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

Liang Xiong, Jianzhou Liu

Abstract

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.

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

Abstract

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 -th order -dimensional tensor and an -dimensional square matrix. Through these established connections, we demonstrate that the classification of the tensor-generated matrix as an -matrix implies the original tensor is also an -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- 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 -eigenvalues, a task that is typically challenging for higher-order tensors. Consequently, we propose representative tensor -eigenvalue inclusion sets, such as modified Brauer's Ovals of Cassini sets, Ostrowski sets, and -type inclusion sets.

Paper Structure

This paper contains 14 sections, 30 theorems, 99 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Preliminaries for tensor
  4. Preliminaries for tensor quantum information
  5. The tensor-generated matrices and $H$-tensors
  6. The tensor-generated matrices and $H$-tensors
  7. The tensor-generated matrix and weakly irreducible diagonally dominant tensor
  8. The tensor-generated matrix and weakly chained diagonally dominant tensor
  9. Application in Classicality for spin states
  10. Classicality for spin states with symmetric coefficient tensors
  11. Classicality for spin states with strongly symmetric coefficient tensors
  12. Application in $H$-Eigenvalue inclusion sets for high order tensors
  13. The applications of tensor-generated in $H$-Eigenvalue inclusion sets for high order tensors
  14. Numerical examples

Key Result

Theorem 1

Let $A\in \mathbb{C}^{n,n}$ be a tensor-generated matrix by tensor $\mathcal{A} \in \mathbb{C}^{[m\times n]}$. If matrix $A$ is an $H$-matrix, then tensor $\mathcal{A}$ is an $H$-tensor.

Theorems & Definitions (57)

  • Definition 1
  • Theorem 1
  • proof
  • remark 1
  • Definition 2
  • Theorem 2
  • proof
  • remark 2
  • remark 3
  • Lemma 1
  • ...and 47 more