论文标题
基质产品状态，几何和不变理论
Matrix product states, geometry, and invariant theory
论文作者
论文摘要
矩阵产品状态在量子信息理论中起重要作用，以代表多体系统的状态。它们可以看作是高维张量空间的低维亚变化。在这些注释中，我们考虑了两个变体：均匀的矩阵乘积状态和均匀的矩阵乘积状态。研究这些品种的线性跨度导致与矩阵不变理论的自然联系。对于均匀的矩阵产物，在2x2矩阵的情况下，矩阵多项式身份的经典结果导致了线性跨度尺寸的公式。 这些笔记部分基于华沙大学作者在主题学期的“ Agates：代数几何形状，并向张量和斜坡上应用”的演讲，部分是基于学期的进一步研究。这仍然是初步版本；更新版本将在2023年的过程中上传。
Matrix product states play an important role in quantum information theory to represent states of many-body systems. They can be seen as low-dimensional subvarieties of a high-dimensional tensor space. In these notes, we consider two variants: homogeneous matrix product states and uniform matrix product states. Studying the linear spans of these varieties leads to a natural connection with invariant theory of matrices. For homogeneous matrix product states, a classical result on polynomial identities of matrices leads to a formula for the dimension of the linear span, in the case of 2x2 matrices. These notes are based partially on a talk given by the author at the University of Warsaw during the thematic semester "AGATES: Algebraic Geometry with Applications to TEnsors and Secants", and partially on further research done during the semester. This is still a preliminary version; an updated version will be uploaded over the course of 2023.
