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Triangular tensor networks, pencils of matrices and beyond

Alessandra Bernardi, Fulvio Gesmundo

TL;DR

This work characterizes tensor network varieties for the triangle graph when one local dimension is $2$, reframing tensors as matrix pencils and achieving a complete Kronecker-invariant classification. It derives precise dimension formulas, normal forms, and necessary membership equations via a Kronecker/pencil framework and coincident-root-locus geometry, coupling algebraic and geometric methods. The paper also elucidates a geometric interpretation through flattenings and Grassmannians, providing set-theoretic equations in key cases (notably the $222$ corner with $333$ physical dimensions) and linking to Ruppert-type invariants. Finally, it extends the construction to arbitrary graphs via graph augmentation, describing defectiveness and the resulting normal forms, thereby offering a blueprint for understanding defectivity and structure in broader tensor-network topologies.

Abstract

We study tensor network varieties associated with the triangular graph, with a focus on the case where one of the physical dimensions is 2. This allows us to interpret the tensors as pencils of matrices. We provide a complete characterization of these varieties in terms of the Kronecker invariants of pencils. We determine their dimension, identifying the cases for which the dimension is smaller than the expected parameter count. We provide necessary conditions for membership in these varieties, in terms of the geometry of classical determinantal varieties, coincident root loci and plane cubic curves. We address some extensions to arbitrary graphs.

Triangular tensor networks, pencils of matrices and beyond

TL;DR

This work characterizes tensor network varieties for the triangle graph when one local dimension is , reframing tensors as matrix pencils and achieving a complete Kronecker-invariant classification. It derives precise dimension formulas, normal forms, and necessary membership equations via a Kronecker/pencil framework and coincident-root-locus geometry, coupling algebraic and geometric methods. The paper also elucidates a geometric interpretation through flattenings and Grassmannians, providing set-theoretic equations in key cases (notably the corner with physical dimensions) and linking to Ruppert-type invariants. Finally, it extends the construction to arbitrary graphs via graph augmentation, describing defectiveness and the resulting normal forms, thereby offering a blueprint for understanding defectivity and structure in broader tensor-network topologies.

Abstract

We study tensor network varieties associated with the triangular graph, with a focus on the case where one of the physical dimensions is 2. This allows us to interpret the tensors as pencils of matrices. We provide a complete characterization of these varieties in terms of the Kronecker invariants of pencils. We determine their dimension, identifying the cases for which the dimension is smaller than the expected parameter count. We provide necessary conditions for membership in these varieties, in terms of the geometry of classical determinantal varieties, coincident root loci and plane cubic curves. We address some extensions to arbitrary graphs.
Paper Structure (15 sections, 18 theorems, 123 equations, 6 figures)

This paper contains 15 sections, 18 theorems, 123 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Outline.
  3. Preliminaries
  4. Graph tensors and tensor network states
  5. Expected dimension
  6. Pencils of matrices
  7. Orbits, isotropy groups and Lie algebras
  8. Triangle TNS for matrix pencils
  9. Dimension of triangle TNS with (m,2,m) corner: critical case
  10. Dimension of triangle TNS with (m,2,m) corner: subcritical case
  11. Equations for tensor network triangles and coincident root loci
  12. Geometry of the flattening image
  13. Pencils and a geometric interpretation of \ref{['main theorem']}
  14. Triangular tensor network with larger physical dimensions
  15. Extension of arbitrary graphs via pencils

Key Result

Theorem 1.1

Let $\mathbf{n} = (2,n_1,n_2)$ and $\mathbf{m} = (m_{01} , m_{12}, m_{02})$ be an assignment of local and physical dimensions of the triangular network $\Delta$, with $n_i \leq m_{ij}m_{ik}$ with $\{i,j,k\} = \{0,1,2\}$. Write $k_i = m_{ij}m_{ik} - n_i$ for $i = 1,2$ and suppose $k_2 \leq k_1$. If set $m=m_{01} = m_{02}$. The following holds: If eqn: nontriviality condition does not hold, then $

Figures (6)

  • Figure 1: Triangular tensor network with bond dimensions $\mathbf{m} = (m_{01},m_{12},m_{02})$ and local dimensions $\mathbf{n} = (n_0,n_1,n_2)$.
  • Figure 2: Triangular tensor network $(\Delta, \mathbf{m},\mathbf{n})$ with bond dimensions $\mathbf{m} = (m_{01},m_{12},m_{02})$ and local dimensions $\mathbf{n} = (2,n_1,n_2)$.
  • Figure 3: Triangular tensor network with bond dimensions $\mathbf{m} = (m,m_{12},m)$ and $\mathbf{n} = (2,m_{12}m,m_{12}m)$.
  • Figure 4: Triangular tensor network with bond dimensions $\mathbf{m} = (m,m_{12},m)$ and $\mathbf{n} = (2,m_{12}m - k_1,m_{12}m-k_2)$.
  • Figure 5: Triangular graph with bond dimensions $\mathbf{m} = (2,m_{12},2)$ and $\mathbf{n} = (2,n_{1},n_2)$.
  • ...and 1 more figures

Theorems & Definitions (36)

  • Theorem 1.1
  • Definition 1
  • Definition 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • Theorem 3.1
  • proof
  • ...and 26 more