Attention: This version is preliminary and contains errors that will be corrected in subsequent publications. Within the FKGP (Fractal Quantum Holographic Paradigm) framework we propose and experimentally verify a scaling law for the second-order Rényi entanglement entropy S2(A) in graph states of varying topological dimension D. The key result is S2(A) = rank_GF2(Gamma) · ln 2, where the GF(2)-rank of the A-B boundary adjacency matrix increases by exactly one at each step of the dimensional ladder. On IBM Heron r2 (ibm_fez) with 65,536 shots per circuit, four steps are confirmed: D=2.00 (ΔSNR=76.9σ), D1=2.05 (34.3σ), D2=e (7.8σ), and D=3 (5.1σ). All detections exceed the 5σ threshold. Cosmological hypotheses are advanced: D1 = 2 + Ω_b (baryon fraction) and D2 = e (informational efficiency). This paper includes a complete reproducibility appendix (Batches 1–10): all QASM circuits, measured P₀ values, noise corrections, and ΔSNR calculations.
Attention: This version is preliminary and contains errors that will be corrected in subsequent publications. Within the FKGP (Fractal Quantum Holographic Paradigm) framework we propose and experimentally verify a scaling law for the second-order Rényi entanglement entropy S2(A) in graph states of varying topological dimension D. The key result is S2(A) = rank_GF2(Gamma) · ln 2, where the GF(2)-rank of the A-B boundary adjacency matrix increases by exactly one at each step of the dimensional ladder. On IBM Heron r2 (ibm_fez) with 65,536 shots per circuit, four steps are confirmed: D=2.00 (ΔSNR=76.9σ), D1=2.05 (34.3σ), D2=e (7.8σ), and D=3 (5.1σ). All detections exceed the 5σ threshold. Cosmological hypotheses are advanced: D1 = 2 + Ω_b (baryon fraction) and D2 = e (informational efficiency). This paper includes a complete reproducibility appendix (Batches 1–10): all QASM circuits, measured P₀ values, noise corrections, and ΔSNR calculations.
Abstract We investigate the scaling of the Rényi entanglement entropies for a particle bipartition of interacting spinless fermions in one spatial dimension. In the Tomonaga–Luttinger liquid regime, we calculate the second Rényi entanglement entropy and show that the leading order finite-size scaling is equal to a universal logarithm of the system size plus a non-universal constant. Higher-order corrections decay as power-laws in the system size with exponents that depend only on the Luttinger parameter. We confirm the universality of our results by investigating the one dimensional t − V model of interacting spinless fermions via exact-diagonalization techniques. The resulting sensitivity of the particle partition entanglement to boundary conditions and statistics supports its utility as a probe of quantum liquids.
Theory of Quantum Entanglement of Particles: Multiscale Graph States, Region Geometry, Quantum Error Correction, and the Effective-Space Hypothesis Abstract This study develops a finite-dimensional, computationally testable framework for examining how multiscale graph structure organizes quantum entanglement and the accessibility of logical information. For the circulant graph family G_N, where N = 2ⁿ, n ≥ 3, and edge offsets are 1, 2, 4, …, N/2, an analytic proof establishes that the binary adjacency matrix across every contiguous block A of length L ≤ N/2 has rank L over F₂. For the associated graph states, this yields S(A) = min(L, N − L). For G₈, the study enumerates 238 proper connected vertex subsets ("tubes") and 32,640 maximal tubings and computes the entanglement entropy of every tube. Explicit bit-flip and phase-dual quantum error-correcting code embeddings are examined under specified Pauli-noise and erasure models, showing that recovery performance depends on the encoding, error model, and decoder class, not on graph-state entropy alone. The dynamical component, M1-Green, models a four-dimensional logical code coupled to eight intermediary bosonic modes and sequential collision-bin output registers. At selected resonantly matched parameter settings and T = 120/K, the computed mutual information I(Ref:R) reaches 3.9414 bits for G₈ and 3.8876 bits for C₈, below the four-bit upper bound. The study also reports a verified finite-step symmetry for C₈ in the tested setup, Petz recovery, and a numerical semidefinite-programming interval of 0.235788–0.245982 for recovery fidelity in a specified 69-dimensional output-space formulation. A one-port Page-like model, a random-unitary scrambling benchmark, and a two-sided mediated channel are treated as distinct models. The accompanying compact archive provides scripts, data tables, figures, and numerical certificates. The effective-space interpretation remains a hypothesis; global recovery optimality and independent experimental validation are not claimed. Keywords: quantum entanglement; graph states; multiscale graphs; circulant graphs; entanglement entropy; graph associahedra; quantum error correction; erasure channels; logical information recovery; collision-bin dynamics; Petz recovery map; semidefinite programming; quantum scrambling; effective geometry.
Внимание: данная версия является предварительной и содержит ошибки, которые будут исправлены в последующих публикациях. В рамках ФКГП (Фрактальной Квантовой Голографической Парадигмы) предложен и экспериментально проверен закон масштабирования энтропии запутанности второго порядка S2(A) в графовых состояниях переменной топологической размерности D. Ключевой результат: S2(A) = rank_{GF2}(Гамма_{AB})·ln 2. На IBMHeron r2 (ibm_fez, 65536 измерений): D=2.00 (76.9σ), D1=2.05 (34.3σ), D2=e (7.8σ), D=3 (5.1σ) — все >5σ. Полное приложение воспроизводимости: все QASM-схемы (батчи 1–10), измеренные P0, коррекция шумов и вычисления ΔSNR.
In quantum spin chains at criticality, two types of scaling for the entanglement entropy exist: one comes from conformal field theory (CFT), and the other is for entanglement support of matrix product state (MPS) approximation. On the other hand, the quantum spin-chain models can be mapped onto two-dimensional (2D) classical ones by the Suzuki-Trotter decomposition. Motivated by the scaling and the mapping, we introduce information entropy for 2D classical spin configurations as well as a spectrum, and examine their basic properties in the Ising and the three-state Potts models on the square lattice. They are defined by the singular values of the reduced density matrix for a Monte Carlo snapshot. We find scaling relations of the entropy compatible with the CFT and the MPS results. Thus, we propose that the entropy is a kind of "holographic" entanglement entropy. At T(c), the spin configuration is fractal, and various sizes of ordered clusters coexist. Then, the singular values automatically decompose the original snapshot into a set of images with different length scales, respectively. This is the origin of the scaling. In contrast to the MPS scaling, long-range spin correlation can be described by only few singular values. Furthermore, the spectrum, which is a set of logarithms of the singular values, also seems to be a holographic entanglement spectrum. We find multiple gaps in the spectrum, and in contrast to the topological phases, the low-lying levels below the gap represent spontaneous symmetry breaking. These contrasts are strong evidence of the dual nature of the holography. Based on these observations, we discuss the amount of information contained in one snapshot.
Entanglement entropy scaling links geometry to quantum correlations and diagnoses quantum criticality. Fractal lattices extend this link to noninteger dimensions, where boundary and bulk dimensions are independent. It remains open which dimension governs entropy scaling and how strongly gap closing can enhance it. Nonlocal occupied-state projectors and the lack of translation symmetry obstruct standard methods. We overcome these obstacles and classify free-fermion entropy scaling with bounds set by the hopping across the subsystem boundary: an upper bound from its rank and the spectral gap, and a lower bound from how much it lowers the ground-state energy. With only negative-energy states filled and this energy lowering proportional to the boundary, the entanglement entropy obeys the boundary law SX=Θ(LXdb) under a uniform gap and satisfies LXdb≲SX≲LXdblnLX when the gap closes no faster than algebraically. Here LX is the subsystem linear size and db the boundary dimension. Half occupation of the chiral zero modes gives a mixed-state entropy SXmix with max{SX,zXln2}⩽SXmix⩽SX+zXln2, where zX is the local zero-mode weight; an extensive zX thus makes SXmix follow the bulk dimension.
Although the leading-order scaling of entanglement entropy is non-universal at a quantum critical point (QCP), sub-leading scaling can contain universal behaviour. Such universal quantities are commonly studied in non-interacting field theories, however it typically requires numerical calculation to access them in interacting theories. In this paper, we use large-scale T = 0 quantum Monte Carlo simulations to examine in detail the second Rényi entropy of entangled regions at the QCP in the transverse-field Ising model in 2 + 1 space–time dimensions—a fixed point for which there is no exact result for the scaling of entanglement entropy. We calculate a universal coefficient of a vertex-induced logarithmic scaling for a polygonal entangled subregion, and compare the result to interacting and non-interacting theories. We also examine the shape-dependence of the Rényi entropy for finite-size toroidal lattices divided into two entangled cylinders by smooth boundaries. Remarkably, we find that the dependence on cylinder length follows a shape-dependent function calculated previously by Stephan et al (2013 New J. Phys. 15 015004) at the QCP corresponding to the 2 + 1 dimensional quantum Lifshitz free scalar field theory. The quality of the fit of our data to this scaling function, as well as the apparent cutoff-independent coefficient that results, presents tantalizing evidence that this function may reflect universal behaviour across these and other very disparate QCPs in 2 + 1 dimensional systems.