Restriction-Based Certificate of Bipartite Schmidt Rank in Hypergraph States

TL;DR

This paper introduces a method to certify bipartite entanglement in hypergraph states by identifying residual-free bilinear cores, providing exponential lower bounds on Schmidt rank through combinatorial conditions and a verification procedure.

Contribution

It develops a restriction-based approach to certify entanglement in hypergraph states, extending cut-rank analysis beyond graph states and introducing residual-free bilinear cores for lower bounds.

Findings

Residual-free bilinear cores yield exponential Schmidt-rank lower bounds.

A combinatorial disjoint bridge matching guarantees large full-rank cores.

A search-and-verify procedure constructs and certifies these cores from hyperedge data.

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.