Geometric–Fractal Structure of Quantum Correlations and Time Delay: A Unified Metric–Measure Framework with Long-Range Dependence Scaling

Overview This preprint develops a unified, mathematically self-contained geometric reinterpretation of quantum entanglement, in which nonlocal correlations arise from the intrinsic geometry of composite Hilbert space rather than from any dynamical propagation between subsystems. The framework is built on: the Bures metric, Quantum Fisher Information (QFI), and completely positive trace-preserving (CPTP) coarse-graining semigroups. This work constitutes the unified synthesis and mathematical completion of two companion preprints: DOI: 10.5281/zenodo.18646759 — Fractal Geometry of Quantum Correlation Space (geometric/LRD foundation) DOI: 10.5281/zenodo.18649326 — Geometric Control of Quantum Time Delay in Non-Smooth Regimes (EWS application) It closes all identified mathematical gaps in both documents and consolidates the results into a single publication-ready manuscript. Main Contributions C1 — Theorem: No-Signaling Under CPTP Coarse-Graining (§3) For any CPTP map acting on subsystem , the reduced state of Alice satisfies: \operatorname{Tr}_B\!\left[(I_A \otimes \mathcal{E}_B)(\rho_{AB})\right] = \rho_A This follows from: linearity of partial trace, trace-preserving property of CPTP maps. Thus, Alice’s measurement statistics are provably independent of Bob’s local operation. The fractal geometry of the entangled-state manifold carries no dynamical signal and preserves relativistic causality. C2 — Canonical Effective Dimension with Explicit Measure (§2, §4) We introduce a scale-dependent effective Bures dimension: D_{\mathrm{eff}}(\ell) = \lim_{\varepsilon \to 0} \frac{\log V(\varepsilon;\ell)}{\log \varepsilon} where is the volume of an -ball in the entangled-state manifold , measured using the canonical Bures volume measure: d\mu_\ell(\theta) = \sqrt{\det g^{\mathrm{B}}(\theta;\ell)}\; d\theta^1 \wedge \cdots \wedge d\theta^n This measure is: 1. coordinate-independent, 2. invariant under local unitaries, 3. monotone non-increasing under CPTP maps (by QFI monotonicity). Non-integer signals corners, cusps, or self-similar structure at arbitrarily fine scales in state space. C3 — Proposition: Closed-Form Scaling Exponent (§6) Assume: Born approximation, stationary zero-mean Gaussian random potential, spectral density S_V(q) \propto |q|^{-(1+2H)}, \quad H \in (1/2,1) We derive the exact RMS EWS time-delay scaling: \tau_{\mathrm{rms}}(k) \propto k^{-\chi(H)}, \quad \boxed{\chi(H)=\frac{3}{2}+H} Derivation steps: 1. Born approximation for scattering phase , 2. computation of via spectral density, 3. differentiation and identification of dominant term in , 4. extraction of \mathrm{Var}[\tau_{\mathrm{EWS}}] \propto k^{-(3+2H)} Monotonicity: \chi'(H)=1>0 Numerical Predictions H χ(H) Physical regime 0.50 2.00 Brownian / diffusive borderline 0.65 2.15 Conjectured infrared attractor 0.75 2.25 Moderate long-range dependence 0.90 2.40 Strong long-range dependence C4 — Explicit Benchmark Calculations (§5) Depolarizing Channel on Bell State For \rho(p) = (1-p)\,|\Psi^-\rangle\langle\Psi^-| + \frac{p}{4}I\otimes I the Bures metric component: g^{\mathrm{B}}_{pp}(p) = \frac{9}{64(1-\tfrac{3}{4}p)} + \frac{3}{16p} A valid CPTP semigroup: p(\ell)=1-e^{-\gamma \ell} This family is smooth, yielding: D_{\mathrm{eff}}=1 This calibrates the formalism without over-claiming fractality. Werner States Near Entanglement Boundary \rho_W(p) = p|\Psi^-\rangle\langle\Psi^-| + (1-p)\frac{I}{4} Separability boundary (PPT): p^*=\frac{1}{3} Near the pure singlet , the metric diverges as: g^{\mathrm{B}}_{pp} \sim (1-p)^{-1/2} yielding: D_{\mathrm{eff}}\approx0.5 Non-integer dimension arises from metric singularity at boundary of . Falsifiable Conjectures Conjecture 1 — Fractal in Many-Body Critical Systems For CPTP block-spin coarse-graining of a 1D quantum critical ground state (central charge ): D_{\mathrm{eff}}(\ell)\notin\mathbb{Z} \quad \text{for } \ell_0