Non-Isometric Quantum Error Correction in Gravity Arjun Kar Department of Physics and Astronomy University of British Columbia

2025-05-02 0 0 799.79KB 43 页 10玖币

侵权投诉

Non-Isometric Quantum Error Correction in Gravity

Arjun Kar

Department of Physics and Astronomy, University of British Columbia,

6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada

E-mail: arjunkar@phas.ubc.ca

Abstract: We construct and study an ensemble of non-isometric error correcting codes in a

toy model of an evaporating black hole in two-dimensional dilaton gravity. In the preferred

bases of Euclidean path integral states in the bulk and Hamiltonian eigenstates in the bound-

ary, the encoding map is proportional to a linear transformation with independent complex

Gaussian random entries of zero mean and unit variance. Using measure concentration, we

show that the typical such code is very likely to preserve pairwise inner products in a set S

of states that can be subexponentially large in the microcanonical Hilbert space dimension

of the black hole. The size of this set also serves as an upper limit on the bulk eﬀective ﬁeld

theory Hilbert space dimension. Similar techniques are used to demonstrate the existence

of state-speciﬁc reconstructions of S-preserving code space unitary operators. State-speciﬁc

reconstructions on subspaces exist when they are expected to by entanglement wedge recon-

struction. We comment on relations to complexity theory and the breakdown of bulk eﬀective

ﬁeld theory.

arXiv:2210.13476v1 [hep-th] 24 Oct 2022

Contents

1 Introduction 1

2 Holographic dictionary 4

2.1 Canonical ensemble 5

2.2 End-of-the-world branes 7

2.3 Microcanonical ensemble 10

3 Overlap preservation 13

3.1 Deviation bound 14

3.2 On a relation to complexity theory 18

4 Bulk reconstruction 20

4.1 State-speciﬁc reconstruction 21

4.2 Entanglement wedge reconstruction 23

5 Discussion 27

5.1 The equilibrium basis 28

5.2 Relative subexponential states 29

5.3 Limitations on complexity 30

5.4 Fundamental averaging 32

5.5 Gravitational operators 33

5.6 Breakdowns of bulk eﬀective ﬁeld theory 34

1 Introduction

In holographic formulations of quantum gravity [1,2], semiclassical spacetime emerges from

the quantum information theory of the microscopic degrees of freedom [3,4]. The ﬁrst hint

of this fundamental role played by information theory appears in the study of gravitational

entropy formulas [5–13] which express von Neumann entropies in quantum gravity in terms of

semiclassical quantities like hypersurface areas and the entropy of quantum ﬁelds propagating

on ﬁxed backgrounds.

The arguments [9,14–17] which lead to these formulas often make use of the Euclidean

gravitational path integral, a somewhat mysterious object which is semiclassical at ﬁrst sight

but has been known nearly since its invention [18,19] to contain some amount of information

– 1 –

about the microscopic degrees of freedom in quantum gravity [20].1Understanding precisely

how much microscopic information is accessible via these Euclidean techniques is a very active

area of inquiry [28–30] in light of recent advances concerning the application of some of the

more powerful of the aforementioned entropy formulas to the black hole information paradox

[11–13,15,16,31].2

On their own, such entropy formulas and Euclidean gravity calculations have taught us

much about the structure of quantum information in quantum gravity (see [42–46] for just a

few examples). But perhaps their deepest implication lies in their relationship with quantum

error correcting codes [47–51]. Quantum error correcting codes were originally created to

allow for robust manipulation and transmission of quantum information [52], and their very

existence is somewhat surprising due to the no-cloning principle of quantum mechanics.

As a consequence of the entropy formulas, a quantum error correcting structure was dis-

covered in holography [47] which consists of a linear map Vthat encodes a bulk semiclassical

“code” Hilbert space Hbinto the microscopic “physical” boundary Hilbert space HB. The

basic idea is that semiclassical bulk states and operators on Hbare encoded redundantly by

Vin the microscopic Hilbert space HB, and losing access to some portions of HBdoes not

necessarily obstruct our ability to reconstruct bulk physics in some portions of Hb. The por-

tions in question are determined by the hypersurfaces appearing in the gravitational entropy

formulas we recalled above [45,49], and this phenomenon is sometimes called “entanglement

wedge reconstruction”.

Since its discovery in this fashion, the error correction structure in holography has come

to essentially supersede the holographic gravitational entropy formulas, as these formulas and

their generalizations are now understood as direct consequences of this structure [50,51]. As

such, the study of gravitational entropy is superseded by the study of the three fundamental

objects in the error correction structure: the code space Hb, the physical space HB, and

the encoding map V. As the behavior of gravitational entropy underpins our picture of the

emergent bulk spacetime, understanding the limits of semiclassical bulk physics is intimately

related to how far this error correction structure may be extended [51,53–55].

The choice of Hbhas involved some degree of arbitrariness, and has generally been taken

to be a ﬁnite subspace of the bulk eﬀective ﬁeld theory Hilbert space, just large enough to

contain the perturbative semiclassical dynamics of interest [56]. On the other hand, HBis

almost always taken to be the complete microscopic Hilbert space of the holographic dual

1Lorentzian approaches to gravitational entropy have historically relied upon a string theoretic or holo-

graphic formulation of quantum gravity where the underlying degrees of freedom are explicitly known [21–26].

However, recent progress in algebraic quantum ﬁeld theory has supplied a new perspective on this issue, one

that may be as general and independent of stringy details as the Euclidean path integral [27].

2The generality and apparent utility of these formulas has even led to eﬀorts to reproduce [32–34] and

analyze [35–41] them in gravitational theories which are not known to be holographic.

– 2 –

boundary theory, perhaps restricted to some ﬁxed charge sector or microcanonical energy

window. The interpretation of Vin holography is as the “bulk-to-boundary map”, and in

Euclidean gravity it roughly corresponds to a rule for transforming bulk path integrals into

boundary path integrals.

Using the Euclidean gravitational path integral [16,47–49] or tensor network models of

holography [57,58], it is possible to explicitly deﬁne the encoding map Vand reconstruct

bulk operators acting on the code space Hbby manipulating the physical degrees of freedom

in HB.3In these explicit situations, we will loosely refer to Vas the holographic dictionary,

despite its apparent diﬀerence from the original notion of the holographic dictionary where

(noting the lack of a non-perturbatively deﬁned bulk theory) Vwas instead thought of as

an isomorphism from a bulk string theory Hilbert space to a boundary gauge theory Hilbert

space [2,59,60].

The standard situation in quantum error correction is for the encoding map Vto be an

isometry. This means Vsatisﬁes V†V=Ib, where Ibis the identity on Hb. In some toy

models and many holographic scenarios, the dictionary Vreally is an isometry and leads

to an approximate quantum error correcting code between the bulk and boundary where all

states in Hblook roughly the same from a semiclassical perspective.4

However, extensions of the standard error correction structure are necessary and indeed

suﬃcient to understand more precise notions of information in bulk subregions [53,54], choices

of Hbcontaining states with highly dissimilar bulk entanglement structure [17], and even

certain extreme situations motivated by the interiors of evaporating or old black holes where

Vcan be arbitrarily far from an isometry [51,55,70,71].5Of these extensions to standard

error correction, the non-isometric extension is the least understood and the most relevant

for situations with strong gravitational eﬀects. In the most detailed study to date of this non-

isometric error correction [55], the model under consideration was a tensor network model

similar in spirit to [58], with no obvious connection to gravity.

As was already explained in [55], the tensor network model has several drawbacks as a

model of gravity. For example, Lorentz and diﬀeomorphism invariance are hard to under-

3Other bulk reconstruction techniques [2,59–64] make use of the causal structure of spacetime and provide

only indirect access to V, often without an explicit choice of code space Hb. This subtlety led to several puzzles

concerning the structure of eﬀective ﬁeld theory within holography, and unraveling these issues led directly to

the quantum error correction ideas we have been reviewing [47,56]. Modular theory has also been employed

as a reconstruction technique [65] which may be versatile enough to handle choices of Hbwhich include bulk

subregions that are separated from HBby a horizon [66–69].

4Technically, this means that bulk subregions are encoded in boundary subregions for every state in Hb

in the same manner. This is sometimes called “complementary recovery” in error correction or “subregion

duality” in holography.

5In the non-isometric situation, the map Vmay annihilate some states in Hb. These null states have been

related to a sort of large diﬀeomorphism invariance [72–74].

– 3 –

stand in such models, and the quantity which plays the role of the microcanonical black hole

Hilbert space dimension has no obvious connection to a geometric area as expected by the

Bekenstein-Hawking formula. On the other hand, the Euclidean path integral naturally allows

for manifest diﬀeomorphism invariance, and the entropy of the black hole is clearly related to

the horizon area. Indeed, computations in Euclidean gravity were the original justiﬁcation for

the Bekenstein-Hawking formula itself. In view of these facts, it is necessary to understand

the extent to which the results obtained by tensor network analysis in [55] may be carried

over to a real gravitational theory deﬁned using the Euclidean path integral.

The purpose of this paper is to construct and study the dictionary Vas a highly non-

isometric quantum error correcting code in a toy model of an evaporating black hole in

two-dimensional Euclidean dilaton gravity. We combine the Euclidean gravity techniques

which allow for a resolution of the information paradox [16] with the notion of non-isometric

error correction introduced in [55]. We ﬁnd gravitational analogues of many of the results

of [55], with interesting diﬀerences in details. In particular, our analogue (3.26) of the main

theorem of [55] dealing with changes in semiclassical state overlaps under the dictionary is

strengthened by diﬀerences in measure concentration between the Haar ensemble and the

complex Gaussian ensemble, which enters our gravitational code. This strengthening carries

over to the bulk reconstruction analyses in e.g. (4.5). Moreover, the derivations themselves

are in fact a bit simpler than in [55], which is surprising as gravitational theories are generally

more complicated than Haar random unitary analyses.

Four sections follow. In Section 2, we deﬁne the dictionary Vin two-dimensional dilaton

gravity and brieﬂy review how the model relates to evaporating black holes. In Section 3,

we prove the gravitational analogue of the fundamental theorem of [55], which allows an esti-

mation of how many states in the code Hilbert space may have preserved overlaps under the

dictionary V. In Section 4, we describe the reconstruction of code operators in a necessarily

state-speciﬁc manner and verify consistency of our results with entanglement wedge recon-

struction. We conclude in Section 5with a discussion of relations to complexity, fundamental

averaging, extensions to gravitational bulk operators, and the breakdown of bulk eﬀective

ﬁeld theory.

2 Holographic dictionary

We will deﬁne the holographic dictionary Vusing the Euclidean gravitational path inte-

gral. Our discussion will apply quite generally to asymptotically anti-de Sitter (AdS) two-

dimensional dilaton gravity theories, but for concreteness we will also spell out the details for

a particular theory: Jackiw-Teitelboim (JT) gravity [75,76], which is a theory of the metric

– 4 –

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

Non-IsometricQuantumErrorCorrectioninGravityArjunKarDepartmentofPhysicsandAstronomy,UniversityofBritishColumbia,6224AgriculturalRoad,Vancouver,BCV6T1Z1,CanadaE-mail:arjunkar@phas.ubc.caAbstract:Weconstructandstudyanensembleofnon-isometricerrorcorrectingcodesinatoymodelofanevaporatingblackholeintwo-d...

展开>> 收起<<

Non-Isometric Quantum Error Correction in Gravity Arjun Kar Department of Physics and Astronomy University of British Columbia.pdf

共43页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Non-Isometric Quantum Error Correction in Gravity Arjun Kar Department of Physics and Astronomy University of British Columbia

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: