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Restriction-Based Certificate of Bipartite Schmidt Rank in Hypergraph States

C. Fajardo, M. Paraschiv

Abstract

We investigate bipartite entanglement in qubit hypergraph states across an arbitrary fixed bipartition. Using the real equally weighted (REW) representation, the Schmidt rank across the cut can be computed as the real rank of a phase-cleaned cross-cut sign matrix. Whereas graph states admit an exact cut-rank rule, because the cross-cut phase is purely bilinear, hypergraph states typically contain higher-degree cross-cut interactions, for which the cut-rank rule fails. Our approach certifies entanglement by fixing a single computational-basis assignment on a subset of qubits, thereby selecting a submatrix on an active slice. When this restriction removes all higher-degree cross-cut residues, the remaining cross-cut phase becomes bilinear up to cut-local terms. We call the resulting submatrices residual-free bilinear cores and show that they yield an exponential Schmidt-rank lower bound in terms of the $\mathbb{F}_2$-rank of an exposed core matrix. We further give a combinatorial sufficient condition, phrased as a disjoint bridge matching, that guarantees the existence of large full-rank cores for broad families of CCZ-type bridge patterns, and we present a search-and-verify procedure that constructs and certifies such cores directly from the hyperedge description.

Restriction-Based Certificate of Bipartite Schmidt Rank in Hypergraph States

Abstract

We investigate bipartite entanglement in qubit hypergraph states across an arbitrary fixed bipartition. Using the real equally weighted (REW) representation, the Schmidt rank across the cut can be computed as the real rank of a phase-cleaned cross-cut sign matrix. Whereas graph states admit an exact cut-rank rule, because the cross-cut phase is purely bilinear, hypergraph states typically contain higher-degree cross-cut interactions, for which the cut-rank rule fails. Our approach certifies entanglement by fixing a single computational-basis assignment on a subset of qubits, thereby selecting a submatrix on an active slice. When this restriction removes all higher-degree cross-cut residues, the remaining cross-cut phase becomes bilinear up to cut-local terms. We call the resulting submatrices residual-free bilinear cores and show that they yield an exponential Schmidt-rank lower bound in terms of the -rank of an exposed core matrix. We further give a combinatorial sufficient condition, phrased as a disjoint bridge matching, that guarantees the existence of large full-rank cores for broad families of CCZ-type bridge patterns, and we present a search-and-verify procedure that constructs and certifies such cores directly from the hyperedge description.
Paper Structure (19 sections, 4 theorems, 51 equations, 1 algorithm)

This paper contains 19 sections, 4 theorems, 51 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hypergraph states and Boolean phase polynomials
  4. Bipartitions, coefficient matrices, and phase cleaning
  5. Bilinear Slice and the Graph-State Cut-Rank Rule
  6. Bilinear slice and cut-incidence matrix $\Gamma_{AB}$
  7. Bilinear case: $\mathrm{SR}_{A|B}=2^{\mathrm{rank}_{\mathbb{F}_2}(\Gamma_{AB})}$ and limitations
  8. Restriction-Based Residual-Free Bilinear Cores
  9. Residual-free bilinear cores and a Schmidt-rank certificate
  10. Disjoint bridge matching: a sufficient combinatorial condition
  11. Algorithm
  12. Inputs and outputs.
  13. Search phase: proposing candidate bridge blocks.
  14. Building a canonical global restriction.
  15. Verification phase: residual-freeness test and core extraction.
  16. ...and 4 more sections

Key Result

Lemma 1

Let $U_A$ and $U_B$ be unitaries acting on $A$ and $B$, respectively. Then the coefficient matrices of $\ket{\psi}$ and $(U_A\otimes U_B)\ket{\psi}$ are related by $M\mapsto U_A M U_B^{\mathsf{T}}$ (in the computational bases), and in particular $M$ and $U_A M U_B^{\mathsf{T}}$ share the same singul

Theorems & Definitions (17)

  • Definition 1
  • Lemma 1
  • proof
  • Theorem 1
  • Example 1: Two independent cross-$CZ$ bridges
  • Example 2: Pure cross $3$-edge: vanishing bilinear slice
  • Definition 2
  • Remark 1
  • Theorem 2
  • proof
  • ...and 7 more