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.
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.
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.
We present a reconstruction of quantum correlation structure from six admissibility principles: compositional closure, locality factorization, norm-bounded growth, convex linearity, closure symmetry (self-duality), and local tomography. We show that these conditions force the state space to be a homogeneous self-dual cone, which by the Koecher–Vinberg theorem corresponds to a Euclidean Jordan algebra. Imposing physical composition constraints reduces this structure uniquely to complex matrix algebras. The Tsirelson bound then follows as a corollary of operator norm boundedness. Superquantum correlations (e.g., PR-box correlations) are excluded not by appeal to relativity but by violation of norm-bounded compositional closure. The result reframes quantum theory as the maximal globally admissible correlation theory under bounded composition
We present a constraint-based axiomatization of quantum correla- tions that derives Bell statistics without invoking Hilbert space, wave functions, or causal mechanisms. Starting from five physically moti- vated axioms—rotational invariance, perfect anti-correlation, uniform marginals (no-signaling), smoothness, and extremality—we prove that the singlet correlation function E(ˆ a,ˆˆb) =−ˆa·b is uniquely deter- mined. The proof proceeds through functional equation analysis: the extremality axiom implies angle additivity, reducing the problem to Cauchy’s equation with boundary conditions. We derive the Tsirelson bound |S|≤2√2 directly from this correlation function without op- erator algebras, and show that CHSH violation follows as a theorem. The no-signaling condition emerges as a structural property of the con- straint, ensuring compatibility with relativistic causality. We extend the framework to tripartite systems, deriving GHZ correlations from analogous axioms and proving maximal Mermin inequality violation. The constraint interpretation dissolves the tension between entangle- ment and causality: correlations are neither caused nor explained by hidden variables—they are specified by the constraint structure itself. This framework recovers all quantum predictions while eliminating causation from the foundational ontology.
This paper constructs an interpretative framework of quantum mechanics thattreats the complex structure as a modal dimension, reconstructing quantum measurement as an information-geometric phase transition rather than an instantaneouswavefunction collapse. By introducing a rigorous decomposition of Jordan algebrasand Lie algebras, the latent-world subspace is proven to form a bi-module structureunder the Jordan product and the projected Lie bracket, thereby being compatible with finite-dimensional algebraic frameworks. In the pure-state limit, the sectional curvature of the Bures metric and the quantum Fisher information satisfy anexact differential-geometric identity; the environmental corrections for the mixedstate case are honestly labeled as scaling conjectures constrained by renormalizationgroup. The information potential difference is rigorously defined via the relative entropy of coherence, and the Landauer principle yields the minimal thermodynamiccost of erasing coherent information in the quasi-static limit, with the ln 2 threshold re‑positioned as a calibratable benchmark in non‑equilibrium, finite‑temperatureenvironments. The physical inadequacy of the real-number framework under thesame Hilbert-space dimension is established by the exclusion principle of Bell-typeexperiments, proving that imaginarity is the physical origin of quantum interferencerather than a notational convenience. By introducing the Frobenius theorem as aselection constraint and Bell-type experiments as empirical constraints, the ternarycircular argument is broken into a unidirectional implication chain, establishingthe epistemological foundation of constraint structural realism. Four experimental predictions with explicit falsification conditions provide an operable interfacefor experiments, completing the proof closure from heuristic program to axiomaticsystem.
The indispensability of the complex structure in quantum mechanics has beenexperimentally confirmed, yet its ontological status—whether complex numbers aremerely a mathematical tool or a constitutive element of physical reality—has longremained undecided. Starting from mathematical necessity, this paper argues thatthe complex structure is not only the descriptive language of quantum mechanicsbut also an ontological constraint condition of physical reality. Based on Frobenius’ theorem on the classification of division algebras over the real numbers, wereveal: if the descriptive framework is required to be both commutative and algebraically closed, then the complex structure is the unique logically necessary choice.This mathematical constraint, when mapped onto physical reality, implies that the”imaginary part” corresponds to a modal dimension that is not directly observablebut constrains observable phenomena. The measurement process is thus reconceptualized as a projection from the complex structure onto a real subspace, whereinformation loss is an inherent feature of the mathematical structure rather thanan additional dynamical assumption. We demonstrate that this framework is compatible with research on imaginarity as a resource in quantum resource theory andwith analyses of Fisher information limits in quantum metrology, and we proposethree testable predictions with quantum metrological precision. This interpretationadvances the quantum measurement problem from a matter of interpretational debate to a level of structural analysis, providing a rigorous mathematical foundationfor understanding the quantum-classical transition.
Historically, quantum mechanics has relied on the assumption of non-communicating parties, enforced via space-like separation, to witness the gap between classical and quantum descriptions. This thesis explores the boundaries of the standard Bell non-local scenario, investigating how to preserve this gap when communication is allowed. In particular, this work focuses on restricted communication, multipartite settings, and operator-algebraic formulations. Key contributions of this work include: • Dimensional Constraints: Establishing mathematical maps from standard Bell scenarios to prepare-and-measure scenarios with dimensionally bounded channels, proving that quantum bounds and self-testing properties are preserved. • Cryptographic Constraints: Analyzing compiled non-local games using quantum homomorphic encryption. The research proves asymptotic quantum soundness for all multipartite compiled games and introduces a quantitative hierarchy, similar to the NPA hierarchy, to characterize sequential correlations. • Operational Equivalences: Demonstrating that correlations arising from arbitrary compositions of non-signalling assemblages in multipartite communication schemes always reduce to standard spatial quantum Bell correlations. • Operator-Algebraic Tools: Introducing a novel chain rule for the Radon-Nikodym derivative to dilate chains of completely-positive maps, alongside the construction of a universal C*-algebra for sequential measurements. Overall, the manuscript unifies concepts from quantum foundations, complexity theory, and operator algebras to provide a rigorous mathematical framework for characterizing and certifying multipartite sequential quantum correlations.