的意义在于即丰富了张量理论，同时为研究保持问题提供理论基础又可以拓宽团队的研究领域，将矩阵空间的保持问题拓展到张量空间。