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arXiv:2411.02825 · Physical review. D/Physical review. D.

Revisiting holographic codes with fractallike boundary erasures

Abhik Bhattacharjee, Joydeep Naskar

Published Jan 14, 2025• 1 citation
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Abstract

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.

DOI
10.1103/physrevd.111.026012

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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.
In this work, we show the robustness of uberholography and its associated quantum error correcting code against the breakdown of entanglement wedge in the presence of highly entropic mixed states in the bulk. We show that for Cantor-setlike erasure in the boundary in AdS3/CFT2{\mathrm{AdS}}_{3}/{\mathrm{CFT}}_{2}AdS3​/CFT2​, the code distance is independent of the mixed-state entropy in the bulk in the $m\ensuremath{\rightarrow}\ensuremath{\infty}$ limit. We also show that for a Sierpinski triangle shaped boundary subregion with fractal boundary erasures in AdS4/CFT3{\mathrm{AdS}}_{4}/{\mathrm{CFT}}_{3}AdS4​/CFT3​, bulk reconstruction is possible in the presence of highly entropic mixed states in the bulk in the large mmm regime.
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.
A bstract We consider holographic entanglement entropy in AdS black hole backgrounds by using the limit of large number of dimensions. By dividing the geometry to two patches (with one patch covering the vicinity of the black hole horizon and another covering the other regions), we are able to obtain fully analytic expressions for the entropy when the entanglement region is a strip. We argue that apart from conformal field theories at finite temperature in high number of dimensions, our method works for nearly critical theories in 3+1 dimensions. In the case of extremal black holes, dual to quantum critical systems, the results take a particularly simple form. We also comment on the case of soliton geometries. Finally, we analyze entanglement entropy for wide strips, and propose a general formula for the first subleading term in the expansion of the entropy in (inverse) system size for generic entanglement regions.
Entanglement entropy of holographic CFTs is expected to play a crucial role in the reconstruction of semiclassical bulk gravity. We consider the entanglement entropy of spherical regions of vacuum, which is known to contain universal contributions. After perturbing the CFT with a relevant scalar operator, also the first order change of this quantity gives a universal term which only depends on a discrete set of basic CFT parameters. We show that in gravity this statement corresponds to the uniqueness of the ghost-free graviton propagator on an AdS background geometry. While the gravitational dynamics in this context contains little information about the structure of the bulk theory, there is a discrete set of dimensionless parameters of the theory which determines the entanglement entropy. We argue that for every (not necessarily holographic) CFT, any reasonable gravity model can be used to compute this particular entanglement entropy. We elucidate how this statement is consistent with AdS/CFT and also give various generalizations. On the one hand this illustrates the remarkable usefulness of geometric concepts for understanding entanglement in general CFTs. On the other hand, it provides hints as to what entanglement data can be expected to provide enough information to distinguish, e.g., bulk theories with different higher curvature couplings.
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.
We study the holographic construction of timelike entanglement entropy (tEE) in black hole backgrounds in Lorentzian geometries. The holographic tEE is realized through extremal surfaces consisting of spacelike and timelike branches that encode its real and imaginary components, respectively. In the BTZ black hole, these surfaces extend into the interior of the black hole and reproduce the field-theoretic results. The analysis is further generalized to higher-dimensional AdS-Schwarzschild black holes, where the characteristics of tEE are obtained with increasing size of the boundary subsystem. Besides, we also show that the boundary subsystem length diverges at a dimension-dependent critical turning point. Notably, this critical point moves closer to the black hole horizon as the dimensionality of the bulk increases. For large subsystem lengths, the finite part of the tEE displays a characteristic volume-plus-area structure, with a real volume term and a complex coefficient of the area term approaching constant values at large dimensions. Besides, we also study the monotonicity of a new quantity, timelike entanglement density, which offers insights into a timelike area theorem in specific limits. Subsequently, we investigate the near-horizon dynamics in various black hole backgrounds, where the spacelike and timelike surfaces exhibit exponential growth of the form e2πβΔte^{\frac{2π}β Δt}eβ2π​Δt with inverse black hole temperature βββ.