The Role of Tensor-Generated Matrices in Analyzing Spin State Classicality and Tensor H-Eigenvalue Distributions

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.