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.
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.
Organising the space of entanglement structures of a multipartite quantum system is a much more challenging task than its bipartite version: while the local unitary (LU) orbit of a bipartite pure state can be conveniently characterized by its entanglement spectrum, invariants of multipartite entanglement structures are comparatively difficult to define and work with. The root cause of this difference is that the bipartite problem can be reduced to the analysis of matrix invariants, while its multipartite version is governed by a much richer space of tensor invariants. The present work explores the latter through the lens of so-called trace-invariants, which are in one-to-one correspondence with combinatorial objects known as colored graphs. We first explain why trace-invariant evaluations can serve as labels of LU-orbits of multipartite pure states, how this strategy extends to random states, and how the effect of local operations (LO) can be analyzed through such data. We then focus on entanglement classification within an (infinite-dimensional) subspace of reference states, whose basic building blocks are GHZ states of various dimensions. We show that relatively simple subclasses of trace-invariants are sufficient to separate the LU-orbits of reference states, and enable a complete (resp. an incomplete) characterization of their relations in the LO (resp. LOCC) resource theory of entanglement. Finally, we investigate how a (still infinite) subclass of reference states of local dimension N can be efficiently distinguished at leading and subleading orders in an asymptotic large-N expansion (among themselves, or from Haar-random states). This analysis relies crucially on combinatorial quantities associated to colored graphs, some of which have already played instrumental roles in the recent literature on random tensors. Results of broader relevance are reported along the way.
Entanglement in quantum graph states is intrinsically linked to rank-width, a graph complexity measure introduced by Oum and Seymour. In this work, we enable the preparation of maximally entangled deterministic graph states in constant depth by developing a general method to derive lower bounds on the rank-width of regular graphs from their edge expansion. By bridging edge-isoperimetric inequalities with the strong chromatic index and Jelínek's approach for lower bounding cut-rank, we systematically establish lower bounds for the rank-width of Cartesian products, including hypercubes, Hamming graphs, and grids. Extending this framework via Boolean function analysis, using a generalization of the Kahn-Kalai-Linial's Theorem, we strengthen the bounds for all Cartesian products by a non-trivial logarithmic factor. These methods result in the discovery of deterministic families of graphs on n vertices with a provably maximum rank-width Θ(n). Our results fill the previous gap in the literature for deterministic graph families of rank-width greater than Θ(n).
Graph states play an important role in quantum information theory through their connection to measurement-based computing and error correction. Prior work revealed elegant connections between the graph structure of these states and their multipartite entanglement. We continue this line of investigation by identifying additional entanglement properties for certain types of graph states. From the perspective of tensor theory, we tighten both upper and lower bounds on the tensor rank of odd ring states ($|{R}_{2n+1}\ensuremath{\rangle}$) to read ${2}^{n}+1\ensuremath{\le}\mathrm{rank}(|{R}_{2n+1}\ensuremath{\rangle})\ensuremath{\le}3\ifmmode\times\else\texttimes\fi{}{2}^{n\ensuremath{-}1}$. Next we show that several multipartite extensions of bipartite entanglement measures are dichotomous for graph states based on the connectivity of the corresponding graph. Finally, we give a simple graph rule for computing the n-tangle ${\ensuremath{\tau}}_{n}$.
Theory of Quantum Entanglement of Particles: Multiscale Graph States, Region Geometry, Quantum Error Correction, and the Effective-Space Hypothesis Abstract This study develops a finite-dimensional, computationally testable framework for examining how multiscale graph structure organizes quantum entanglement and the accessibility of logical information. For the circulant graph family G_N, where N = 2ⁿ, n ≥ 3, and edge offsets are 1, 2, 4, …, N/2, an analytic proof establishes that the binary adjacency matrix across every contiguous block A of length L ≤ N/2 has rank L over F₂. For the associated graph states, this yields S(A) = min(L, N − L). For G₈, the study enumerates 238 proper connected vertex subsets ("tubes") and 32,640 maximal tubings and computes the entanglement entropy of every tube. Explicit bit-flip and phase-dual quantum error-correcting code embeddings are examined under specified Pauli-noise and erasure models, showing that recovery performance depends on the encoding, error model, and decoder class, not on graph-state entropy alone. The dynamical component, M1-Green, models a four-dimensional logical code coupled to eight intermediary bosonic modes and sequential collision-bin output registers. At selected resonantly matched parameter settings and T = 120/K, the computed mutual information I(Ref:R) reaches 3.9414 bits for G₈ and 3.8876 bits for C₈, below the four-bit upper bound. The study also reports a verified finite-step symmetry for C₈ in the tested setup, Petz recovery, and a numerical semidefinite-programming interval of 0.235788–0.245982 for recovery fidelity in a specified 69-dimensional output-space formulation. A one-port Page-like model, a random-unitary scrambling benchmark, and a two-sided mediated channel are treated as distinct models. The accompanying compact archive provides scripts, data tables, figures, and numerical certificates. The effective-space interpretation remains a hypothesis; global recovery optimality and independent experimental validation are not claimed. Keywords: quantum entanglement; graph states; multiscale graphs; circulant graphs; entanglement entropy; graph associahedra; quantum error correction; erasure channels; logical information recovery; collision-bin dynamics; Petz recovery map; semidefinite programming; quantum scrambling; effective geometry.
Abstract The entanglement properties of quantum states associated with directed graphs are investigated. Using a measure derived from the Fubini–Study metric, multipartite entanglement is quantitatively related to the local connectivity of the graph. In Entanglement in Directed Graph States (2025), arXiv:2505.10716, it is demonstrated that the vertex degree distribution fully determines this entanglement measure and remains invariant under vertex relabeling, highlighting its topological character. As a consequence, the measure depends only on the total degree of each vertex, making it independent of the distinction between incoming and outgoing edges. This framework is applied to several specific graph structures, including hierarchical networks, neural network–inspired graphs, full binary tree and linear bridged cycle graphs, demonstrating how their combinatorial properties influence entanglement distribution. These results provide a geometric perspective on quantum correlations in complex systems, offering potential applications in the design and analysis of quantum networks.
This work studies the global multipartite entanglement width generated by fixed pairwise weighted-phase networks acting on full-support product inputs. It develops the global-width consequences of the fixed-cut weighted-phase theory established in the preceding study, Schmidt-Rank Capacity and Spectral Robustness in Fixed Weighted Phase Networks; the corresponding public research record is available at DOI 10.5281/zenodo.21858429. For a graph G=(V,E) with a fixed real edge-weight template W=(We)e∈E, we consider the commuting weighted-phase evolution UW(t)=exp(it∑e={u,v}∈EWenunv), with a single scalar evolution parameter t. For every bipartition A∣Ac, let νG(A) be the maximum matching number of the crossing graph G[A,Ac]. The main result globalizes the fixed-cut matching law. There exists an open, dense, full-Lebesgue-measure class of fixed templates W such that, outside one locally finite exceptional set of evolution times, the same scalar t simultaneously saturates the matching capacity of every nontrivial bipartition: log2SRA∣Ac(∣ΨW(t)⟩)=νG(A) for all nontrivial cuts A∣Ac. Consequently, χwd(∣ΨW(t)⟩)=mmw(G) for generic fixed weighted-phase networks, where χwd is Schmidt-rank width and mmw(G) is the maximum matching width of the interaction graph. This provides a continuous weighted counterpart to the standard graph-state relation between Schmidt-rank width and rank-width. For complete support Kn, mmw(Kn)=⌈3n⌉, so almost every nonuniform fixed template attains χwd=⌈3n⌉, which is the largest Schmidt-rank width permitted for any n-qubit pure state. Uniform complete-graph templates behave very differently. Their exact width is χwdunif=log2min{⌈3n⌉+1,ord(eiθ)}, with ord(eiθ)=∞ allowed. Thus the same complete interaction support and the same one-dimensional run-time control exhibit a sharp template hierarchy: Clifford⟶χwd=1,generic uniform⟶χwd=Θ(logn),generic nonuniform⟶χwd=Θ(n). This separation is not restricted to dense complete graphs. Since rw(G)≤mmw(G), graph families with linear rank-width transfer directly to fixed weighted-phase families with linear generic Schmidt-rank width. In particular, known deterministic bounded-degree high-rank-width constructions yield sparse O(n)-coupler networks with generic Schmidt-rank width Θ(n). The width separation has an exact tensor-network consequence. For generic uniform complete templates, an explicit subcubic tree tensor network exists with optimal maximal bond dimension Dunif=⌈3n⌉+1=Θ(n), whereas generic nonuniform complete templates require Dgen=2⌈n/3⌉=2Θ(n). Thus fixed template diversity, rather than run-time control dimension alone, can change exact global tree-tensor complexity from polynomial to exponential while the interaction support and the number of run-time control parameters remain unchanged. The computational interpretation is deliberately limited. Superlogarithmic Schmidt-rank width lies outside the logarithmic-width hypothesis underlying the standard low-width exact-TTN simulation guarantee, and linear width is a particularly strong realization of this separation. However, large or linear Schmidt-rank width is neither asserted to be necessary nor sufficient for universal measurement-based quantum computation, hard classical simulation in general, fault tolerance, or quantum advantage. Version v0.2r2 is the frozen theorem-core release. It preserves the mathematical results of v0.2r1 while finalizing the claim boundary concerning tensor-network simulation, measurement-based quantum computation, and computational advantage.