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arXiv:2209.15026 · Physical review. D/Physical review. D.

Reconstruction wedges in AdS/CFT with boundary fractallike structures

Ning Bao, Joydeep Naskar

Published Mar 17, 2023• 3 citations
arXivHTMLPDF

Abstract

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 ${\mathrm{AdS}}_{3}/{\mathrm{CFT}}_{2}$, 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 ${\mathrm{AdS}}_{4}/{\mathrm{CFT}}_{3}$, bulk reconstruction is possible in the presence of highly entropic mixed states in the bulk in the large $m$ regime.

DOI
10.1103/physrevd.107.066014

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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→\ensuremath∞m\ensuremath{\rightarrow}\ensuremath{\infty}m\ensuremath→\ensuremath∞ 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.
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.
Bulk reconstruction is a central problem in AdS/CFT, and entanglement wedge reconstruction is its subregion version. We argue that this subregion statement should be separated from the stronger holographic quantum error correction interpretation, in which one region-independent logical bulk operator has code-preserving representatives in several boundary regions. A simple locality argument shows that such a common reconstruction must commute with the code-preserving local algebras in the complementary regions. This is the mechanism realized in HaPPY-type codes: the erased regions are blind to a protected logical algebra. An ordinary finite NNN holographic CFT does not have such a protected invisible sector for supergravity fields. Its low-energy local observables, in particular, suitably smeared stress tensors, detect the physical support and gravitational dressing of ordinary bulk operators, up to possible center or superselection data. Thus, there is no such holographic quantum error correction and the N=∞N=\inftyN=∞ agreement of global and subregion HKLL formulae is a free-theory statement. What remains is entanglement wedge reconstruction without holographic quantum error correction, or subregion complementarity: each boundary region has its own code-preserving low-energy algebra and its own region-adapted bulk interpretation, rather than a shared logical operator.
In the context of quantum theories of spacetime, one overarching question is how quantum information in the bulk spacetime is encoded holographically in boundary degrees of freedom.It is particularly interesting to understand the correspondence between bulk subregions and boundary subregions in order to address the emergence of locality in the bulk quantum spacetime.For the AdS/CFT correspondence, it is known that this bulk information is encoded redundantly on the boundary in the form of an error-correcting code.Having access only to a subregion of the boundary is as if part of the holographic code has been damaged by noise and rendered inaccessible.In quantum-information science, the problem of recovering information from a damaged code is addressed by the theory of universal recovery channels.We apply and extend this theory to address the problem of relating bulk and boundary subregions in AdS/CFT, focusing on a conjecture known as entanglement wedge reconstruction.Existing work relies on the exact equivalence between bulk and boundary relative entropies, but these are only approximately equal in bulk effective field theory, and in similar situations it is known that predictions from exact entropic equalities can be qualitatively incorrect.We show that the framework of universal recovery channels provides a robust demonstration of the entanglement wedge reconstruction conjecture as well as new physical insights.Most notably, we find that a bulk operator acting in a given boundary region's entanglement wedge can be expressed as the response of the boundary region's modular Hamiltonian to a perturbation of the bulk state in the direction of the bulk operator.This formula can be interpreted as a noncommutative version of Bayes's rule that attempts to undo the noise induced by restricting to only a portion of the boundary.To reach these conclusions, we extend the theory of universal recovery channels to finite-dimensional operator algebras and demonstrate that recovery channels approximately preserve the multiplicative structure of the operator algebra.
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.
In this note we prove a simple theorem in quantum information theory, which implies that bulk operators in the Anti-de Sitter / Conformal Field Theory (AdS/CFT) correspondence can be reconstructed as CFT operators in a spatial subregion AAA, provided that they lie in its entanglement wedge. This is an improvement on existing reconstruction methods, which have at most succeeded in the smaller causal wedge. The proof is a combination of the recent work of Jafferis, Lewkowycz, Maldacena, and Suh on the quantum relative entropy of a CFT subregion with earlier ideas interpreting the correspondence as a quantum error correcting code.
The AdS/CFT correspondence is an explicit realization of the holographic principle relating a theory of gravity in a volume of space to a lower dimensional quantum field theory on its boundary. By exploiting elements of quantum error correction, qubit toy models of this correspondence have been constructed for which the bulk logical operators are representable by operators acting on the boundary. Given a boundary subregion, wedges in the volume space are used to enclose the bulk qubits for which logical operators are reconstructable on that boundary subregion. In this thesis a number of different wedges, such as the causal wedge, greedy entanglement wedge and minimum entanglement wedge, are examined. More specifically, Monte-Carlo simulations of boundary erasure are performed with various toy models to study the differences between wedges and the effect on these wedge by the type of the model, non-uniform boundaries and stacking of models. It has been found that the minimum entanglement wedge is the best approximate for the true geometric wedge. This is illustrated by an example toy model for which an operator beyond the greedy entanglement wedge was also reconstructed. In addition, by calculating the entropy of these subregions, the viability of a mutual information wedge is rejected. Only for particular connected boundary subregions was the inclusion of the central tensor by the geometric wedge associated to a rise in mutual information.