Spacetime and Gravity as Emergent Quantum Entanglement - A Proof of Concept: Paper I — The Binary Tree Network, Exact Entropy Formula and AdS₂ Metric Convergence
This paper establishes the geometric and entropic foundations of an exactly solvable holographic model for two-dimensional quantum gravity. Taking the binary tree perfect tensor network — a hierarchical quantum system with N=2LN = 2^L N=2L boundary sites and bond dimension χ\chi χ — as its setting, it proves three main results: First, an exact closed-form formula S(Am)=f∗(m)logχS(A_m) = f^*(m)\log\chi S(Am)=f∗(m)logχ is derived for the entanglement entropy of any boundary interval, where f∗(m)=min(popcount2(m), 1+popcount2(Nmin(m)−m))f^*(m) = \min(\mathrm{popcount}_2(m),\ 1 + \mathrm{popcount}_2(N_{\min}(m)-m)) f∗(m)=min(popcount2(m), 1+popcount2(Nmin(m)−m)) is a combinatorial function of the interval size, establishing the Ryu–Takayanagi formula as a theorem of the network rather than a conjecture. Second, the tree is shown to embed isometrically into the hyperbolic plane H2\mathbb{H}^2 H2 with every edge of equal hyperbolic length η=arccosh(21/16)\eta = \mathrm{arccosh}(21/16) η=arccosh(21/16), and the rescaled boundary metric is proved quasi-isometric to AdS2\mathrm{AdS}_2 AdS2 geodesics with explicit tight constants. Third, Newton's constant G=1/(4logχ)G = 1/(4\log\chi) G=1/(4logχ) and the AdS2\mathrm{AdS}_2 AdS2 radius R=δ/ln2R = \delta/\ln 2 R=δ/ln2 are derived independently from two routes — the RT formula and the embedding geometry respectively — yielding R/G=4log2χR/G = 4\log_2\chi R/G=4log2χ as a proved theorem, with no free parameters once χ\chi χ is specified. The paper additionally derives the exact modular Hamiltonian of the vacuum state from the perfect tensor isometry, and recovers the bulk stress-energy tensor from boundary entropy perturbations via an explicit inversion formula, completing the chain from a discrete quantum network to a bulk metric with gravitational coupling and matter content.
This paper establishes the geometric and entropic foundations of an exactly solvable holographic model for two-dimensional quantum gravity. Taking the binary tree perfect tensor network — a hierarchical quantum system with N=2LN = 2^L N=2L boundary sites and bond dimension χ\chi χ — as its setting, it proves three main results: First, an exact closed-form formula S(Am)=f∗(m)logχS(A_m) = f^(m)\log\chi S(Am)=f∗(m)logχ is derived for the entanglement entropy of any boundary interval, where f∗(m)=min(popcount2(m), 1+popcount2(Nmin(m)−m))f^(m) = \min(\mathrm{popcount}_2(m),\ 1 + \mathrm{popcount}2(N{\min}(m)-m)) f∗(m)=min(popcount2(m), 1+popcount2(Nmin(m)−m)) is a combinatorial function of the interval size, establishing the Ryu–Takayanagi formula as a theorem of the network rather than a conjecture. Second, the tree is shown to embed isometrically into the hyperbolic plane H2\mathbb{H}^2 H2 with every edge of equal hyperbolic length η=arccosh(21/16)\eta = \mathrm{arccosh}(21/16) η=arccosh(21/16), and the rescaled boundary metric is proved quasi-isometric to AdS2\mathrm{AdS}_2 AdS2 geodesics with explicit tight constants. Third, Newton's constant G=1/(4logχ)G = 1/(4\log\chi) G=1/(4logχ) and the AdS2\mathrm{AdS}_2 AdS2 radius R=δ/ln2R = \delta/\ln 2 R=δ/ln2 are derived independently from two routes — the RT formula and the embedding geometry respectively — yielding R/G=4log2χR/G = 4\log_2\chi R/G=4log2χ as a proved theorem, with no free parameters once χ\chi χ is specified. The paper additionally derives the exact modular Hamiltonian of the vacuum state from the perfect tensor isometry, and recovers the bulk stress-energy tensor from boundary entropy perturbations via an explicit inversion formula, completing the chain from a discrete quantum network to a bulk metric with gravitational coupling and matter content.
This is the thrid paper of the series of three: Spacetime and Gravity as Emergent Quantum Entanglement — A Proof of Concept The paper closes the three principal open questions of the series by identifying the physical boundary Hamiltonian, fixing all free thermodynamic parameters, and establishing that the boundary theory is maximally chaotic. Four results are proved: Theorem A identifies the boundary Hamiltonian as the Sachdev–Ye–Kitaev model: an explicit Collins–Śniady Weingarten calculation applied to the binary tree encoder's unitary 2-design property shows that the tree ensemble average and the SYK disorder average produce identical Gaussian coupling variance at leading order in 1/χ, yielding the same Schwinger–Dyson equations and the same conformal Green function G(τ) = −b·J⁻¹/²·sgn(τ)/|τ|¹/², establishing H_R = H_SYK as an equality of disorder-averaged effective field theories. Theorem B derives the previously free Schwarzian coefficient and topological dilaton — φ_r = (πb²N/16J)·log(J/Δ) and φ_0 = N/(2log₂χ) — making the near-extremal black hole entropy a parameter-free prediction. Theorem C proves that the quantum Lyapunov exponent saturates the Maldacena–Shenker–Stanford chaos bound at λ_L = 2πT, with scrambling time t* ~ β log N, placing the boundary theory in the fast-scrambling and GUE random-matrix universality class. Theorem D confirms the boundary stress-tensor two-point function ⟨T(x)T(0)⟩ = 6log₂χ/(2x²) as an unconditional theorem, consistent with an effective central charge c_eff = 6log₂χ. A corollary shows that the Bisognano–Wichmann assumption of Paper II is unnecessary once the explicit boundary Hamiltonian is in hand. Together with Papers I and II, this paper completes the derivation of the full holographic dictionary for the binary tree model — from a discrete quantum network to emergent geometry, physical time, gravitational field equations, and quantum chaos — entirely from first principles.
This paper constructs an exactly solvable holographic model of de Sitter quantum gravity from the {4,3,3} tessellation of S³. Eight cubes tile the three-sphere; placing a perfect tensor at each cell, contracting across shared faces, and designating one cell as the observer produces a holographic model in 3+1 dimensions. Every result is exact: the Ryu-Takayanagi entropy formula holds with zero bulk entanglement, the de Sitter entropy is S = 6 log χ, the Hilbert space dimension is χ⁶ (confirming Banks's finite-dimensionality conjecture as a theorem), the reduced state is maximally mixed (reproducing the Gibbons-Hawking temperature), the Page curve is reproduced exactly, and the observer algebra is the full matrix algebra Mat(χ⁶) with a unique trace state. The proofs use only the perfect tensor property and the graph structure of the tessellation. No approximation, no large-N limit, and no holographic dictionary beyond the tensor network itself is required.
Holographic tensor networks serve as toy models for the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence, capturing many of its essential features in a concrete manner. However, existing holographic tensor network models remain far from a complete theory of quantum gravity. A key obstacle is their discrete structure, which only approximates the semi-classical geometry of gravity in a qualitative sense. In \cite{Lin:2024dho}, it was shown that a network of partial-entanglement-entropy (PEE) threads, which are bulk geodesics with a specific density distribution, generates a perfect tessellation of AdS space. Moreover, such PEE-network tessellations can be constructed for more highly symmetric geometries using the Crofton formula. In this paper, we assign a quantum state to each vertex in the PEE network and develop several holographic tensor network models: (1) the factorized PEE tensor network, which takes the form of a tensor product of EPR pairs; (2) the HaPPY-like PEE tensor network constructed from perfect tensors; and (3) the random PEE tensor network. In all these models, we reproduce the exact Ryu-Takayanagi formula by showing that the minimal number of cuts along a homologous surface in the network exactly equals the area of that surface.
The relation between loop quantum gravity (LQG) and tensor networks is explored from the perspectives of the bulk-boundary duality and holographic entanglement entropy. We find that the LQG spin-network states in a space $\mathrm{\ensuremath{\Sigma}}$ with boundary $\ensuremath{\partial}\mathrm{\ensuremath{\Sigma}}$ is an exact holographic mapping similar to the proposal in [X.-L. Qi, Exact holographic mapping and emergent space-time geometry, arXiv:1309.6282]. The tensor network, understood as the boundary quantum state, is the output of the exact holographic mapping emerging from a coarse-graining procedure of spin networks. Furthermore, when a region A and its complement A are specified on the boundary $\ensuremath{\partial}\mathrm{\ensuremath{\Sigma}}$, we show that the boundary entanglement entropy S(A) of the emergent tensor network satisfies the Ryu-Takayanagi formula in the semiclassical regime, i.e., S(A) is proportional to the minimal area of the bulk surface attached to the boundary of A in $\ensuremath{\partial}\mathrm{\ensuremath{\Sigma}}$.
Update from prior version. Abstract The Quantum Family Tree (QFT) theory proposes that spatial distance emerges as a measure of kinship difference within a quantum genealogical tree: sites that share a recent common ancestor are near, while sites whose lineages diverge deep in the tree are far apart. In this framework, geometry is not fundamental but inherited – the entanglement structure of a randomly branching quantum process encodes a notion of distance that reproduces, in a precise quantitative sense, the entropy–area relation of holographic gravity. We realize this proposal in a concrete solvable model – a binary Bruhat–Tits tree whose bonds carry independent Haar-random U(4) gates. For an interval A of size 2^k boundary sites, the quenched von Neumann entropy S_VN(A) grows linearly with k in the deep-tree limit. We measure the slope c_VN and find c_VN = m² × c_RT, m² = 3/10 where c_RT = 3 log₂(5/2) is the classical mutual-information (Ryu–Takayanagi-like) coefficient. The measurement uses six independent tree depths (D = 20–150) and yields c_VN = 1.18845 ± 0.00116 bits confirming the ratio β₁ = c_VN/(3c_RT) = 1/10 within 1.1σ, and strongly disfavoring the nearest alternative β₁ = 10/99. We prove: (i) η₂ = 2/5 exactly via Weingarten calculus; (ii) a rank-2 bound on ρ_A for aligned intervals within the QFT construction; (iii) the first-law identity δ⟨H_A⟩/δS = 1 for k=1; and (iv) an exact 2-component purity recursion in the {SWAP, TrTr} basis that captures the full position-averaged Rényi-2 structure of the boundary state without Monte Carlo. The purity kernel at each tree node decomposes as M = b·SWAP + c·TrTr, reducing the 16-dimensional recursion to two components with exact rational coefficients. The carry eigenvalue η₂ = 2/5 is proved to be q-independent (verified at q = 2, 3, 4 with multiplicities C(q,2)), and the straddle algebra at O(ε) near q = 1 is additive rather than bilinear. These two facts – q-independence of dynamics and O(ε) additivity – imply that the entire proof of c_VN = m²·c_RT reduces to a single remaining representation-theoretic computation: the sector weight w₂ governing how the SWAP sector contributes to von Neumann entropy at q → 1. We predict w₂ = (1 − η₂)/2 = 3/10 = m². We identify the Choi-boundary state as a two-chain Matrix Product Density Operator (MPDO) with bond dimension 2 and transfer matrix spectrum {1, η₂ = 2/5}. We reduce the entropy-rate problem exactly to a single scalar quantity – the sector weight w₂ – eliminating the need for large-depth recursion or asymptotic analysis. This model reproduces both the von Neumann and Rényi-2 entropies across all tested scales and depths to sub-percent accuracy with no adjustable parameters beyond η₂ fixed by Weingarten calculus. The quenched Rényi-2 rate is consistent with the exact rational value h_S2 = 14/45 (0.07σ) and the annealed Rényi-2 rate with 8/27, with Jensen gap ratio 21/20 exactly.
In this Dissertation, different aspects of holographic entanglement entropy are explored. In quantum information theory, entanglement is a useful resource needed to perform quantum operations. Entanglement entropy quantifies this resource and even if computable in quantum field theories, it is quite hard to calculate explicitly. However, in the context of gauge/gravity duality, (holographic) entanglement entropy was proposed by Ryu and Takayanagi (RT) to be captured by the area of a minimal surface in Anti DeSitter (AdS) space, which is extremely simple to compute. We begin by using the holographic dictionary to derive the RT formula. This is done by first considering smooth bulk geometries which are dual to n copies of the boundary geometry glued in a particular way. The action of these geometries computes the Renyi entropies S_n for any integer n. We give a prescription to analytically continue the action to non-integer n and argue that, as n goes to 1,S_1, the entanglement entropy, is given by the area of a minimal surface, reproducing the RT conjecture. That is, we reduced the RT proposal to the equality between bulk and boundary partition functions, a standard entry in the dictionary. We then generalize the previous formalism to account for corrections due to bulk quantum fields (1/N corrections in the boundary). This allows us to derive a new formula: boundary entanglement entropy is given by the area of the minimal surface plus bulk entanglement entropy. This extends RT beyond the planar limit and we present several predictions of the proposal. Our exploration of holographic entanglement entropy continues by considering different states: by comparing the quantum corrected entropy for nearby states we obtain the modular hamiltonian operator in bulk perturbation theory. From this expression of the modular hamiltonian, we derive that relative entropy is bulk relative entropy and that the modular flow is the bulk modular flow.