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.
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 F2-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.
This work studies bipartite Schmidt rank in continuously weighted diagonal phase networks, with particular emphasis on the distinction between interaction support, parameter freedom, and higher-order algebraic dependencies. For a fixed bipartition and a full-support product input, the Schmidt rank is reduced exactly to the complex rank of a cross-cut phase kernel. In the independently weighted pairwise qubit sector, the main result gives an exact generic matching law: rgen=2ν(G×), where ν(G×) is the maximum matching number of the crossing bipartite interaction graph. The symbolic rank, maximum attainable rank, and almost-everywhere physical rank coincide. The paper then shows that the interaction support alone does not determine the generic Schmidt rank once the parameter family is constrained. For a uniformly weighted complete bipartite network Kp,q, the exact rank is rankKθ=min{p+1,q+1,ord(eiθ)}, with infinite multiplicative order interpreted in the natural way. The K2,2 example therefore exhibits a rank stratification (4→3→2) when passing from independently weighted phases to a uniform one-parameter family and then to the Clifford point. For genuine higher-order phase interactions, the cross-cut kernel is factorized through a union-coefficient matrix. This reveals a different obstruction: even with independent hyperedge weights, repeated algebraic use of the same phase parameters can force the symbolic rank below the structural matching rank. The paper proves that no such obstruction occurs with at most two crossing hyperedges and gives an explicit three-hyperedge example with rstr=4,rsym=3. Exact finite enumeration further certifies, within the stated simple crossing-hypergraph model, that five total cut vertices are the smallest size at which this structural-versus-symbolic rank mismatch occurs. The results distinguish three mechanisms controlling cross-cut Schmidt-rank capacity: pairwise interaction combinatorics, external parameter tying, and intrinsic higher-order union-algebraic dependencies. The manuscript is accompanied by exact verification code and reproducibility checks for the finite classification statements. Version: v0.2r3
Graph states provide a bridge between graph theory and multipartite quantumentanglement, with cut-rank, rank-width, and related parameters playing a centralrole in both structural graph theory and the characterization of stabilizer states.Building on this interplay, we introduce a new scalar invariant—the S M NazmuzSakib entanglement rank entropy—defined as the Shannon entropy of the cut-rankdistribution over vertex subsets of fixed cardinality. For a simple graph G = (V, E)of order n, and k ∈ {1, . . . , n − 1}, we consider the random variable ρG(X) whereX ⊆ V is a uniformly random k-subset, and we define the Sakib entanglementrank entropy H(k)ρ (G) as the Shannon entropy (in bits) of its distribution.Forgraph states, where ρG(X) coincides with the Schmidt rank exponent across thebipartition (X, V \ X), H(k)ρ (G) is precisely the Shannon entropy of the bipartiteentanglement (in ebits) seen by random size-k bipartitions.We formally define the S M Nazmuz Sakib entanglement rank entropy, provethat it is invariant under local complementation and hence a local-Clifford invariantfor graph states, and study a balanced version Hbalρ (G) = H(n/2)ρ(G) (for even n).We then perform a data-based analysis on explicit graph families (paths, cycles,stars, complete graphs, and 2 × m grid graphs) of size up to n = 12, computingHbalρ (G), the average cut-rank, and the full cut-rank distributions. The resultingdataset shows that, while complete and star graphs have vanishing Sakib entangle-ment rank entropy, more “distributed” structures such as paths, cycles, and gridsexhibit increasing entropy with system size, despite similar ranges of cut-rank.Motivated by these observations, we formulate the S M Nazmuz Sakib entan-glement heterogeneity hypothesis, proposing that, at fixed rank-width and averagecut-rank, a larger Sakib entanglement rank entropy correlates with higher classicalsimulation complexity of the corresponding graph states.We outline how pub-licly available graph-state orbit datasets and average cut-rank techniques can becombined with the proposed invariant to further test this hypothesis.
Abstract We study the entanglement properties of quantum hypergraph states of n qubits, focusing on multipartite entanglement . We compute multipartite entanglement for hypergraph states with a single hyperedge of maximum cardinality, for hypergraph states endowed with all possible hyperedges of cardinality equal to n − 1 and for those hypergraph states with all possible hyperedges of cardinality greater than or equal to n − 1 . We then find a lower bound to the multipartite entanglement of a generic quantum hypergraph state. We finally apply the multipartite entanglement results to the construction of entanglement witness operators , able to detect genuine multipartite entanglement in the neighbourhood of a given hypergraph state. We first build entanglement witnesses of the projective type, then propose a class of witnesses based on the stabilizer formalism, hence called stabilizer witnesses , able to reduce the experimental effort from an exponential to a linear growth in the number of local measurement settings with the number of qubits.
We investigate multipartite entanglement in a particular family of pure n-qubit hypergraph states through a statistical-mechanics framework, where the average bipartite purity maps onto an effective Hamiltonian of 2n classical binary spins. In this correspondence, each hypergraph state uniquely corresponds to a classical spin configuration, while temperature serves as a control parameter that continuously interpolates between a uniform ensemble of random hypergraph states at high temperature and maximally multipartite entangled states (MMES) at zero temperature. Remarkably, the exponential of the zero-temperature entropy directly gives the number of MMES within the set of hypergraph states. For small system sizes (n≤5), we perform an exact enumeration, fully characterizing the energy landscape and associated thermodynamic observables, and validating known MMES counts. For larger systems (n=6 and 7), where exact methods become computationally infeasible, we employ simulated annealing and parallel tempering algorithms to efficiently sample the exponentially large state space. Our analysis yields quantitative predictions of the number of MMES and reveals how entanglement is statistically distributed across the sets of hypergraph states. These results establish hypergraph states as an ideal platform for investigating multipartite entanglement through thermodynamic methods, offering both computational advances and physical insights into the structure of quantum entanglement in restricted families of quantum states.
The hypergraph states are pure multipartite quantum states corresponding to a hypergraph. It is an equal superposition of the states belonging to the computational basis. Given any hypergraph, we can construct a hypergraph state determined by a Boolean function. In contrast, we can find a hypergraph, corresponding to a Boolean function. This investigation develops a number of combinatorial structures concerned with the hypergraph states. For instance, the elements of the computational basis generate a lattice. The chains and antichains in this lattice assist us to find the equation of the Boolean function explicitly as well as to find a hypergraph. In addition, we investigate the entanglement property of the hypergraph states in terms of their combinatorial structures. We demonstrate several classes of hypergraphs, such that every cut of equal length on the corresponding hypergraph states has an equal amount of entanglement.
Abstract We study entanglement properties of hypergraph states in arbitrary finite dimension. We compute multipartite entanglement of elementary qudit hypergraph states, namely those endowed with a single maximum-cardinality hyperedge. We show that, analogously to the qubit case, also for arbitrary dimension there exists a lower bound for multipartite entanglement of connected qudit hypergraph states; this is given by the multipartite entanglement of an equal-dimension elementary hypergraph state featuring the same number of qudits as the largest-cardinality hyperedge. We highlight interesting differences between prime and non-prime dimension in the entanglement features.