Paper VI — Entanglement Information Locality and Many-Body Structure

DOI: 10.5281/zenodo.17925692 Entanglement Information Locality and Many-Body Structure Paper VI of the Ordered-Dynamics Reconstruction Program Paul Cooneya aIndependent Researcher, Innisfil, Ontario, Canada E-mail: paul.co[email protected]to.ca Contents 1 Introduction and Logical Position 1 2 Operational Independence as Additive Capacity 2 2.1 Information Capacity and Composition 2 2.2 Direct sum vs Tensor product 2 3 Entanglement as Interaction Memory 3 4 Entanglement Information Locality (EIL) 3 5 Lieb–Robinson Bound as Saturation of EIL 3 6 No-Signaling as a Theorem 3 7 Tsirelson Bound from EIL 4 8 Monogamy as Correlation Budgeting 4 9 Conclusion 4 DOI: 10.5281/zenodo.17925683 apers I–V reconstructed single-particle quantum mechanics from bounded information, operational locality, and relativistic causality. The present paper extends the program to composite systems. We introduce Operational Independence as an additive informationcapacity principle for spatially separated subsystems and prove that it forces tensor-product composition uniquely. Entanglement then arises not as a postulate but as the unavoidable memory of past interactions. We formulate Entanglement Information Locality (EIL), a bound on the rate at which interaction-generated correlation can be created. EIL is shown to saturate as a Lieb–Robinson bound, yielding an explicit entanglement light cone and operational no-signaling as a theorem. Quantum correlation bounds, including the Tsirelson limit, emerge as maximal correlations compatible with EIL. Finally, entanglement monogamy is derived as a finite correlation-budget constraint. Entanglement is thus reconstructed as a dynamical, resource-limited consequence of local interaction history rather than nonlocal influence. Contents 1 Introduction and Logical Position Papers I–V established that quantum dynamics is the saturation of sharp informational constraints. The remaining conceptual gap concerns many-body structure. Standard quantum theory postulates: •Tensor products for composite systems, – 1 – •Entanglement as a primitive, •Nonlocal correlations subject to ad hoc bounds. This paper shows that none of these are independent assumptions. Central claim. Once information capacity is finite and interactions are local, correlations can only be created by shared interaction history. Entanglement is therefore a form of interaction memory, dynamically limited by locality. 2 Operational Independence as Additive Capacity 2.1 Information Capacity and Composition The information capacity of a system with Hilbert space dimension dscales as C∼log d. For independent systems, capacities must add. [Operational Independence (OI)] For two spacelike-separated laboratories Aand B, the total operational information capacity satisfies C(A∪B)=C(A) + C(B). Local preparations and manipulations can be performed simultaneously without mutual interference. Remark 1.This is an informational, not algebraic, axiom. It expresses the ability of Alice and Bob to fill their local information buffers concurrently. 2.2 Direct sum vs Tensor product Direct-sum composition violates additive capacity. Proof. Consider the direct sum AB =A⊕B. The dimension is dA+dB. The capacity scales as: C⊕∼log(dA+dB)≈max(log dA,log dB). This implies the system operationally resides in Aor B, but not both simultaneously. Thus, C⊕=CA+CB. [Tensor product necessity] Operational Independence uniquely forces AB ∼ =A⊗B. Proof. The tensor product yields dimension dA·dB. The capacity scales as: C⊗∼log(dA·dB) = log dA+ log dB=CA+CB. This is the unique associative composition rule preserving linearity that satisfies capacity additivity. Operationally, it is the only structure allowing independent local degrees of freedom to be fully occupied simultaneously. – 2 – 3 Entanglement as Interaction Memory Definition 1 (Entanglement).A state ρAB is entangled if it cannot be prepared by independent local operations (LOCC). Entanglement cannot be created without interaction. Proof. Local operations map product states to product states (or mixtures thereof). Only an interaction Hamiltonian VAB coupling the tensor factors can generate non-product correlations from a product initial state. Remark 2 (Interpretation).Entanglement records the fact that two systems once interacted. It is not influence at a distance but the persistence of shared history. 4 Entanglement Information Locality (EIL) Entanglement is a physical resource generated by interactions. Its growth must be bounded. [Entanglement Information Locality] Let Aand Binteract via a bounded Hamiltonian VAB. Then for any admissible entanglement measure E(e.g., Logarithmic Negativity, Entanglement Entropy), there exists a universal constant ΛEsuch that:     d dtE(ρAB(t))    ≤ΛE∥VAB∥. Remark 3.EIL states that correlations cannot be generated faster than interaction strength allows. It acts as an “information drag” force, resisting the instantaneous delocalization of information. 5 Lieb–Robinson Bound as Saturation of EIL [Entanglement light cone] For finite-range interactions on a metric graph, EIL implies the Lieb–Robinson bound: ∥[OX(t), OY(0)]∥ ≤ C∥OX∥∥OY∥e−µ(d(X,Y )−vEt), where vEis the entanglement velocity determined by ΛE. Proof. The growth of correlations is governed by the commutator expansion of the Heisenberg evolution eiHt. The norm of these commutators is bounded by the interaction strength integrated over time. Summing the series yields the exponential cone structure. Outside this cone (d>vEt), correlations are exponentially suppressed. Remark 4.The entanglement cone is the spacetime boundary of shared interaction history. 6 No-Signaling as a Theorem [Operational no-signaling] Local operations on Acannot influence outcome statistics in B outside the entanglement cone. Proof. Signaling requires the generation of distinguishability between ρBand ρ′ Bconditioned on operations at A. By EIL, the capacity to generate this distinguishability propagates at vE. Thus, outside the cone, ρBremains invariant. – 3 – 7 Tsirelson Bound from EIL [Tsirelson bound] For local measurements on entangled pairs, the CHSH correlation is bounded: |⟨CHSH⟩| ≤ 2√2. Proof. Hypothetical “super-quantum” correlations (e.g., PR boxes achieving 4) would imply an information capacity exceeding the bound imposed by EIL. Specifically, they would require the “interaction memory” to contain more information than could have been generated by the finite Hamiltonian during the interaction window. The Tsirelson bound represents the saturation of the EIL capacity. 8 Monogamy as Correlation Budgeting [Monogamy of entanglement] For a tripartite system ABC, E(A:B)+E(A:C)≤ E(A:BC). Remark 5.A finite-capacity system (like a qubit) possesses a finite “correlation budget.” Maximal entanglement with Bconsumes the entire capacity of A, leaving no degrees of freedom available to correlate with C. Monogamy is simply the conservation of correlation capacity. 9 Conclusion We have shown that: 1. Tensor products are forced by additive information capacity (OI). 2. Entanglement is the unavoidable memory of interaction history. 3. EIL limits the growth of this memory, yielding Lieb–Robinson cones. 4. Quantum limits (No-signaling, Tsirelson, Monogamy) are conservation laws for this finite interaction resource. Many-body quantum structure is therefore forced by finite information and locality, not postulated. Next step. Paper VII derives variable particle number and quantum fields as the necessary local information buffers required to maintain additive capacity in the continuum limit. – 4 –