Bulk locality and entanglement swapping in AdS/CFT

Kelly, William R.

doi:10.1007/jhep03(2017)153

Cited by 6 publications

(14 citation statements)

References 68 publications

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“…Another interesting research direction is to try to study in detail how some apparent paradoxes raised by the framework of exact entanglement wedge reconstruction are resolved by going to the approximate case. This was done in finite-dimensions by Penington [44] in the case of the information paradox, but for instance, the tension between the lack of additivity of entanglement wedges and the boundary Reeh-Schlieder theorem, first uncovered by Kelly [35] and precisely stated in the operator-algebraic context by Faulkner [16], still seems quite mysterious, and can only be treated in an infinite-dimensional setting. Ultimately, we hope that our approximate statements for infinite-dimensional boundary Hilbert spaces will be a first step towards a fully consistent formulation of the emergence of the bulk from the entanglement structure of the CFT.…”

Section: Discussionmentioning

confidence: 99%

“…In other words, making Newton's constant G N finite breaks the exact quantum error correcting code structure. This is not a mere technicality and it has been pointed out in [29,35,44] that handling approximation is crucial for understanding the semiclassical limit of AdS/CFT and accounts for important physical aspects of the correspondence. In particular,…”

Section: Limitations Of the Exact Approachmentioning

confidence: 99%

“…• The redundancy in the bulk-to-boundary encoding can only be consistent with the Reeh-Schlieder theorem on the boundary if entanglement wedge reconstruction is approximate (with an error that may be nonperturbative in G N ) [35].…”

Section: Limitations Of the Exact Approachmentioning

confidence: 99%

“…There is an important subtlety in studying some crucial features of the emergence of the bulk in AdS/CFT: the quantum extremal surface formula can only be exact approximately, albeit to nonperturbative order in G N [29]. This subtlety is essential for resolving some apparent paradoxes in AdS/CFT, such as aspects of the information paradox [44] and nonadditivity of entanglement wedges [35]. For studying black holes in the G N → 0 limit, we allow the code subspace to have an arbitrarily large dimension.…”

Section: Introductionmentioning

confidence: 99%

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Nonperturbative gravity corrections to bulk reconstruction

Gesteau¹,

Kang²

2021

Preprint

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We introduce a new algebraic framework for understanding nonperturbative gravitational aspects of bulk reconstruction with a finite or infinite-dimensional boundary Hilbert space. We use relative entropy equivalence between bulk and boundary with an inclusion of nonperturbative gravitational errors, which give rise to approximate recovery. We utilize the privacy/correctability correspondence to prove that the reconstruction wedge, the intersection of all entanglement wedges in pure and mixed states, manifestly satisfies bulk reconstruction. We explicitly demonstrate that local operators in the reconstruction wedge of a given boundary region can be recovered in a state-independent way for arbitrarily large code subspaces, up to nonperturbative errors in G N . We further discuss state-dependent recovery beyond the reconstruction wedge and the use of the twirled Petz map as a universal recovery channel. We discuss our setup in the context of quantum islands and the information paradox.

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Section: Discussionmentioning

confidence: 99%

Section: Limitations Of the Exact Approachmentioning

confidence: 99%

Section: Limitations Of the Exact Approachmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Nonperturbative gravity corrections to bulk reconstruction

Gesteau¹,

Kang²

2021

Preprint

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“…This assumption is mild for infinite dimensional algebras, and indeed if the CFT vacuum is in the code-subspace it must be true. So the resolution, as discussed in [21], is to move beyond exact recovery where cyclic and separating vectors lose their power.…”

Section: Introductionmentioning

confidence: 99%

The holographic map as a conditional expectation

Faulkner

2020

Preprint

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We study the holographic map in AdS/CFT, as modeled by a quantum error correcting code with exact complementary recovery. We show that the map is determined by local conditional expectations acting on the operator algebras of the boundary/physical Hilbert space. Several existing results in the literature follow easily from this perspective. The Black Hole area law, and more generally the Ryu-Takayanagi area operator, arises from a central sum of entropies on the relative commutant. These entropies are determined in a state independent way by the conditional expectation. The conditional expectation can also be found via a minimization procedure, similar to the minimization involved in the RT formula. For a local net of algebras associated to connected boundary regions, we show the complementary recovery condition is equivalent to the existence of a standard net of inclusions -an abstraction of the mathematical structure governing QFT superselection sectors given by Longo and Rehren. For a code consisting of algebras associated to two disjoint regions of the boundary theory we impose an extra condition, dubbed dual-additivity, that gives rise to phase transitions between different entanglement wedges. Dual-additive codes naturally give rise to a new split code subspace, and an entropy bound controls which subspace and associated algebra is reconstructable. We also discuss known shortcomings of exact complementary recovery as a model of holography. For example, these codes are not able to accommodate holographic violations of additive for overlapping regions. We comment on how approximate codes can fix these issues.

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Learning the Alpha-bits of black holes

Hayden

Penington

2019

J. High Energ. Phys.

110

221

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When the bulk geometry in AdS/CFT contains a black hole, the boundary reconstruction of a given bulk operator will often necessarily depend on the choice of black hole microstate, an example of state dependence. As a result, whether a given bulk operator can be reconstructed on the boundary at all can depend on whether the black hole is described by a pure state or thermal ensemble. We refine this dichotomy, demonstrating that the same boundary operator can often be used for large subspaces of black hole microstates, corresponding to a constant fraction α of the black hole entropy. In the Schrödinger picture, the boundary subregion encodes the α-bits (a concept from quantum information) of a bulk region containing the black hole and bounded by extremal surfaces. These results have important consequences for the structure of AdS/CFT and for quantum information. Firstly, they imply that the bulk reconstruction is necessarily only approximate and allow us to place non-perturbative lower bounds on the error when doing so. Second, they provide a simple and tractable limit in which the entanglement wedge is state-dependent, but in a highly controlled way. Although the state dependence of operators comes from ordinary quantum error correction, there are clear connections to the Papadodimas-Raju proposal for understanding operators behind black hole horizons. In tensor network toy models of AdS/CFT, we see how state dependence arises from the bulk operator being 'pushed' through the black hole itself. Finally, we show that black holes provide the first 'explicit' examples of capacity-achieving α-bit codes. Unintuitively, Hawking radiation always reveals the α-bits of a black hole as soon as possible. In an appendix, we apply a result from the quantum information literature to prove that entanglement wedge reconstruction can be made exact to all orders in 1/N .

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Bulk locality and entanglement swapping in AdS/CFT

Cited by 6 publications

References 68 publications

Nonperturbative gravity corrections to bulk reconstruction

Nonperturbative gravity corrections to bulk reconstruction

The holographic map as a conditional expectation

Learning the Alpha-bits of black holes

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