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Competing informational and dynamical criteria for subsystem structure

Yoshifumi Maruko

Published Apr 18, 2026• 0 citations
Publisher

Abstract

We study subsystem-structure selection in small interacting quantum systems. We compare two criteria: (i) maximization of cross-boundary correlations and (ii) minimization of dynamical coupling across subsystem boundaries. Using exact diagonalization of asymmetric spin chains, we show that these criteria can favor different tensor factorizations of Hilbert space. We derive the crossover parameter λ* at which the preferred factorization switches. The existence of this regime demonstrates that subsystem structure need not be uniquely determined by dynamical stability alone. This paper is part of the Relational Emergence Model (REM) series (REM4 of 5).

DOI
10.5281/zenodo.19637583

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Relational Emergence Model (REM 5.12): Variational Selection of Tensor-Product Structure in Quantum Systems

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Quantum theory does not uniquely determine subsystem structure. This paper proposes a variational principle on factorization space to select preferred tensor-product structure. The optimization objective balances informational articulation (mutual information across subsystem boundaries) and dynamical separability (interaction coupling across the partition). The framework is formulated on the coset manifold F ≅ U(N)/(U(n_A)⊗U(n_B)) and provides a generative extension of relational quantum mechanics (Rovelli 1996), in which subsystem structure itself is treated as a variational degree of freedom rather than a primitive assumption. A three-qubit illustration demonstrates regime competition with crossover at λ* ≈ 0.28, consistent with results in the REM series (REM2–REM4). This paper is the fifth in the Relational Emergence Model (REM) series.

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The Relational Emergence Model (REM) is a generative framework for the articulation of relational structure in quantum theory. While Relational Quantum Mechanics (RQM) defines physical states relative to other systems, it does not specify how subsystem structure itself becomes determined. REM addresses this gap by modeling subsystem differentiation as a variational selection process over candidate tensor-product factorizations of Hilbert space. The model introduces a three-layer architecture: Layer 0 — a pre-relational Hilbert space without a privileged tensor-product structureLayer 1 — differentiation as relational articulationLayer 2 — decoherence as stabilization of articulated structure Differentiation is formulated as an optimization problem on a factorization manifold, F ≅ U(N)/(U(n_A) ⊗ U(n_B)), with selection governed by a variational functional Φ = Φ_S + λ Φ_H, combining informational articulation (mutual information) and dynamical stability (boundary Hamiltonian contribution). The parameter λ controls the balance between informational correlation and dynamical coherence, admitting interpretations analogous to inverse temperature, timescale ratio, or renormalization-scale dependence. A three-qubit toy model demonstrates nontrivial factorization selection and a crossover between informationally favored and dynamically stabilized regimes. The framework connects relational quantum mechanics, decoherence theory, information geometry, subsystem structure in gauge theory, and variational optimization on quotient manifolds. The present work is not proposed as a complete physical theory but as a mathematically structured foundational proposal for studying how relational structure itself may become selected, stabilized, and rendered physically meaningful. The REM framework aims to clarify the generative layer underlying relational description and provides a basis for further development including continuous factorization flow, tensor-network formulations, categorical structure, and operational implementations.

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On the emergence of preferred structures in quantum theory

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Competing Informational and Dynamical Criteria for Subsystem Structure: Relational Emergence Model — Revised Version 5

Yoshifumi Maruko

REM_lambda Version 4 Version 4 presents a corrected numerical implementation of the variational selection framework used in the Relational Emergence Model (REM). The model evaluates candidate Hilbert-space factorisations through competition between informational differentiation and dynamical boundary cost. $$\Phi(F)=\Phi_S(F)-\lambda C_H(F)$$ The informational term is $$\Phi_S(F)=I(A:B)$$ and the sign-definite quadratic dynamical cost is $$C_H(F)=\mathrm{Tr}(\rho H_{\partial F}^{2})=\langle H_{\partial F}^{2}\rangle$$ Numerical correction in Version 4 Version 4 corrects two implementation issues in the numerical analysis accompanying REM_lambda Version 3. The discrete-cut implementation previously calculated $$ \langle H_{\partial F}\rangle^{2} $$ instead of the intended quadratic boundary-fluctuation cost $$ \langle H_{\partial F}^{2}\rangle $$ The previous continuous-factorisation routine transformed the state and the full Hamiltonian together, making the dynamical cost invariant under the candidate unitary transformation. Consequently, the Version 3 optimiser effectively maximised mutual information rather than the complete REM functional. Version 4 corrects both issues and recomputes all affected numerical results. Three-qubit XY-chain benchmark The numerical benchmark uses the asymmetric three-qubit XY-chain Hamiltonian $$H=J_{12}(X_1X_2+Y_1Y_2)+J_{23}(X_2X_3+Y_2Y_3)+h\sum_i Z_i$$ The parameters are $$J_{12}=1.5,\qquad J_{23}=0.6,\qquad h=0.2$$ The ground state is obtained through exact diagonalisation. For the two site-contiguous candidate partitions, the corrected dynamical cost is evaluated as $$C_H=\langle H_{\partial F}^{2}\rangle$$ At $$\lambda=0.2$$ the best site-contiguous partition achieves $$\Phi=0.560$$ The crossover between the information-dominated and dynamics-dominated contiguous partitions remains $$\lambda^*\approx0.165$$ Continuous factorisation optimisation For each candidate unitary $U$, the rotated Hamiltonian is $$H_U=UHU^\dagger$$ It is decomposed into local and interaction components: $$H_U=H_A(U)+H_B(U)+H_{\partial}(U)$$ The local components are obtained through partial-trace projection with an identity-component correction. The continuous optimisation evaluates the complete REM functional: $$\Phi(U)=I(A:B)_{U\psi}-\lambda\langle\psi_U|H_{\partial}(U)^2|\psi_U\rangle$$ A search over a four-parameter generator submanifold recovers $$\Phi=1.05$$ This exceeds the best site-contiguous result: $$\Phi=0.560$$ The corrected numerical improvement is therefore $$0.560\rightarrow1.05$$ The previously reported Version 3 result $$0.68\rightarrow1.32$$ is withdrawn because it was obtained using an incorrect objective function. Factorisation-space dimension The full factorisation space is represented by $$U(8)/(U(4)\otimes U(2))$$ The tensor-product representation of $U(4)\times U(2)$ has a one-dimensional kernel. The effective local-unitary subgroup therefore has dimension $$16+4-1=19$$ Since $$\dim U(8)=64$$ the quotient manifold has real dimension $$64-19=45$$ The present numerical implementation explores a four-parameter generator submanifold of this 45-dimensional quotient manifold. The recovered factorisation is a higher-scoring solution within the explored submanifold. It is not claimed to be the global optimum over the full factorisation manifold. Finite-size scaling Finite-size calculations were performed for $$N=3,4,6,8$$ under weak-center, linear, and step coupling profiles. All finite-size values were recomputed using the corrected quadratic dynamical cost $$C_H=\langle H_{\partial F}^{2}\rangle$$ Validation and reproducibility The accompanying reproducibility package includes: exact diagonalisation for asymmetric XY chains; evaluation of site-contiguous candidate cuts; factorisation-dependent reconstruction of the interaction Hamiltonian; continuous optimisation on a four-parameter generator submanifold; finite-size scaling calculations; deterministic benchmark values; 16 automated tests; GitHub Actions continuous integration for Python 3.12 and 3.13. The regression tests verify: exact agreement with $\langle H_{\partial F}^{2}\rangle$; the distinction between $\langle H_{\partial F}^{2}\rangle$ and $\langle H_{\partial F}\rangle^{2}$; factorisation dependence of the reconstructed boundary cost; consistency at the identity transformation; reproduction of corrected benchmark values; successful finite-size and output-generation workflows. Scope Version 4 supersedes the numerical results reported in Version 3. The theoretical REM functional and the use of the sign-definite quadratic boundary-fluctuation cost remain unchanged. Source and reproducibility GitHub repository: https://github.com/malko73/rem Version 4 DOI: https://doi.org/10.5281/zenodo.21427776

•Apr 18, 2026•0 citations

Selection Principles for the Differentiation Map in the Relational Emergence Model

Yoshifumi Maruko

The Relational Emergence Model (REM) extends Relational Quantum Mechanics (RQM) by treating relational structure as *generated* rather than assumed. This companion paper addresses the central open question left in the main REM framework: what selects the differentiation map 𝒟 that articulates a specific tensor-product decomposition out of the pre-relational Hilbert space (Layer 0 → Layer 1). We introduce a variational formulation over candidate factorizations ℱ, with functionals based on von Neumann entropy maximization (Φ_S), mutual-information maximization (Φ_I), and energetic stability (Φ_H). A combined functional Φ = Φ_S + λ Φ_H allows continuous interpolation between purely informational and dynamically stabilized regimes. A concrete 3-qubit numerical example (with entropy landscape visualization) demonstrates that informational extremization induces nontrivial bipartition selection. The framework is further embedded in three mathematical structures: information geometry on factorization space, categorical quantum mechanics (Frobenius algebra selection), and gauge-theoretic subsystem construction (boundary degrees of freedom analogy). Connections to recent integrated-information approaches in relational quantum dynamics (Zaghi 2025) are discussed as natural extensions. This work complements the main REM paper by supplying the missing selection mechanism, transforming REM from a conceptual proposal into a mathematically structured generative model of relational quantum reality. Related publication: "Relational Emergence Model: A Generative Extension of Relational Quantum Mechanics" (Maruko 2026, Zenodo DOI: [https://doi.org/10.5281/zenodo.19123314]) This is the companion paper providing the selection mechanism for the differentiation map 𝒟 introduced in the main REM framework.

•Mar 22, 2026•0 citations

Complementarity as interaction-dependent decomposition: a relational mechanism for a foundational Quantum principle

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This paper proposes that quantum complementarity arises from interaction-dependent tensor-product-structure (TPS) selection. Different measurement interactions impose distinct decompositions on the joint system–apparatus Hilbert space, each stabilizing a different pointer algebra through decoherence. Complementary descriptions emerge when the TPS induced by one interaction is incompatible with another, so no single coupling can realize both observable algebras simultaneously. The framework stays within standard quantum mechanics, requiring no new postulates. A minimal qubit model demonstrates how incompatible measurement couplings stabilize different subsystem splits and suppress coherence pathways needed for alternative descriptions. The account clarifies Bohr's original claim that phenomena depend on the whole experimental arrangement, grounding it in interaction geometry rather than interpretive principle. Testable predictions include tunable complementarity regimes, TPS interference revival, and multi-detector incompatibility thresholds.

•Jan 13, 2026•2 citations
We study subsystem-structure selection in small interacting quantum systems. We compare two criteria: (i) maximization of cross-boundary correlations and (ii) minimization of dynamical coupling across subsystem boundaries. Using exact diagonalization of asymmetric spin chains, we show that these criteria can favor different tensor factorizations of Hilbert space. We derive the crossover parameter λ* at which the preferred factorization switches. The existence of this regime demonstrates that subsystem structure need not be uniquely determined by dynamical stability alone. This paper is part of the Relational Emergence Model (REM) series (REM4 of 5).
Quantum theory does not uniquely determine subsystem structure. This paper proposes a variational principle on factorization space to select preferred tensor-product structure. The optimization objective balances informational articulation (mutual information across subsystem boundaries) and dynamical separability (interaction coupling across the partition). The framework is formulated on the coset manifold F ≅ U(N)/(U(n_A)⊗U(n_B)) and provides a generative extension of relational quantum mechanics (Rovelli 1996), in which subsystem structure itself is treated as a variational degree of freedom rather than a primitive assumption. A three-qubit illustration demonstrates regime competition with crossover at λ* ≈ 0.28, consistent with results in the REM series (REM2–REM4). This paper is the fifth in the Relational Emergence Model (REM) series.
The Relational Emergence Model (REM) is a generative framework for the articulation of relational structure in quantum theory. While Relational Quantum Mechanics (RQM) defines physical states relative to other systems, it does not specify how subsystem structure itself becomes determined. REM addresses this gap by modeling subsystem differentiation as a variational selection process over candidate tensor-product factorizations of Hilbert space. The model introduces a three-layer architecture: Layer 0 — a pre-relational Hilbert space without a privileged tensor-product structureLayer 1 — differentiation as relational articulationLayer 2 — decoherence as stabilization of articulated structure Differentiation is formulated as an optimization problem on a factorization manifold, F ≅ U(N)/(U(n_A) ⊗ U(n_B)), with selection governed by a variational functional Φ = Φ_S + λ Φ_H, combining informational articulation (mutual information) and dynamical stability (boundary Hamiltonian contribution). The parameter λ controls the balance between informational correlation and dynamical coherence, admitting interpretations analogous to inverse temperature, timescale ratio, or renormalization-scale dependence. A three-qubit toy model demonstrates nontrivial factorization selection and a crossover between informationally favored and dynamically stabilized regimes. The framework connects relational quantum mechanics, decoherence theory, information geometry, subsystem structure in gauge theory, and variational optimization on quotient manifolds. The present work is not proposed as a complete physical theory but as a mathematically structured foundational proposal for studying how relational structure itself may become selected, stabilized, and rendered physically meaningful. The REM framework aims to clarify the generative layer underlying relational description and provides a basis for further development including continuous factorization flow, tensor-network formulations, categorical structure, and operational implementations.
We assess the possibilities offered by Hilbert space fundamentalism, an attitude towards quantum physics according to which all physical structures (e.g. subsystems, locality, spacetime, preferred observables) should emerge from minimal quantum ingredients (typically a Hilbert space, Hamiltonian, and state). As a case study, we first mainly focus on the specific question of whether the Hamiltonian can uniquely determine a tensor product structure, a crucial challenge in the growing field of quantum mereology. The present paper reviews, clarifies, and critically examines two apparently conflicting theorems by Cotler et al. and Stoica. We resolve the tension, show how the former has been widely misinterpreted and why the latter is correct only in some weaker version. We then propose a correct mathematical way to address the general problem of preferred structures in quantum theory, relative to the characterization of emergent objects by unitary-invariant properties. Finally, we apply this formalism in the particular case we started with, and show that a Hamiltonian and a state are enough structure to uniquely select a preferred tensor product structure.
REM_lambda Version 4 Version 4 presents a corrected numerical implementation of the variational selection framework used in the Relational Emergence Model (REM). The model evaluates candidate Hilbert-space factorisations through competition between informational differentiation and dynamical boundary cost. Φ(F)=ΦS(F)−λCH(F)\Phi(F)=\Phi_S(F)-\lambda C_H(F)Φ(F)=ΦS​(F)−λCH​(F) The informational term is ΦS(F)=I(A:B)\Phi_S(F)=I(A:B)ΦS​(F)=I(A:B) and the sign-definite quadratic dynamical cost is CH(F)=Tr(ρH∂F2)=⟨H∂F2⟩C_H(F)=\mathrm{Tr}(\rho H_{\partial F}^{2})=\langle H_{\partial F}^{2}\rangleCH​(F)=Tr(ρH∂F2​)=⟨H∂F2​⟩ Numerical correction in Version 4 Version 4 corrects two implementation issues in the numerical analysis accompanying REM_lambda Version 3. The discrete-cut implementation previously calculated ⟨H∂F⟩2 \langle H_{\partial F}\rangle^{2} ⟨H∂F​⟩2 instead of the intended quadratic boundary-fluctuation cost ⟨H∂F2⟩ \langle H_{\partial F}^{2}\rangle ⟨H∂F2​⟩ The previous continuous-factorisation routine transformed the state and the full Hamiltonian together, making the dynamical cost invariant under the candidate unitary transformation. Consequently, the Version 3 optimiser effectively maximised mutual information rather than the complete REM functional. Version 4 corrects both issues and recomputes all affected numerical results. Three-qubit XY-chain benchmark The numerical benchmark uses the asymmetric three-qubit XY-chain Hamiltonian H=J12(X1X2+Y1Y2)+J23(X2X3+Y2Y3)+h∑iZiH=J_{12}(X_1X_2+Y_1Y_2)+J_{23}(X_2X_3+Y_2Y_3)+h\sum_i Z_iH=J12​(X1​X2​+Y1​Y2​)+J23​(X2​X3​+Y2​Y3​)+h∑i​Zi​ The parameters are J12=1.5,J23=0.6,h=0.2J_{12}=1.5,\qquad J_{23}=0.6,\qquad h=0.2J12​=1.5,J23​=0.6,h=0.2 The ground state is obtained through exact diagonalisation. For the two site-contiguous candidate partitions, the corrected dynamical cost is evaluated as CH=⟨H∂F2⟩C_H=\langle H_{\partial F}^{2}\rangleCH​=⟨H∂F2​⟩ At λ=0.2\lambda=0.2λ=0.2 the best site-contiguous partition achieves Φ=0.560\Phi=0.560Φ=0.560 The crossover between the information-dominated and dynamics-dominated contiguous partitions remains λ∗≈0.165\lambda^*\approx0.165λ∗≈0.165 Continuous factorisation optimisation For each candidate unitary UUU, the rotated Hamiltonian is HU=UHU†H_U=UHU^\daggerHU​=UHU† It is decomposed into local and interaction components: HU=HA(U)+HB(U)+H∂(U)H_U=H_A(U)+H_B(U)+H_{\partial}(U)HU​=HA​(U)+HB​(U)+H∂​(U) The local components are obtained through partial-trace projection with an identity-component correction. The continuous optimisation evaluates the complete REM functional: Φ(U)=I(A:B)Uψ−λ⟨ψU∣H∂(U)2∣ψU⟩\Phi(U)=I(A:B)_{U\psi}-\lambda\langle\psi_U|H_{\partial}(U)^2|\psi_U\rangleΦ(U)=I(A:B)Uψ​−λ⟨ψU​∣H∂​(U)2∣ψU​⟩ A search over a four-parameter generator submanifold recovers Φ=1.05\Phi=1.05Φ=1.05 This exceeds the best site-contiguous result: Φ=0.560\Phi=0.560Φ=0.560 The corrected numerical improvement is therefore 0.560→1.050.560\rightarrow1.050.560→1.05 The previously reported Version 3 result 0.68→1.320.68\rightarrow1.320.68→1.32 is withdrawn because it was obtained using an incorrect objective function. Factorisation-space dimension The full factorisation space is represented by U(8)/(U(4)⊗U(2))U(8)/(U(4)\otimes U(2))U(8)/(U(4)⊗U(2)) The tensor-product representation of U(4)×U(2)U(4)\times U(2)U(4)×U(2) has a one-dimensional kernel. The effective local-unitary subgroup therefore has dimension 16+4−1=1916+4-1=1916+4−1=19 Since dim⁡U(8)=64\dim U(8)=64dimU(8)=64 the quotient manifold has real dimension 64−19=4564-19=4564−19=45 The present numerical implementation explores a four-parameter generator submanifold of this 45-dimensional quotient manifold. The recovered factorisation is a higher-scoring solution within the explored submanifold. It is not claimed to be the global optimum over the full factorisation manifold. Finite-size scaling Finite-size calculations were performed for N=3,4,6,8N=3,4,6,8N=3,4,6,8 under weak-center, linear, and step coupling profiles. All finite-size values were recomputed using the corrected quadratic dynamical cost CH=⟨H∂F2⟩C_H=\langle H_{\partial F}^{2}\rangleCH​=⟨H∂F2​⟩ Validation and reproducibility The accompanying reproducibility package includes: exact diagonalisation for asymmetric XY chains; evaluation of site-contiguous candidate cuts; factorisation-dependent reconstruction of the interaction Hamiltonian; continuous optimisation on a four-parameter generator submanifold; finite-size scaling calculations; deterministic benchmark values; 16 automated tests; GitHub Actions continuous integration for Python 3.12 and 3.13. The regression tests verify: exact agreement with ⟨H∂F2⟩\langle H_{\partial F}^{2}\rangle⟨H∂F2​⟩; the distinction between ⟨H∂F2⟩\langle H_{\partial F}^{2}\rangle⟨H∂F2​⟩ and ⟨H∂F⟩2\langle H_{\partial F}\rangle^{2}⟨H∂F​⟩2; factorisation dependence of the reconstructed boundary cost; consistency at the identity transformation; reproduction of corrected benchmark values; successful finite-size and output-generation workflows. Scope Version 4 supersedes the numerical results reported in Version 3. The theoretical REM functional and the use of the sign-definite quadratic boundary-fluctuation cost remain unchanged. Source and reproducibility GitHub repository: https://github.com/malko73/rem Version 4 DOI: https://doi.org/10.5281/zenodo.21427776
The Relational Emergence Model (REM) extends Relational Quantum Mechanics (RQM) by treating relational structure as *generated* rather than assumed. This companion paper addresses the central open question left in the main REM framework: what selects the differentiation map 𝒟 that articulates a specific tensor-product decomposition out of the pre-relational Hilbert space (Layer 0 → Layer 1). We introduce a variational formulation over candidate factorizations ℱ, with functionals based on von Neumann entropy maximization (Φ_S), mutual-information maximization (Φ_I), and energetic stability (Φ_H). A combined functional Φ = Φ_S + λ Φ_H allows continuous interpolation between purely informational and dynamically stabilized regimes. A concrete 3-qubit numerical example (with entropy landscape visualization) demonstrates that informational extremization induces nontrivial bipartition selection. The framework is further embedded in three mathematical structures: information geometry on factorization space, categorical quantum mechanics (Frobenius algebra selection), and gauge-theoretic subsystem construction (boundary degrees of freedom analogy). Connections to recent integrated-information approaches in relational quantum dynamics (Zaghi 2025) are discussed as natural extensions. This work complements the main REM paper by supplying the missing selection mechanism, transforming REM from a conceptual proposal into a mathematically structured generative model of relational quantum reality. Related publication: "Relational Emergence Model: A Generative Extension of Relational Quantum Mechanics" (Maruko 2026, Zenodo DOI: [https://doi.org/10.5281/zenodo.19123314]) This is the companion paper providing the selection mechanism for the differentiation map 𝒟 introduced in the main REM framework.
This paper proposes that quantum complementarity arises from interaction-dependent tensor-product-structure (TPS) selection. Different measurement interactions impose distinct decompositions on the joint system–apparatus Hilbert space, each stabilizing a different pointer algebra through decoherence. Complementary descriptions emerge when the TPS induced by one interaction is incompatible with another, so no single coupling can realize both observable algebras simultaneously. The framework stays within standard quantum mechanics, requiring no new postulates. A minimal qubit model demonstrates how incompatible measurement couplings stabilize different subsystem splits and suppress coherence pathways needed for alternative descriptions. The account clarifies Bohr's original claim that phenomena depend on the whole experimental arrangement, grounding it in interaction geometry rather than interpretive principle. Testable predictions include tunable complementarity regimes, TPS interference revival, and multi-detector incompatibility thresholds.