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.
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.
We study entanglement reachability in quantum many-body systems evolving under physically admissible local dynamics. While Lieb--Robinson bounds constrain the propagation of information, and complexity-based approaches characterize the resources required to represent or prepare quantum states, these frameworks do not by themselves determine which entanglement structures are dynamically attainable. The central question addressed in this work is therefore: which entanglement structures can arise under constrained quantum evolution, and which remain inaccessible despite being kinematically allowed? We introduce a unified axiomatic framework for entanglement reachability based on the notion of \emph{structural confinement}. The framework incorporates finite-range locality, finite-dimensional memory degrees of freedom, and abstract representability constraints expressed through operator subsystems. Rather than characterizing accessibility directly in Hilbert space, we formulate the problem on an induced structural space of entanglement configurations and quantum information patterns. We establish a structural confinement theorem showing that admissible dynamics generate only a restricted reachable region in structural space. The theorem provides a general criterion for identifying forbidden structures whenever their defining information cannot be preserved within the admissible representational framework. Thus, ocality, memory, and representability constraints do not merely limit the rate of information propagation; they impose additional dynamical restrictions on the set of quantum structures that can actually be realized. The generality of the framework is demonstrated through three explicit realizations involving multipartite coherence, tensor-network representability, and protected logical nformation in quantum error-correcting codes. Although these realizations involve distinct physical structures, they are governed by the same underlying confinement mechanism: dynamical accessibility is determined by the compatibility between physical evolution and the structural information retained by the chosen representation. These results establish structural confinement as a general organizing principle for entanglement reachability under constrained quantum dynamics. The framework provides a unified language for deriving dynamical accessibility limits beyond propagation bounds alone and for classifying which quantum structures are reachable, forbidden, or conditionally accessible in locally constrained many-body systems.
(This version has been superceeded, see v.3) Contemporary approaches to physical unification typically emphasize dynamical laws, interpretive frameworks, or information-theoretic reconstructions. Less attention is given to the structural constraints that determine which physical descriptions can remain coherent and stable under interaction, scale transition, and aggregation. This paper develops a constraint-first analysis of coherence, arguing that persistence and unification require restrictions on admissible physical continuations that are not reducible to dynamics alone. Within the framework referred to here as Quantum Coherence Theory (QCT), coherence is treated not as a fragile quantum resource but as a structural property governed by interaction geometry and constraint density. On this view, classical behavior emerges when accumulated constraints sufficiently narrow admissible evolution, while quantum behavior re-emerges when constraints are relaxed. The account does not introduce new dynamics, collapse mechanisms, or modifications of quantum formalism, but instead clarifies the conditions under which existing formalisms yield stable, unified descriptions. The resulting constraints distinguish QCT from observer-centric, intrinsic-collapse, and purely descriptive decoherence accounts, while remaining compatible with globally unitary dynamics. Formal analysis in later sections develops operational redundancy conditions, admissibility constraints on joint states, and rigidity-style arguments linking redundant record structure to the stability of classical descriptions. These results do not constitute a complete theory of unification, but they narrow the space of physically admissible descriptions and clarify the structural role of coherence in bridging quantum and classical regimes.
Efficient representations of multipartite quantum states play a fundamental role in quantum information theory, providing both conceptual insight and practical tools for characterizing entanglement. Motivated by the axiomatic framework for graph states [Phys. Rev. A 85, 062313 (2012)] and its subsequent extension to hypergraph states [Phys. Rev. A 87, 022311 (2013)], we introduce an axiomatic construction of \emph{matroid states}, a new family of multipartite quantum states associated with matroids. Our constructions are based on a set of axioms analogous to those that define graph and hypergraph states, yielding a consistent quantum representation of arbitrary matroids. Two ways of constructing matroid states are proposed: the first is defined in terms of circuits, and the second in terms of independent sets. In both approaches, we establish the existence of universal global operators that satisfy desirable properties such as locality, symmetry, commutativity, and are associated with the combinatorial structure of matroids. Furthermore, we establish a hierarchy connecting graph, matroid, and hypergraph states within a unified framework. Additionally, we show how to obtain an arbitrary graph state by applying suitable families of matroid states, whose corresponding operators are the generators of the stabilizer subgroup of the graph state. These results identify matroid theory as a natural combinatorial language for the efficient description of multipartite quantum states and open new perspectives for the investigation of quantum entanglement and related combinatorial structures.
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.
Version 2: Replaced Word file with properly formatted PDF. No changes to scientific content. --- We argue that quantum entanglement is best understood as the structural non-factorizability of joint continuation measures in a pregeometric framework — one that presupposes neither a background Hilbert space nor a background spacetime. This thesis has direct implications for the interpretation of Bell non-locality, the debate between ontic and epistemic structural realism, and the question of whether entanglement is a fundamental feature of reality or an artifact of quantum-mechanical representation. Within Temporal Rate Ontology (TRO), physical systems are modelled as directed acyclic graphs (DAGs) of continuation events, and entanglement is defined as the non-existence of a product decomposition of the joint continuation measure — a purely combinatorial condition that requires no wavefunction and no shared configuration space. We prove that this structural condition entails the standard correlated measurement outcomes of quantum mechanics, under a mild non-degeneracy condition on the joint DAG that is generically satisfied. The quadratic probability rule is derived from an aggregation-stability argument within the CLT regime rather than postulated. Bell non-locality is reinterpreted as global DAG coherence rather than superluminal influence. Decoherence is modelled as constraint-induced suppression of continuation multiplicity. The framework is situated against relational quantum mechanics, causal set theory, and ontic structural realism, and philosophical objections concerning subsystem identity, underdetermination, and the emergence of the Hilbert space are addressed directly.
Abstract We introduce a geometric framework for representing quantum entanglementthrough reduced entanglement coordinates derived from the Schmidt decompo-sition. For pure two-qubit states, the framework yields an entanglement conein which separability corresponds to geometric collapse to the cone apex. Thisconstruction extends naturally to higher-rank bipartitions through Schmidt-simplex geometries associated with Schmidt-rank reduction. For multipartitesystems, we introduce entanglement-graph coordinates that use pairwise bipar-tite reductions as geometric probes of global quantum states. The resultingrepresentation defines a local-unitary-invariant reduced geometry whose dimen-sion scales quadratically with subsystem number, compared with the exponentialscaling of the underlying pure-state space. Although not a complete invariantof multipartite entanglement, the graph geometry preserves important rela-tional and connectivity structure while remaining computationally tractable.Building on this representation, we develop a geometric interpretation of vari-ational quantum dynamics in which optimization trajectories evolve throughstructured regions of entanglement-graph space. We introduce Hamiltonian-selected entanglement-graph sectors, establish exact and perturbative sectorconstructions, derive local graph-distance energy relations near nondegenerateoptima, and develop learned geometric diagnostics for variational optimization.Numerical experiments on molecular and spin-chain Hamiltonians support thehypothesis that low-energy states concentrate near Hamiltonian-dependent graphsectors and demonstrate predictive information in the resulting geometric rep-resentations. Together, these results provide a unified geometric framework foranalyzing entanglement structure, quantum dynamics, and variational quantumoptimization.