This work adopts the structural perspective of dual constraints to extend nonlocal decomposability to many-body systems. Under the premise of connectivity,we establish a hierarchical framework for constraint analysis centered on k-localdecomposability and the subset participation number. The core contributions include: introducing k-local decomposability as a unified constraint language formany-body entanglement, proving the reverse inclusion chain derived from coarsening invariance and the forward hierarchy of indecomposable depth ℓ; rigorouslydemonstrating the structural difference between GHZ and W states in terms ofsubset participation numbers—the GHZ state has P(A) = 2 on any proper subset, indicating centralized entanglement, while the W state has P(A) → 1 in thesingle-particle limit and P(A) → 2−1 on macroscopic subsets, reflecting individual dilution and collective emergence, i.e., distributed entanglement; establishing ahierarchical criterion for genuine many-body entanglement: if for some connectedk-body subset A, P(A) > 2, then the system possesses at least k-body genuineentanglement; revealing the exact relation P(A) = 2∥∂A∥ for graph states under bipartition, and providing a heuristic derivation based on topological order effectivefield theory, yielding a logarithmic-linear scaling law P(A) ∼ ∥∂A∥ for the boundary region of the surface code; from the information-geometric perspective, embedding the many-body Schmidt parameter space into a high-dimensional probabilitysimplex and establishing an asymptotic equivalence between the subset participation number and the Fisher geodesic distance in the weak entanglement limit.The theoretical limitations and future experimental pathways are also explicitlydiscussed.
This work adopts the structural perspective of dual constraints to extend nonlocal decomposability to many-body systems. Under the premise of connectivity,we establish a hierarchical framework for constraint analysis centered on k-localdecomposability and the subset participation number. The core contributions include: introducing k-local decomposability as a unified constraint language formany-body entanglement, proving the reverse inclusion chain derived from coarsening invariance and the forward hierarchy of indecomposable depth ℓ; rigorouslydemonstrating the structural difference between GHZ and W states in terms ofsubset participation numbers—the GHZ state has P(A) = 2 on any proper subset, indicating centralized entanglement, while the W state has P(A) → 1 in thesingle-particle limit and P(A) → 2−1 on macroscopic subsets, reflecting individual dilution and collective emergence, i.e., distributed entanglement; establishing ahierarchical criterion for genuine many-body entanglement: if for some connectedk-body subset A, P(A) > 2, then the system possesses at least k-body genuineentanglement; revealing the exact relation P(A) = 2∥∂A∥ for graph states under bipartition, and providing a heuristic derivation based on topological order effectivefield theory, yielding a logarithmic-linear scaling law P(A) ∼ ∥∂A∥ for the boundary region of the surface code; from the information-geometric perspective, embedding the many-body Schmidt parameter space into a high-dimensional probabilitysimplex and establishing an asymptotic equivalence between the subset participation number and the Fisher geodesic distance in the weak entanglement limit.The theoretical limitations and future experimental pathways are also explicitlydiscussed.
The structural origin of quantum entanglement has been a longstanding controversy. This paper adopts a constraint-theoretic approach, not presupposing anyontological status for entanglement, but rigorously demonstrating that the necessary and sufficient condition for quantum correlations to exceed classical Bellbounds lies in the non-local decomposability of dual pairings. By distinguishingformal duality from physical duality, we show that the positive-definiteness of theinner product on Hilbert space—rather than the complex structure—is the mathematical root of non-local decomposability. Within the dual-constraint framework,the Schmidt rank of a pure state, the mixedness of its reduced density matrix, theviolation of Bell inequalities, and the non-local decomposability of the dual pairing constitute equivalent reformulations. For mixed states, we further establishthe equivalence between convex decompositions of dual pairings and separability,and provide an algebraic construction of entanglement witnesses. The frameworkreveals the Tsirelson bound as an intrinsic upper limit imposed jointly by thepositive-definiteness of the inner product and the noncommutativity of local algebras, clarifies the effective-dimension significance of the participation number oninformation-geometric manifolds, and yields testable quantum metrological predictions under open-system decoherence. Moreover, we establish a rigorous translationdictionary between dual constraints and categorical quantum mechanics, and discuss heuristic connections to holographic duality. The overall framework shiftsquantum correlations from being a property of states to the algebraic indecomposability of dual pairings, providing a systematic language for a unified understandingof the structural conditions of quantum entanglement.
Understanding what limits many-body quantum entanglement is a central problem in physics. Spatial locality has long provided a fundamental mechanism: correlations between a region and its complement must be mediated through a small spatial interface, thereby constraining their entanglement. Here we show that universal constraints on entanglement can persist even when interactions are strongly nonlocal, through collectivity: many weak interactions generate fluctuations controlled by their square-summed, rather than total, strength. We establish this mechanism rigorously for generic gapped Hamiltonians with Kac-normalized power-law interactions r−α on a D-dimensional lattice. For arbitrary bipartitions, we prove that the ground-state entanglement scales at most logarithmically with system size for α<D/2 and subextensively for D/2<α<D, due to suppressed collective fluctuations around individual sites. For spatially regular bipartitions with codimension-one boundaries, we use a renormalization-group construction to extend this suppression to larger length scales, yielding polylogarithmic scaling for D/2<α<(D+1)/2 and parametrically stronger subvolume bounds for (D+1)/2<α<D. We also establish the corresponding optimality results: for arbitrary bipartitions, the logarithmic scaling for α<D/2 is optimal and α=D/2 marks the optimal threshold for universal logarithmic bounds, while for regular bipartitions the threshold at α=(D+1)/2 is likewise optimal for universal polylogarithmic bounds when D≥2. Together, these results reveal collectivity as a fundamental mechanism for constraining many-body entanglement alongside spatial locality.
Dual constraint entropy, as a natural emergence of the information cost ofnode condensation in constraint networks, provides a fine information-theoreticmetric connecting boundary entanglement structure and bulk geometry for holographic duality. Starting from the two axioms of information conservation andcomputability, this work establishes constraint networks as the discrete skeletonof the bulk-boundary correspondence, and proves that the dual constraint entropySdual = −lnTr(ρ2 A) is the inevitable information cost incurred when the node condensation operator is triggered at the information density threshold. In the HaPPYholographic error-correcting code, the area law Sdual(A) = Amin/αH + γ is rigorously derived, where the effective holographic parameter αH = a/ln2 is uniquelydetermined by the lattice spacing and the fundamental unit of information, eliminating the free parameter nature of Newton’s constant. It is further proved thatwhen the subsystem dimension is no less than 3, dual constraint entropy can distinguish pure states that have identical von Neumann entropy but different entanglement distribution structures, revealing its sensitivity to the kurtosis of theSchmidt spectrum. The analysis of fluctuation scaling ∆Sdual ∼ √lndA in completely random tensor networks provides an information-theoretic interpretationof holographic quantum corrections and yields testable numerical predictions. Thecorrespondence between the information density gradient in the discrete-continuumlimit and bulk curvature is formulated as the constraint network coarse-grainingconjecture, charting the direction for a microscopic mechanism of gravity emergingfrom quantum information.
Quantum entanglement is traditionally described as a statistical correlation encoded in a many-body wavefunction. While this framework successfully predicts measurement outcomes, it leaves unresolved whether entanglement reflects purely probabilistic structure or an intrinsic physical organization of quantum states. Recent experiments have demonstrated that high-dimensional entangled states possess thousands of distinct, experimentally accessible topological invariants that persist under basis changes, noise, and local perturbations. These features are difficult to reconcile with a purely statistical interpretation of the wavefunction, but arise naturally if entanglement corresponds to intrinsic geometric structure within state space. This work analyzes experimental results reported in Nature Communications (2025) showing robust, high-dimensional topological structure embedded in conventional entangled states. We show that the multiplicity, stability, and basis-independence of these invariants support an interpretation of entanglement as intrinsic geometry rather than as observer-dependent probability. Under this interpretation, nonlocal correlations reflect shared geometric structure rather than information transfer, and robustness arises from topological protection rather than statistical averaging. The paper does not modify the formal machinery of quantum mechanics, but reframes the physical meaning of the quantum state. Entanglement is interpreted as a geometric configuration of collective modes whose topology constrains allowed correlations. This perspective provides a natural physical explanation for robustness, nonlocality, and measurement independence, and places strong constraints on interpretations in which the wavefunction is purely epistemic. Together with independent results in condensed matter and gravitational physics, these findings support a geometry-first interpretation of quantum phenomena and contribute to a growing body of experimental evidence that quantum states possess intrinsic physical structure beyond probabilistic description.
Quantum entanglement is commonly characterized through global state descriptions on tensor product spaces, correlation measures or algebraic constructions, while local consistency constraints play no explicit structural role. We formulate entanglement as a combinatorial structure of overlapping local descriptions, drawing on De Bruijn graphs, where nodes represent overlapping contexts and paths encode globally coherent assemblies. We construct a graph whose nodes represent reduced quantum states on subsystems of fixed size and whose edges encode admissible extensions consistent with quantum mechanical compatibility conditions. Global many body states correspond to paths on this graph, while entanglement is reinterpreted as a property of graph connectivity and path multiplicity, rather than as a standalone numerical quantity. This formalism allows a separation between constraints imposed purely by local quantum consistency and additional structure introduced by dynamics, symmetries or boundary conditions, also clarifying how large-scale structural features may arise from local compatibility alone. Our graph-based formulation provides several advantages over conventional approaches. Supporting a unified treatment of static entanglement structure and dynamical evolution, it incorporates finite order locality and memory effects. Entanglement growth can be interpreted as path proliferation, while decoherence and noise correspond to the removal of admissible transitions. Our approach leads to testable hypotheses concerning the scaling of admissible state extensions, the robustness of entangled structures under local perturbations and the emergence of effective geometry from overlap constraints. Potential future directions include applications to many body reconstruction problems and comparative analysis of different classes of quantum states within a single combinatorial language.
The interplay of the constituents of interacting many-body systems may reveal emergent properties on the macroscopic scale which are not inherent to the individual constituents. These properties are expressed in macroscopic observables \ndescribing the state — denoted as the phase of the system. Continuous phase transition between phases are generically manifested in critical behavior, for example, a divergence of a macroscopic observable. The identification of the present phase of a system and the classification of critical phenomena into universality classes are exciting challenges of condensed-matter physics. \nIn this thesis, we use entanglement entropies as macroscopic quantities for the characterization of phases of quantum matter and critical theories. For groundstates of quantum many-body systems the entanglement entropy is a measure of \nthe amount of entanglement between two subsystems. The generic dependence of the entanglement entropy on the size and shape of the subsystems is contained in the well-known boundary law — stating a scaling of the entanglement entropy with the boundary between the two subsystems. We numerically investigate how the coefficient of this dependence reflects quantum phase transitions in simple spin-half bilayer models. \nSubleading terms to the boundary law such as a logarithmic contribution provide universal numbers for the criticality of field theories. We examine free and interacting theories from an entanglement entropy perspective in order to assess the role of the coefficient of the logarithmic correction induced by corners in the subsystems. Beyond its universality, this coefficient also quantifies degrees of freedom of low-lying excitations in the conformal field theory describing a critical point. \nA constant contribution to the boundary law indicates the presence of so-called topological order in the ground state of a many-body system. This extremely useful property can also be identified in classical counterparts of entanglement entropy which we study at the example of various toric code models. To this endeavor, we have designed Monte Carlo techniques which allow for an efficient numerical computation of the constant contribution. In particular, we analyze via entanglement entropies under which conditions remnants of topological order are present in the quantum system at finite temperature and at perturbations from a magnetic field. The major motivation behind this effort is to use topological order for the robust storage of a quantum information — a basic need for the construction of quantum \ncomputers.