Multipartite quantum scenarios are a significant and challenging resource in quantum information science. Tensors provide a powerful framework for representing multipartite quantum systems. In this work, we introduce the role of tensor-generated matrices that can broadly be defined as the relationships between an $m$-th order $n$-dimensional tensor and an $n$-dimensional square matrix. Through these established connections, we demonstrate that the classification of the tensor-generated matrix as an $H$-matrix implies the original tensor is also an $H$-tensor. We also explore various similar properties exhibited by both the original tensor and the tensor-generated matrix, including weak irreducibility, weakly chained diagonal dominance, and (strong) symmetry. These findings provide a method to transform intricate tensor problems into matrices in specific contexts, which is especially pertinent due to the NP-hard complexity of the majority of tensor problems. Subsequently, we explore the application of tensor-generated matrices in analyzing the classicality of spin states. Leveraging the tensor representation, we introduce classicality criteria for (strongly) symmetric spin-$j$ states, which potentially provide fresh perspectives on the study of multipartite quantum resources. Finally, we extend classical matrix eigenvalue inclusion sets to higher-order tensor $H$-eigenvalues, a task that is typically challenging for higher-order tensors. Consequently, we propose representative tensor $H$-eigenvalue inclusion sets, such as modified Brauer's Ovals of Cassini sets, Ostrowski sets, and $S$-type inclusion sets.

翻译：多体量子场景是量子信息科学中重要且具有挑战性的资源。张量为表示多体量子系统提供了强大的框架。本文引入了张量生成矩阵的作用，其可广义定义为$m$阶$n$维张量与$n$维方阵之间的关联关系。通过建立的关联性，我们证明若张量生成矩阵被分类为$H$-矩阵，则原张量亦为$H$-张量。我们还探讨了原张量与张量生成矩阵共同表现出的多种相似性质，包括弱不可约性、弱链对角占优性及（强）对称性。这些发现为在特定情境下将复杂张量问题转化为矩阵问题提供了方法，鉴于大多数张量问题具有NP-hard复杂度，此方法尤具意义。随后，我们探索了张量生成矩阵在分析自旋态经典性中的应用。借助张量表示，我们提出了（强）对称自旋-$j$态的经典性判据，这可能为多体量子资源的研究提供新视角。最后，我们将经典矩阵特征值包含集推广至高阶张量$H$-特征值，这对高阶张量通常是具有挑战性的任务。基于此，我们提出了代表性的张量$H$-特征值包含集，例如修正的Brauer卡西尼卵形集、Ostrowski集及$S$型包含集。