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Holographic Spacetime from Quantum Feedback and Topological Averaging: Quadratic area response, RT phase transitions, and numerical predictions

Alex De Giuseppe

Published Mar 16, 2026• 0 citations
Publisher

Abstract

This manuscript develops a unified theoretical framework for entanglement response in holographic quantum gravity, structured around three interconnected pillars. **1. Ancilla-mediated feedback protocol and exact quadratic formula.**We construct a completely positive, trace-preserving (CPTP) channel in AdS₃/CFT₂ using an external ancilla. The ancilla weakly measures the integrated energy in a boundary region A and, via a classical feedback signal, applies a conditional unitary generated by the area operator of the Ryu–Takayanagi surface associated with a complementary region B. The protocol respects standard quantum mechanics and bulk causality, eliminating any concern about nonlinearity or superluminal signalling. For small energy injections δE, the linear response vanishes by modular symmetry; the leading quadratic contribution yields an explicit, parameter‑free formula for the variation of entanglement entropy: δS_B = (8π R_B² / L_A) [ a / (R_B² (R_B² – a²)) + (1/(2 R_B³)) arctan(a/R_B) ] (δE)². The formula depends only on the geometry of the two boundary intervals. When the regions are tuned to a critical configuration where the RT surface is tangent to the bulk null cone of A, the quadratic coefficient diverges, signalling a first‑order geometric phase transition governed by a Landau–Ginzburg normal form (cusp catastrophe). Three falsifiable numerical predictions for MERA tensor‑network simulations are provided, along with a concrete quantum circuit proposal for NISQ devices. The protocol also injects magic (non‑stabilizerness) into the boundary state, with a susceptibility that diverges at the critical point. **2. Angular non‑injectivity, topological averaging, and UV regularization.**We show that massive particles on bound orbits in AdS₃ possess an angular coordinate φ(τ) that is non‑injective: it intersects any constant‑angle hypersurface in N > 1 distinct points. The intersection multiplicity is derived from first principles and scales as N(ε) ∼ ε^{-(d-2)} in terms of the UV cutoff ε. Averaging the RT entropy over the N sheets generated by this multiplicity cancels the leading UV divergence exactly, without external counterterms. A no‑go argument demonstrates that topological averaging is the unique mechanism achieving this cancellation while respecting locality, causality, and the area law. The same averaging mechanism reduces the bare vacuum energy to the observed cosmological constant via Λ_obs = Λ_bare / N, with a refined estimate based on Standard Model degrees of freedom, and regularises the energy density near classical spacetime singularities (Big Bang and black hole). **3. TPST Master Equation and Observer–Geometry Identity.**Assembling the feedback protocol and the topological averaging yields an effective Master Equation for the bulk metric, derived from an effective action (Appendix E) and verified in all known limits (Einstein gravity, linearised response, Newtonian limit, entanglement first law): G_μν + Λ[ρ*] g_μν = 8π G_N T_μν + R_μν, where Λ[ρ*] = 4π G_N λ² ⟨T_{00}⟩_A[ρ*] is a dynamical cosmological constant depending on the observer's quantum state, and R_μν is a source term localised on the RT surface, proportional to the metric perturbation and to the geometric kernel K(a,R_B) derived from the quadratic response. The coupling λ = 2/√L is fixed uniquely by the Brown–Henneaux relation, leaving no free parameters. A Proposition establishes the uniqueness of this equation under assumptions of diffeomorphism invariance, the entanglement first law, and locality of the feedback on the RT surface. At the self‑consistent fixed point ρ*, the equation implies the Observer–Geometry Identity: ρ* = G[ρ*] = O[ρ*], which states that the boundary quantum state, the bulk geometry it generates, and the observer that probes it are three representations of the same entity. This identity is promoted to a variational principle that actively generates the gravitational dynamics. The core results (CPTP protocol, quadratic formula, phase transition, MERA predictions, no‑go argument, magic injection) are derived with full rigour within standard AdS/CFT and quantum information theory. The broader implications (Master Equation, cosmological constant, singularity resolution) are presented as a motivated research programme. All technical details, including the ancilla evolution, double‑commutator evaluation, Landau–Ginzburg derivation, orbital multiplicity calculation, effective action, and Newtonian limit, are provided in six appendices. This manuscript is current in Official Peer Review. Not final version.Copyright©2026 Alex De Giuseppe.All rights reserved. This work is protected by copyright. Any form of plagiarism, unauthorized reproduction, or misappropriation of ideas, mathematically results, or text without proper citation constitutes a violation of academic and intellectual property standards and common laws. No commercial use, adaptation, or derivative works are permitted without explicit written permission from the author. For correspondence, citations, collaboration inquiries, or feedback please contact:degiuseppealex@gmail.com The hash files that determine ownership have been created

DOI
10.5281/zenodo.20546179

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This manuscript develops a unified theoretical framework for entanglement response in holographic quantum gravity, structured around three interconnected pillars. **1. Ancilla-mediated feedback protocol and exact quadratic formula.**We construct a completely positive, trace-preserving (CPTP) channel in AdS₃/CFT₂ using an external ancilla. The ancilla weakly measures the integrated energy in a boundary region A and, via a classical feedback signal, applies a conditional unitary generated by the area operator of the Ryu–Takayanagi surface associated with a complementary region B. The protocol respects standard quantum mechanics and bulk causality, eliminating any concern about nonlinearity or superluminal signalling. For small energy injections δE, the linear response vanishes by modular symmetry; the leading quadratic contribution yields an explicit, parameter‑free formula for the variation of entanglement entropy: δS_B = (8π R_B² / L_A) [ a / (R_B² (R_B² – a²)) + (1/(2 R_B³)) arctan(a/R_B) ] (δE)². The formula depends only on the geometry of the two boundary intervals. When the regions are tuned to a critical configuration where the RT surface is tangent to the bulk null cone of A, the quadratic coefficient diverges, signalling a first‑order geometric phase transition governed by a Landau–Ginzburg normal form (cusp catastrophe). Three falsifiable numerical predictions for MERA tensor‑network simulations are provided, along with a concrete quantum circuit proposal for NISQ devices. The protocol also injects magic (non‑stabilizerness) into the boundary state, with a susceptibility that diverges at the critical point. *2. Angular non‑injectivity, topological averaging, and UV regularization.We show that massive particles on bound orbits in AdS₃ possess an angular coordinate φ(τ) that is non‑injective: it intersects any constant‑angle hypersurface in N > 1 distinct points. The intersection multiplicity is derived from first principles and scales as N(ε) ∼ ε^{-(d-2)} in terms of the UV cutoff ε. Averaging the RT entropy over the N sheets generated by this multiplicity cancels the leading UV divergence exactly, without external counterterms. A no‑go argument demonstrates that topological averaging is the unique mechanism achieving this cancellation while respecting locality, causality, and the area law. The same averaging mechanism reduces the bare vacuum energy to the observed cosmological constant via Λ_obs = Λ_bare / N, with a refined estimate based on Standard Model degrees of freedom, and regularises the energy density near classical spacetime singularities (Big Bang and black hole). 3. TPST Master Equation and Observer–Geometry Identity.Assembling the feedback protocol and the topological averaging yields an effective Master Equation for the bulk metric, derived from an effective action (Appendix E) and verified in all known limits (Einstein gravity, linearised response, Newtonian limit, entanglement first law): G_μν + Λ[ρ] g_μν = 8π G_N T_μν + R_μν, where Λ[ρ] = 4π G_N λ² ⟨T_{00}⟩_A[ρ] is a dynamical cosmological constant depending on the observer's quantum state, and R_μν is a source term localised on the RT surface, proportional to the metric perturbation and to the geometric kernel K(a,R_B) derived from the quadratic response. The coupling λ = 2/√L is fixed uniquely by the Brown–Henneaux relation, leaving no free parameters. A Proposition establishes the uniqueness of this equation under assumptions of diffeomorphism invariance, the entanglement first law, and locality of the feedback on the RT surface. At the self‑consistent fixed point ρ, the equation implies the Observer–Geometry Identity: ρ = G[ρ] = O[ρ], which states that the boundary quantum state, the bulk geometry it generates, and the observer that probes it are three representations of the same entity. This identity is promoted to a variational principle that actively generates the gravitational dynamics. The core results (CPTP protocol, quadratic formula, phase transition, MERA predictions, no‑go argument, magic injection) are derived with full rigour within standard AdS/CFT and quantum information theory. The broader implications (Master Equation, cosmological constant, singularity resolution) are presented as a motivated research programme. All technical details, including the ancilla evolution, double‑commutator evaluation, Landau–Ginzburg derivation, orbital multiplicity calculation, effective action, and Newtonian limit, are provided in six appendices. This manuscript is current in Official Peer Review. Not final version.Copyright©2026 Alex De Giuseppe.All rights reserved. This work is protected by copyright. Any form of plagiarism, unauthorized reproduction, or misappropriation of ideas, mathematically results, or text without proper citation constitutes a violation of academic and intellectual property standards and common laws. No commercial use, adaptation, or derivative works are permitted without explicit written permission from the author. For correspondence, citations, collaboration inquiries, or feedback please contact:degiuseppealex@gmail.com The hash files that determine ownership have been created
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.
In 2006, Ryu and Takayanagi (RT) pointed out that (with a suitable cutoff) the entanglement entropy between two complementary regions of an equal-time surface of a d+1-dimensional conformal field theory on the conformal boundary of AdS_{d+2} is, when the AdS radius is appropriately related to the parameters of the CFT, equal to 1/4G times the area of the d-dimensional minimal surface in the AdS bulk which has the junction of those complementary regions as its boundary, where G is the bulk Newton constant. We point out here that the RT-equality implies that, in the quantum theory on the bulk AdS background which is related to the boundary CFT according to Rehren's 1999 algebraic holography theorem, the entanglement entropy between two complementary bulk Rehren wedges is equal to 1/4G times the (suitably cut off) area of their shared ridge. (This follows because of the geometrical fact that, for complementary ball-shaped regions, the RT minimal surface is precisely the shared ridge of the complementary bulk Rehren wedges which correspond, under Rehren's bulk-wedge to boundary double-cone bijection, to the complementary boundary double-cones whose bases are the RT complementary balls.) This is consistent with the Bianchi-Meyers conjecture -- that, in a theory of quantum gravity, the entanglement entropy, S, between the degrees of freedom of a given region with those of its complement is S = A/4G (+ lower order terms) -- but only if the phrase 'degrees of freedom' is replaced by 'matter degrees of freedom'. It also supports related previous arguments of the author -- consistent with the author's 'matter-gravity entanglement hypothesis' -- that the AdS/CFT correspondence is actually only a bijection between just the matter (i.e. non-gravity) sector operators of the bulk and the boundary CFT operators.
Recent work showed holographic error correcting codes to have simple universal features at O(1/G). In particular, states of fixed Ryu-Takayanagi (RT) area in such codes are associated with flat entanglement spectra indicating maximal entanglement between appropriate subspaces. We extend such results to one-loop order (O(1) corrections) by controlling both higher-derivative corrections to the bulk effective action and dynamical quantum fluctuations below the cutoff. This result clarifies the relation between the bulk path integral and the quantum code, and implies that i) simple tensor network models of holography continue to match the behavior of holographic CFTs beyond leading order in G, ii) the relation between bulk and boundary modular Hamiltonians derived by Jafferis, Lewkowycz, Maldacena, and Suh holds as an operator equation on the code subspace and not just in code-subspace expectation values, and iii) the code subspace is invariant under an appropriate notion of modular flow. A final corollary requires interesting cancelations to occur in the bulk renormalization-group flow of holographic quantum codes. Intermediate technical results include showing the Lewkowycz-Maldacena computation of RT entropy to take the form of a Hamilton-Jacobi variation of the action with respect to boundary conditions, corresponding results for higher-derivative actions, and generalizations to allow RT surfaces with finite conical angles.
Tensor networks provide a natural framework for exploring holographic duality because they obey entanglement area laws. They have been used to construct explicit toy models realizing many of the interesting structural features of the AdS/CFT correspondence, including the non-uniqueness of bulk operator reconstruction in the boundary theory. In this article, we explore the holographic properties of networks of random tensors. We find that our models naturally incorporate many features that are analogous to those of the AdS/CFT correspondence. When the bond dimension of the tensors is large, we show that the entanglement entropy of all boundary regions, whether connected or not, obey the Ryu-Takayanagi entropy formula, a fact closely related to known properties of the multipartite entanglement of assistance. We also discuss the behavior of Rényi entropies in our models and contrast it with AdS/CFT. Moreover, we find that each boundary region faithfully encodes the physics of the entire bulk entanglement wedge, i.e., the bulk region enclosed by the boundary region and the minimal surface. Our method is to interpret the average over random tensors as the partition function of a classical ferromagnetic Ising model, so that the minimal surfaces of Ryu-Takayanagi appear as domain walls. Upon including the analog of a bulk field, we find that our model reproduces the expected corrections to the Ryu-Takayanagi formula: the bulk minimal surface is displaced and the entropy is augmented by the entanglement of the bulk field. Increasing the entanglement of the bulk field ultimately changes the minimal surface behavior topologically, in a way similar to the effect of creating a black hole. Extrapolating bulk correlation functions to the boundary permits the calculation of the scaling dimensions of boundary operators, which exhibit a large gap between a small number of low-dimension operators and the rest. While we are primarily motivated by the AdS/CFT duality, the main results of the article define a more general form of bulk-boundary correspondence which could be useful for extending holography to other spacetimes.
Abstract Recent work showed holographic error correcting codes to have simple universal features at O(1/G). In particular, states of fixed Ryu-Takayanagi (RT) area in such codes are associated with flat entanglement spectra indicating maximal entanglement between appropriate subspaces. We extend such results to one-loop order (O(1) corrections) by controlling both higher-derivative corrections to the bulk effective action and dynamical quantum fluctuations below the cutoff. This result clarifies the relation between the bulk path integral and the quantum code, and implies that i) simple tensor network models of holography continue to match the behavior of holographic CFTs beyond leading order in G, ii) the relation between bulk and boundary modular Hamiltonians derived by Jafferis, Lewkowycz, Maldacena, and Suh holds as an operator equation on the code subspace and not just in code-subspace expectation values, and iii) the code subspace is invariant under an appropriate notion of modular flow. A final corollary requires interesting cancelations to occur in the bulk renormalization-group flow of holographic quantum codes. Intermediate technical results include showing the Lewkowycz-Maldacena computation of RT entropy to take the form of a Hamilton-Jacobi variation of the action with respect to boundary conditions, corresponding results for higher-derivative actions, and generalizations to allow RT surfaces with finite conical angles.
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)log⁡χS(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/(4log⁡χ)G = 1/(4\log\chi) G=1/(4logχ) and the AdS2\mathrm{AdS}_2 AdS2 radius R=δ/ln⁡2R = \delta/\ln 2 R=δ/ln2 are derived independently from two routes — the RT formula and the embedding geometry respectively — yielding R/G=4log⁡2χR/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.