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arXiv:2607.06870

Phase transitions and uberholography of holographic pure-state geometries

Ning Bao, Keiichiro Furuya, Jacob March

hep-thquant-phSubmitted Jul 8, 2026• Announced Jul 9, 2026• 0 citations
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Abstract

We study the error-correcting properties of pure-state holographic geometries, in which mixed boundary subregions are replaced, via the surface/state correspondence, by the Ryu--Takayanagi (RT) geodesic bounding their entanglement wedges. In AdS$_3$/CFT$_2$ we derive a cross-ratio threshold relation $η'/η= e^{ΔH/2}$ for the connected/disconnected transition of the entanglement wedge when two holes are punched in such a geometry. The quantity $ΔH$ is sourced entirely by geodesics ending on RT boundaries. It shifts the standard two-interval threshold $η= 1/2$, and we classify when its sign is fixed by the pattern of hole endpoints. Turning to code properties, we show that the recursive hole-punching underlying uberholography cannot start within an RT-boundary, while an untouched asymptotic boundary can still fractalize, and we find numerically that in the configurations we study it does so with the universal fractal dimension $α\approx 0.786$. The resulting upper bounds on price and distance are nevertheless procedure dependent. In the configurations we study, punching holes on the asymptotic boundary while retaining the RT-boundary yields strictly tighter bounds than first tracing out the RT-boundary and then fractalizing.

Comments
25 pages, 12 figures

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We study the error-correcting properties of pure-state holographic geometries, in which mixed boundary subregions are replaced, via the surface/state correspondence, by the Ryu--Takayanagi (RT) geodesic bounding their entanglement wedges. In AdS3_33​/CFT2_22​ we derive a cross-ratio threshold relation η′/η=eΔH/2η'/η= e^{ΔH/2}η′/η=eΔH/2 for the connected/disconnected transition of the entanglement wedge when two holes are punched in such a geometry. The quantity ΔHΔHΔH is sourced entirely by geodesics ending on RT boundaries. It shifts the standard two-interval threshold η=1/2η= 1/2η=1/2, and we classify when its sign is fixed by the pattern of hole endpoints. Turning to code properties, we show that the recursive hole-punching underlying uberholography cannot start within an RT-boundary, while an untouched asymptotic boundary can still fractalize, and we find numerically that in the configurations we study it does so with the universal fractal dimension α≈0.786α\approx 0.786α≈0.786. The resulting upper bounds on price and distance are nevertheless procedure dependent. In the configurations we study, punching holes on the asymptotic boundary while retaining the RT-boundary yields strictly tighter bounds than first tracing out the RT-boundary and then fractalizing.
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We explore the fine structure of the holographic entanglement entropy proposal (the Ryu-Takayanagi formula) in AdS3_33​/CFT2_{2}2​. With the guidance from the boundary and bulk modular flows we find a natural slicing of the entanglement wedge with the modular planes, which are co-dimension one bulk surfaces tangent to the modular flow everywhere. This gives an one-to-one correspondence between the points on the boundary interval A\mathcal{A}A and the points on the Ryu-Takayanagi (RT) surface EA\mathcal{E}_{\mathcal{A}}EA​. In the same sense an arbitrary subinterval A2\mathcal{A}_2A2​ of A\mathcal{A}A will correspond to a subinterval E2\mathcal{E}_2E2​ of EA\mathcal{E}_{\mathcal{A}}EA​. We propose that the length of E2\mathcal{E}_2E2​ captures the contribution from A2\mathcal{A}_2A2​ to the entanglement entropy SAS_{\mathcal{A}}SA​, and compare it with the mutual information between A2\mathcal{A}_2A2​ and the complement Ac\mathcal{A}^cAc.
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.
We propose a framework for preparing quantum states with a holographic entanglement structure, in the sense that the entanglement entropies are governed by minimal surfaces in a chosen bulk geometry. We refer to such entropies as holographic because they obey a relation between entropies and bulk minimal surfaces, known as the Ryu-Takayanagi (RT) formula, that is a key feature of holographic models of quantum gravity. Typically in such models, the bulk geometry is determined by solving Einstein's equations. Here, we simply choose a bulk geometry, then discretize the geometry into a coupling graph comprising bulk and boundary nodes. Evolving under this graph of interactions and measuring the bulk nodes leaves behind the desired pure state on the boundary. We numerically demonstrate that the resulting entanglement properties approximately reproduce the predictions of the Ryu-Takayanagi formula in the chosen bulk geometry. We consider graphs associated with hyperbolic disk and wormhole geometries, but the approach is general. We also introduce a procedure for decorating the graph which, when applied to the hyperbolic disk, yields a genuine CFT ground state with power-law correlations but which only partially obeys RT. The minimal ingredients in our proposal involve only Gaussian operations and measurements and are readily implementable in photonic and cold-atom platforms.
In this paper we investigate the code properties of holographic fractal geometries initiated in Pastawski []. We study reconstruction wedges in AdS 3 / CFT 2 for black hole backgrounds, which are in qualitative agreement with the vacuum-AdS approximation using generalized entanglement entropy in Bao []. In higher dimensions, we study reconstruction wedges for the infinite, straight strip in AdS d + 1 / CFT d and clarify the roles of “straight” and “infinite” in their code properties. Lastly, we comment on uberholography from the perspective of complexity transfer and one-shot holography.