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Algebraic approach to spacetime bulk reconstruction

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Pith's one-line read This paper proves that exact bulk reconstruction in AdS/CFT is the same as matching Connes cocycle flow between bulk and boundary algebras and as preserving relative entropies, and it uses this equivalence to establish subregion-subregion…

desk verdict Solid general recovery theorems with a genuinely new cocycle-flow characterization; the AdS subregion-duality claim is a promising but under-specified application that needs a fleshed-out Section 4.3. read the letter →

arxiv 2412.00298 v1 pith:WIHIILK3 submitted 2024-11-30 math.OA hep-thmath-phmath.FAmath.MP

Jason Crann , Monica Jinwoo Kang This is my paper · ORCID
classification math.OAhep-thmath-phmath.FAmath.MP MSC 46L1046L5581T4081P4581T20
keywords AdS/CFTcorrespondencebulkreconstructionvonNeumannalgebrasConnescocycleflowrelativeentropycomplementaryrecoverysubregion-subregiondualityKlein-Gordonfields
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes an operator-algebraic criterion for when a bulk region of anti-de Sitter space can be reconstructed from a boundary region: exact bulk recovery is equivalent to the bulk and boundary modular flows agreeing on every state through the Connes cocycle derivatives, and also equivalent to the equality of bulk and boundary relative entropies. The equivalence is proved for general von Neumann algebras with a cyclic-separating vacuum vector, and then applied to Klein-Gordon fields on the universal cover of AdS, where boundary causal diamonds and bulk causal wedges are shown to satisfy complementary recovery. A companion approximate statement ties approximate bulk recovery to the large-N vanishing of the cocycle-intertwining error, with full equivalence in the hyperfinite case. The result supplies an infinite-dimensional, field-theoretic foundation for holographic quantum error correction and identifies the kink transform with bulk cocycle flow in exact recovery settings, with boundary modular flow inducing bulk geometry and a notion of time.

What carries the argument

The load-bearing object is the Connes cocycle derivative $[D\varphi:D\omega]_t=\Delta(\varphi/\omega_0)^{it}\Delta(\omega/\omega_0)^{-it}$, a bounded operator that measures how the modular flow of one state is tilted relative to another; it is the state-dependent local extension of modular time. The proof of Theorem 1.1 is carried by recovery maps $R:\mathcal{B}\to\mathcal{A}$ satisfying $R(b)V=Vb$, by the sufficiency theory that forces cocycles to lie in sufficient subalgebras, and by the uniqueness of the KMS condition, which pins the pushed-forward cocycle to the bulk cocycle. An analytic continuation argument transfers the intertwining identity from the real line to the imaginary axis, yielding equality of the spatial derivatives and hence of relative entropies, even when the bulk relative entropy is infinite. In the AdS application, the geometric modular action of the wedge-preserving isometry group $\Lambda_{p,q}(t)$ on boundary diamonds and bulk wedges supplies the local notion of time, matching the algebraic cocycle flow.

What would settle it

A concrete test is to compute the kernel of the Fock-field map $u\mapsto \varphi_F(u)\Omega$ for $u$ in the symplectic subspace $X_{\mathrm{bulk}}(W_{p,q})$ using the global AdS vacuum two-point function (4.26); a nonzero kernel would mean the vacuum vector is not separating for $\pi_\omega(A_{\mathrm{bulk}}(W_{p,q}))''$, so the hypothesis of Theorem 4.3 fails and the subregion-duality conclusion would not follow from this argument.

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Extended reading notes

Core claim

The paper's central discovery is Theorem 1.1. Given an isometry $V:\mathcal{H}\hookrightarrow\mathcal{K}$, von Neumann algebras $\mathcal{A}\subseteq B(\mathcal{K})$, $\mathcal{B}\subseteq B(\mathcal{H})$, and a vector $\Omega$ that is cyclic and separating for $\mathcal{B}$ such that $V\Omega$ is cyclic and separating for $\mathcal{A}$, the following are equivalent: complementary recovery of $\mathcal{B}$ and $\mathcal{B}'$ for the complementary channels $V^*(\cdot)V$ on $\mathcal{A}$ and $\mathcal{A}'$; the intertwining identity $V[D\omega_\Omega:D\omega_\psi]_t=[D\omega_{V\Omega}:D\omega_{V\psi}]_tV$ (and the same for the commutants) for all states $\psi$ and all $t\in\mathbb{R}$; and preservation of bulk-boundary relative entropies for all states. Theorem 1.2 is the approximate analogue: a sequence of isometries admitting approximate recovery channels forces approximate cocycle intertwining, which in turn forces approximate privacy, and all three conditions become equivalent when the bulk algebra is hyperfinite. The paper then constructs a concrete instance from the symplectic solution space of the Klein-Gordon equation on the universal cover of AdS, with $\mathcal{A}=\pi_\omega(A_{\mathrm{bd}}(D_{p,q}))''$ and $\mathcal{B}=\pi_\omega(A_{\mathrm{bulk}}(W_{p,q}))''$, and shows, using the geometric modular flow of the vacuum, that the equivalent conditions hold. A by-product is that a boundary type III$_1$ factor with an ergodic vacuum forces the dual bulk algebra to be either $\mathbb{C}1$ (with one-dimensional Hilbert space) or a type III$_1$ factor.

Load-bearing premise

The load-bearing premise is that the global vacuum vector is cyclic and separating for the bulk wedge algebra, and that its image under the bulk-to-boundary isometry is cyclic and separating for the boundary diamond algebra; this no-local-annihilation property is taken from the literature and not proved here for the universal cover of AdS.

Editorial extensions

If this is right

  • Exact complementary recovery, preservation of Connes cocycle flow, and equality of relative entropies are three co-equal descriptions of bulk reconstruction, so a holographic code that satisfies one satisfies all three.
  • Boundary causal diamonds in the universal cover of AdS are dual to bulk causal wedges of Klein-Gordon fields, realizing operator-algebraic subregion-subregion duality in an infinite-dimensional setting.
  • The kink transform conjecture gains a concrete realization: in exact recovery settings the kink transform is bulk cocycle flow, with boundary modular flow implementing bulk time and geometry.
  • In the large-$N$ limit, approximate recovery implies vanishing cocycle error, which in turn implies approximate privacy; for hyperfinite bulk algebras these conditions are equivalent, making the cocycle error a useful order parameter for approximate reconstruction.
  • The type III$_1$ ergodicity structure propagates from boundary subregions to bulk subregions: a dual bulk algebra is trivial or type III$_1$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same equivalence could be used as a detection tool: deviation from cocycle intertwining in an approximate code can be read as the amount of bulk locality that finite-$N$ effects destroy, which may be easier to compute than the recovery error itself.
  • The framework suggests a testable route to extend subregion duality beyond Klein-Gordon fields: if a holographic Hadamard state admits geometric modular flow on both sides, the same argument should yield complementary recovery for interacting or higher-spin fields.
  • One concrete corollary one could test in models: if a dual bulk algebra were of type III$_\lambda$ with $\lambda\neq 1$, then the ergodic-vacuum assumption would have to fail in the corresponding holographic dual.
  • The approximate theorem suggests quantifying the $N$-dependence of the cocycle error as a proxy for the semi-classical expansion; a finite-$N$ code with exactly vanishing cocycle error would be a candidate for all-orders bulk reconstruction.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper develops operator-algebraic characterizations of bulk reconstruction in AdS/CFT. The main exact result, Theorem 1.1, states that for an isometry V:H→K and von Neumann algebras A, B with a cyclic separating vector Ω for B such that VΩ is cyclic separating for A, complementary recovery of B and B′ is equivalent to (i) intertwining of Connes cocycle flows on B and B′ with those on A and A′, and (ii) equality of relative entropies on the two algebras and their commutants. Theorem 1.2 gives an approximate version: asymptotic recovery implies asymptotic cocycle intertwining, which implies asymptotic commutativity of the encoded algebras with the bulk commutants, and these three conditions are equivalent when B is hyperfinite. In Section 4 the paper applies Theorem 1.1 to Klein–Gordon fields on the universal cover of AdS, using the holography framework of Dybalski–Wrochna [30], and claims an operator-algebraic subregion–subregion duality between boundary causal diamonds and bulk causal wedges (Theorem 4.3). The final section also derives, under the modular-theoretic hypotheses, that bulk duals of type III_1 boundary factors with ergodic vacuum are either trivial or type III_1.

Significance. If the results are correct, Theorem 1.1 is a genuinely useful addition to the operator-algebraic quantum error correction literature: it characterizes exact complementary recovery through the physically motivated cocycle flow, and the approximate version adds a new asymptotic viewpoint. The paper also provides a detailed, largely rigorous proof of the exact theorem using standard modular theory, spatial derivatives, and ultraproduct techniques. The application in Section 4 aims at an infinite-dimensional, continuum subregion–subregion duality, a goal of clear interest to mathematical physics. However, the strongest advertised physical conclusion, Theorem 4.3, is presented as a proof sketch and depends on two unstated black-box theorems ([37, Theorem 1.1] and [51, Theorem 1.1]) as well as on unproved Reeh–Schlieder-type hypotheses; this is the main weakness of the paper.

major comments (4)
  1. [§4.3 (Theorem 4.3)] The decisive step in the proof of Theorem 4.3 is the sentence "We are therefore in position to apply [37, Theorem 1.1] to achieve complementary recovery of the inclusion." The cited theorem is never stated, so the reader cannot verify that its hypotheses are met or that its conclusion is exactly the existence of recovery channels R:B→A and R′:B′→A′ with V*R(b)V=b and V*R′(b′)V=b′. Since complementary recovery is a pair of existence statements, the proof should either state [37, Theorem 1.1] (in an appendix or in the body), construct the recovery maps explicitly, or cite a stated theorem with matching hypotheses and conclusion.
  2. [§4.3 (Reeh–Schlieder hypothesis)] Theorem 1.1 requires Ω to be cyclic and separating for B=πω(Abulk(Wp,q))′′ and VΩ to be cyclic and separating for A=πω(Abd(Dp,q))′′. The paper does not prove either property. For the boundary net, the paper asserts that W∞2 defines a causal, conformally covariant pre-cosheaf and then invokes [17, Theorem 2.3(ii)], but it does not verify the axioms of that theorem (e.g., Reeh–Schlieder property, locality, modular covariance) for the algebra πω(Abd(Dp,q))′′. For the bulk wedge algebra, no argument is given that ωΩ is separating for πω(Abulk(Wp,q))′′ on the universal cover of AdS. Without these hypotheses, the modular theory of Theorem 1.1 cannot be applied to conclude subregion–subregion duality.
  3. [§3.1 (Theorem 1.1 proof)] The proof of Theorem 1.1 relies on two external results without stating their hypotheses: [37, Theorem 1.1(4)] is used to justify the intertwining of modular flows under the recovery map, and [51, Theorem 1.1] supplies the implication (3)⇒(1). These results are load-bearing for the equivalence claimed in Theorem 1.1. The paper should state these theorems with their exact assumptions and conclusions, or give self-contained proofs of the specific consequences used, so that the central equivalence can be checked independently of the cited papers.
  4. [§4.3 (boundary 2-point function)] The proof that the boundary limit W∞2 is the 2-point function of a causal conformal net is only sketched. The paper shows that W2 coincides with the known AdS vacuum 2-point function, but the passage from the restricted net {πF(Abd(O))′′}O⊆Min(r) to a net satisfying the hypotheses of [17, Theorem 2.3(ii)] requires more than convergence of the 2-point function: it requires positive definiteness, locality, conformal covariance, and the Reeh–Schlieder property for the vacuum. The paper should spell out how these properties follow from the quasi-free construction and the cited references [13,17].
minor comments (5)
  1. [Theorem 1.1] In the statement of condition 3, the last equality reads "SB′(ω′ψ, ω′φ)"; the final vector should presumably be Ω, not φ. Please correct this typo.
  2. [§3.3 (Proof of Theorem 1.2)] Near the end of the proof, the sentence "Therefore, all three conditions are are equivalent" contains a duplicated "are".
  3. [§3.3 (Comment after Theorem 1.2)] The phrase "Theorem 1.2 (11)" appears to refer to condition (1) of Theorem 1.2; please renumber or reword to avoid confusion with an equation number.
  4. [§4.3] The notation "pertinent eG0 action" is unclear; it should presumably be the action of the universal cover of SO(2,d), e.g., Ĝ0 or the tilde version introduced earlier.
  5. [§4] Typographical issues in Section 4 include "retarted/advanced propagators" (should be "retarded/advanced") and the inconsistent use of "M o" and "M o" for the interior of M.

Circularity Check

0 steps flagged · score 2.0 of 10

No equation-level circularity: the central equivalences are substantive modular-theoretic results; minor author-overlap citations and an unstated black-box theorem create verification gaps, not circular reductions.

full rationale

The paper's main equivalence (Theorem 1.1) is not circular. Complementary recovery (condition 1) is a channel-theoretic statement about correctable subalgebras; the cocycle-derivative intertwining (condition 2) and relative-entropy preservation (condition 3) are separately defined quantities. The proof connects them through Takesaki's modular theory, Petz's sufficient subalgebra theorem [63, Theorem 9.3], and the prior result [51, Theorem 1.1], not by definitional identification. The use of [51] involves an author of the present paper, but it is a published, distinct theorem on holographic relative entropy and complementary recovery in infinite dimensions, and the current proof adds the cocycle-flow implication and an analytic-continuation argument for infinite relative entropies. The use of [37, Theorem 1.1(4)] in the (1) implies (2) step is also author-overlapping, but the paper reproduces the needed modular-flow argument via [75] and [76, Corollary VIII.1.4], so the step does not reduce to the citation alone. For the physical application, Theorem 4.3 depends on [37, Theorem 1.1] as a black box: after identifying the global AdS vacuum two-point function, obtaining KMS and geometric modular flow for the boundary diamonds, and showing modular invariance of the bulk wedge algebra, the paper concludes complementary recovery by appeal to that unstated theorem. This is a dependence on a prior result by an overlapping author, and the paper would be easier to verify if [37]'s exact hypotheses were stated; it is nevertheless an application of a general theorem rather than a renaming of the target conclusion. The Reeh-Schlieder-type cyclic/separating hypothesis required by Theorem 1.1 is not proved in Section 4.3; it is a physical premise cited to the literature, so its failure would undermine the application, but a missing or merely cited hypothesis is a correctness risk, not circularity. No fitted parameter is renamed as a prediction, and no quantity is defined in terms of the result it is used to derive. I therefore find no significant circularity; the score reflects only the presence of author-overlapping citations in the proof chain and the unstated [37] black box.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters are fitted to data; the mass ν is a physical input, not a fit. The key assumptions are the holographic framework of [30], the geometric modular structure of [16,17], and the author's own recovery theorem [37], all invoked as black boxes. The Reeh-Schlieder property is the least explicitly justified input.

assumptions (7)
  • standard math Standard relative modular theory and sufficiency of subalgebras, including Connes cocycle derivatives, relative modular operators, the KMS condition, and sufficiency theorems of Petz.
    Used throughout Sections 2 and 3, especially in the proof of Theorem 1.1 and in Lemma 3.2, following [63] and [76].
  • domain assumption The global AdS vacuum ω is a ground state and KMS state for the Klein-Gordon dynamics, and a holographic Hadamard state, yielding the covariance η.
    Section 4.3 adopts this from [26, 35, 55, 82] to define the symplectic completion and the GNS representation.
  • domain assumption The bulk-to-boundary map ∂ of [30] is continuous, intertwines isometries with conformal transformations (Lemma 4.1), and gives the inclusion Abulk(V(O)) ⊆ Abd(O) for domains of dependence.
    Core mechanism of Section 4, taken from [30, Theorem 3.7] and Holmgren's theorem; Proposition 4.2 uses this to prove W_{p,q} ⊆ V(D_{p,q}).
  • domain assumption For vacuum AdS, the modular automorphism of the vacuum state on boundary diamonds and bulk wedges is geometrically implemented by the one-parameter isometry group preserving the causal region.
    Invoked in Section 4.3 via [16] and [17, Theorem 2.3(ii)] to obtain the geometric KMS structure needed for [37].
  • domain assumption The theorem of Gesteau and Kang [37, Theorem 1.1] that matching modular flows imply complementary recovery for inclusions of von Neumann algebras.
    Used as a black box in the proof of Theorem 4.3; it is a prior result by one of the present authors, distinct from the current claim.
  • domain assumption The Breitenlohner-Freedman bound ν > 0, so that the conformal weight Δ = ν+ = d/2 + ν is the relevant scaling dimension.
    Assumed in Section 4.2 to ensure the bulk-to-boundary map ∂ has the stated properties and the limit defining the boundary two-point function exists.
  • domain assumption The vacuum vector Ω is cyclic and separating for the boundary diamond algebras and, via the isometry, for the bulk wedge algebras (Reeh-Schlieder property).
    Required in Theorem 1.1 and assumed in Section 4.3; the paper cites [16] and [17] but does not prove this property for the specific AdS setting.

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Pith. "Pith review of Algebraic approach to spacetime bulk reconstruction." pith.science (2026). https://pith.science/paper/WIHIILK3

@misc{pith2026241200298,
  author       = {Pith},
  title        = {Pith review of: Algebraic approach to spacetime bulk reconstruction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WIHIILK3}},
  note         = {Machine review of arXiv:2412.00298}
}
abstract

Motivated by the theory of holographic quantum error correction in the anti-de Sitter/conformal field theory (AdS/CFT) correspondence, together with the kink transform conjecture on the bulk AdS description of boundary cocycle flow, we characterize (approximate) complementary recovery in terms of (approximate) intertwining of bulk and boundary cocycle derivatives. Using the geometric modular structure in vacuum AdS, we establish an operator algebraic subregion-subregion duality of boundary causal diamonds and bulk causal wedges for Klein-Gordon fields in the universal cover of AdS. Our results suggest that, from an algebraic perspective, the kink transform is bulk cocycle flow, which (in the above case) induces the bulk geometry via geometric modular action and the corresponding notion of time. As a by-product, we find that if the von Neumann algebra of a boundary CFT subregion is a type $\mathrm{III}_1$ factor with an ergodic vacuum, then the von Neumann algebra of the corresponding dual bulk subregion, is either $\mathbb{C}1$ (with a one-dimensional Hilbert space) or a type $\mathrm{III}_1$ factor.

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Figure 1.1
Figure 1.1. Geometric depiction of the Ryu–Takayanagi surface [PITH_FULL_IMAGE:figures/full_fig_p003_1_1.png] view at source ↗

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Reference graph

Works this paper leans on

86 extracted references · 75 canonical work pages · cited by 1 Pith paper

  1. [51]

    Holographic tensor network models and quantum error correc- tion: a topical review,

    A. Jahn and J. Eisert, “Holographic tensor network models and quantum error correc- tion: a topical review,” Quantum Sci. Technol. 6 (2021) 033002

    work page 2021

  2. [37]

    Thermal states are vital: Entanglement Wedge Reconstruction from Operator-Pushing

    E. Gesteau and M. J. Kang, “Thermal states are vital: Entanglement Wedge Recon- struction from Operator-Pushing,” arXiv:2005.07189 [hep-th]

    work page Pith review arXiv 2005

  3. [30]

    A mechanism for holography for non-interacting fields on anti-de Sitter spacetimes,

    W. Dybalski and M. Wrochna, “A mechanism for holography for non-interacting fields on anti-de Sitter spacetimes,” Class. Quantum Grav. 36 (2019), 085006 (14pp)

    work page 2019

  4. [1]

    Bulk Locality and Quantum Error Correction in AdS/CFT,

    A. Almheiri, X. Dong and D. Harlow, “Bulk Locality and Quantum Error Correction in AdS/CFT,” JHEP 04, 163 (2015)

    2015

  5. [2]

    The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole,

    A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield, “The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole,” JHEP 12, 063 (2019)

    2019

  6. [3]

    The Page curve of Hawking radiation from semiclassical geometry,

    A. Almheiri, R. Mahajan, J. Maldacena and Y. Zhao, “The Page curve of Hawking radiation from semiclassical geometry,” JHEP 03 (2020), 149. 30

    2020

  7. [4]

    Ultraproducts of von Neumann algebras,

    H. Ando and U. Haagerup, “Ultraproducts of von Neumann algebras,” J. Funct. Anal. 266 (2014), no. 12, 6842-6913

    2014

  8. [5]

    Pseudodifferential calculus on a singular foliation,

    I. Androulidakis and G. Skandalis, “Pseudodifferential calculus on a singular foliation,” J. Noncommut. Geom. 5 (2011), no. 1, 125-152

    2011

Show all 86 references
  1. [6]

    Relative Hamiltonian for faithful normal states of a von Neumann algebra,

    H. Araki, “Relative Hamiltonian for faithful normal states of a von Neumann algebra,” Publ. RIMS, Kyoto Univ. 9 (1973), 165-209

    1973

  2. [7]

    Relative Entropy of States of von Neumann Algebras,

    H. Araki, “Relative Entropy of States of von Neumann Algebras,” Publ. RIMS, Kyoto Univ. 11 (1976), 809-833

    1976

  3. [8]

    Relative entropy of states of von Neumann algebras II,

    H. Araki, “Relative entropy of states of von Neumann algebras II,” Publ. RIMS, Kyoto Univ. 13 (1977), 173-192

    1977

  4. [9]

    Subalgebras of C ∗-algebras,

    W. B. Arveson,, “Subalgebras of C ∗-algebras,” Acta Math. 123 (1969), 141-224

    1969

  5. [10]

    Quantum field theory in Anti-de Sitter space- time,

    S. J. Avis, C. J. Isham and D. Storey, “Quantum field theory in Anti-de Sitter space- time,” Phys. Rev. D, 18:3565-3576, (1978)

    1978

  6. [11]

    Generalization of quantum error correction via the Heisenberg picture,

    C. B´ eny, A. Kempf and D.W. Kribs, “Generalization of quantum error correction via the Heisenberg picture,” Phys. Rev. Lett. 98 (2007), 100502

    2007

  7. [12]

    Quantum error correction on infinite-dimensional Hilbert spaces

    C. B´ eny, A. Kempf and D. W. Kribs, “Quantum error correction on infinite-dimensional Hilbert spaces”, J. Math. Phys. 50 (2009), no. 6, 062108, 24 pp

    2009

  8. [13]

    A general construction of conformal field theories from scalar anti-de Sitter quantum field theories,

    M. Bertola, J. Bros, U. Moschella and R. Schaeffer, “A general construction of conformal field theories from scalar anti-de Sitter quantum field theories,” Nuclear Physics B 587 (2000), 619-644

    2000

  9. [14]

    Gravity dual of Connes cocycle flow,

    R. Bousso, V. Chandrasekaran, P. Rath, A. Shahbazi-Moghaddam, “Gravity dual of Connes cocycle flow,” Phys. Rev. D 102 (2020), no. 6, 066008, 21 pp

    2020

  10. [15]

    Positive energy in anti-de Sitter backgrounds and gauged extended supergravity,

    P. Breitenlohner and D. Z. Freedman, “Positive energy in anti-de Sitter backgrounds and gauged extended supergravity,” Phys. Lett. B, 115(3):197-201, (1982)

    1982

  11. [16]

    Towards a General Theory of Quantized Fields on the Anti-de Sitter Space-Time,

    J. Bros, H. Epstein and U. Moschella, “Towards a General Theory of Quantized Fields on the Anti-de Sitter Space-Time,” Commun. Math. Phys. 231 (2002), 481-528

    2002

  12. [17]

    Modular structure and duality in conformal quantum field theory,

    R. Brunetti, D. Guido and R. Longo, “Modular structure and duality in conformal quantum field theory,” Comm. Math. Phys. 156 (1993), no. 1, 201-219

    1993

  13. [18]

    The current algebra on the circle as a germ of local field theories,

    D. Buchholz, G. Mack and I. Todorov, “The current algebra on the circle as a germ of local field theories,” Nuclear Phys. B Proc. Suppl. 5, no. 2, 20-56, 1988

    1988

  14. [19]

    Connes, On the spatial theory of von Neumann algebras, Journal of Functional Anal- ysis 35 (1980), no

    A. Connes, On the spatial theory of von Neumann algebras, Journal of Functional Anal- ysis 35 (1980), no. 2, 153-164. 31

    1980

  15. [20]

    Connes, Une classification des facteurs de type III, In Annales Scientifiques de l’ ´Ecole Normale Sup´ erieure,6 (1973), no

    A. Connes, Une classification des facteurs de type III, In Annales Scientifiques de l’ ´Ecole Normale Sup´ erieure,6 (1973), no. 2, pp. 133-252

    1973

  16. [21]

    Entanglement wedge reconstruction via universal recovery channels,

    J. Cotler, P. Hayden, G. Penington, G. Salton, B. Swingle and M. Walter, “Entanglement wedge reconstruction via universal recovery channels,” Phys. Rev. X 9 (2019) no.3, 031011

    2019

  17. [22]

    Private algebras in quantum information and infinite-dimensional complementarity,

    J. Crann, D. W. Kribs, R. H. Levene and I. G. Todorov, “Private algebras in quantum information and infinite-dimensional complementarity,” J. Math. Phys. 57 (2016), no. 1, 015208

    2016

  18. [23]

    Hadamard states for a scalar field in anti-de Sitter spacetime with arbitrary boundary conditions,

    C. Dappiaggi and H. R. Ferreira, “Hadamard states for a scalar field in anti-de Sitter spacetime with arbitrary boundary conditions,” Phys. Rev. D 94 (2016), no. 12, 125016, 19 pp

    2016

  19. [24]

    Propagators in curved spacetimes from operator theory,

    J. Derezi´ nski, C. Ga β, “Propagators in curved spacetimes from operator theory,” arXiv:2409.03279v1

  20. [25]

    Generalized integrals in the theory of Bessel and Gegenbauer functions,

    J. Derezi´ nski, C. Gaβ and B. Ruba, “Generalized integrals in the theory of Bessel and Gegenbauer functions,” Proceedings of Contemporary Mathematics, (to appear), 2023

    2023

  21. [26]

    Derezi´ nski and C

    J. Derezi´ nski and C. G´ erard,Mathematics of Quantization and Quantum Fields , Cam- bridge Monographs in Mathematical Physics, Cambridge University Press (2013)

    2013

  22. [27]

    Vecteurs totalisateurs d’une alg` ebre de von Neumann

    J. Dixmier and O. Mar´ echal, “Vecteurs totalisateurs d’une alg` ebre de von Neumann”, Commun. Math. Phys. 22 (1971), 44-50

    1971

  23. [28]

    Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality,

    X. Dong, D. Harlow and A. C. Wall, “Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality,” Phys. Rev. Lett. 117, no. 2, 021601 (2016)

    2016

  24. [29]

    Generalized free fields and the AdS-CFT correspon- dence,

    M. D¨ utsch and K. -H. Rehren, “Generalized free fields and the AdS-CFT correspon- dence,” Ann. Henri Poincare 4 (2003) 613-635

    2003

  25. [31]

    The holographic map as a conditional expectation,

    T. Faulkner, “The holographic map as a conditional expectation,” arXiv:2008.04810 [hep-th]

    2008 arXiv

  26. [32]

    Asymptotically isometric codes for holography

    T. Faulkner and M. Li, “Asymptotically isometric codes for holography”, arXiv:2211.12439 [hep-th]

    arXiv

  27. [33]

    Elementary particles in a curved space. II,

    C. Fronsdal, “Elementary particles in a curved space. II,” Phys. Rev. D 10, 589-598 (1974). 32

    1974

  28. [34]

    Operator algebras and conformal field theory,

    F. Gabbiani and J. Frohlich, “Operator algebras and conformal field theory,” Commun. Math. Phys. 155, 569-640 (1993)

    1993

  29. [35]

    Microlocal analysis of quantum fields on curved spacetimes,

    C. G´ erard, “Microlocal analysis of quantum fields on curved spacetimes,” ESI Lect. Math. Phys. EMS Publishing House, Berlin, (2019) viii+220 pp

    2019

  30. [36]

    Holographic baby universes: an observable story,

    E. Gesteau and M. J. Kang, “Holographic baby universes: an observable story,” arXiv:2006.14620 [hep-th]

    2006 arXiv

  31. [38]

    The infinite-dimensional HaPPY code: entanglement wedge reconstruction and dynamics,

    E. Gesteau and M. J. Kang, “The infinite-dimensional HaPPY code: entanglement wedge reconstruction and dynamics,” arXiv:2005.05971 [hep-th]

    2005 arXiv

  32. [39]

    Nonperturbative gravity corrections to bulk reconstruc- tion,

    E. Gesteau and M. J. Kang, “Nonperturbative gravity corrections to bulk reconstruc- tion,” Journal of Physics A: Mathematical and Theoretical 56, no. 38, 385401 (2023)

    2023

  33. [40]

    Uniform ergodic theorems for identity preserving Schwarz maps on W ∗- algebras,

    U. Groh, “Uniform ergodic theorems for identity preserving Schwarz maps on W ∗- algebras,” J. Operator Theory. 11 (1984), no. 2, 395-404

    1984

  34. [41]

    Haag, Local Quantum Physics: Fields, particles, algebras , Texts and Monographs in Physics, Springer-Verlag, 1992

    R. Haag, Local Quantum Physics: Fields, particles, algebras , Texts and Monographs in Physics, Springer-Verlag, 1992

    1992

  35. [42]

    The standard from of von Neumann algebras,

    U. Haagerup, “The standard from of von Neumann algebras,” Math. Scand. 37 (1975), 271-283

    1975

  36. [43]

    Holographic representation of local bulk operators,

    A. Hamilton, D. N. Kabat, G. Lifschytz and D. A. Lowe, “Holographic representation of local bulk operators,” Phys. Rev. D 74, 066009 (2006)

    2006

  37. [44]

    The Ryu–Takayanagi Formula from Quantum Error Correction,

    D. Harlow, “The Ryu–Takayanagi Formula from Quantum Error Correction,” Commun. Math. Phys. 354, no. 3, 865-912 (2017)

    2017

  38. [45]

    TASI Lectures on the Emergence of Bulk Physics in AdS/CFT,

    D. Harlow, “TASI Lectures on the Emergence of Bulk Physics in AdS/CFT,” PoS TASI2017, 002 (2018)

    2018

  39. [46]

    Concise lectures on selected topics of von Neumann algebras

    F. Hiai, “Concise lectures on selected topics of von Neumann algebras”, arXiv:2004.02383 [math.OA]

    2004 arXiv

  40. [47]

    L. H¨ ormander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis , Grundlehren der mathematischen Wissenschaften 256, Springer Berlin Heidelberg, 1990

    1990

  41. [48]

    Dynamics in non-globally-hyperbolic static spacetimes. II. General analysis of prescriptions for dynamics

    A. Ishibashi and R. M. Wald, “Dynamics in non-globally-hyperbolic static spacetimes. II. General analysis of prescriptions for dynamics” Classical Quantum Gravity20 (2003), no. 16, 3815-3826. 33

    2003

  42. [49]

    Ishibashi and R

    A. Ishibashi and R. M. Wald, “Dynamics in nonglobally hyperbolic static space-times:

  43. [50]

    Anti-de Sitter space-time,” Class. Quant. Grav., 21, (2004), 2981-3014.bi

    2004

  44. [52]

    Holographic Relative Entropy in Infinite- Dimensional Hilbert Spaces,

    M. J. Kang and D. K. Kolchmeyer, “Holographic Relative Entropy in Infinite- Dimensional Hilbert Spaces,” Commun. Math. Phys. 400, no.3, 1665-1695 (2023)

    2023

  45. [53]

    Entanglement wedge reconstruction of infinite- dimensional von Neumann algebras using tensor networks,

    M. J. Kang and D. K. Kolchmeyer, “Entanglement wedge reconstruction of infinite- dimensional von Neumann algebras using tensor networks,” Phys. Rev. D 103, no.12, 126018 (2021)

    2021

  46. [54]

    Modelling the spacetime lattice: discrete to continuum,

    M. J. Kang and S. Qasim, “Modelling the spacetime lattice: discrete to continuum,” To appear

  47. [55]

    Holographic tensor network in 3d spacetime,

    M. J. Kang and S. Qasim, “Holographic tensor network in 3d spacetime,” To appear

  48. [56]

    Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate Killing horizon,

    B. S. Kay and R. M. Wald. “Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate Killing horizon,” Physics Reports 207 (1991), no. 2, 49-136

    1991

  49. [57]

    Bulk Locality and Entanglement Swapping in AdS/CFT,

    W. R. Kelly, “Bulk Locality and Entanglement Swapping in AdS/CFT,” JHEP 03, 153 (2017)

    2017

  50. [58]

    Theory of quantum error-correcting codes,

    E. Knill and R. Laflamme, “Theory of quantum error-correcting codes,” Phys. Rev. A (3) 55 (1997), no.2, 900-911

    1997

  51. [59]

    About the convolution of distributions on groupoids,

    J.-M. Lescure, D. Manchon and S. Vassout, “About the convolution of distributions on groupoids,” J. Noncommut. Geom. 11 (2017), no. 2, 757-789

    2017

  52. [60]

    Causal connectability between quantum systems and the black hole interior in holographic duality,

    S. Leutheusser and H. Liu, “Causal connectability between quantum systems and the black hole interior in holographic duality,” Phys. Rev. D 108, no.8, 086019 (2023)

    2023

  53. [61]

    Longo, Notes on algebraic invariants for noncommutative dynamical systems , Comm

    R. Longo, Notes on algebraic invariants for noncommutative dynamical systems , Comm. Math. Phys. 69 (1979), 195-207

    1979

  54. [62]

    On Landauer’s principle and bound for infinite systems,

    R. Longo, “On Landauer’s principle and bound for infinite systems,” Comm. Math. Phys. 363 (2018), no. 2, 531-560

    2018

  55. [63]

    The Large N limit of superconformal field theories and supergravity,

    J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys. (1998) 2, 231-252

    1998

  56. [64]

    Ohya and D

    M. Ohya and D. Petz, Quantum entropy and its use , Texts and Monographs in Physics, Springer-Verlag, Berlin, 1993, viii+335 pp. 34

    1993

  57. [65]

    Holographic quantum error- correcting codes: toy models for the bulk/boundary correspondence,

    F. Pastawski, B. Yoshida, D. Harlow and J. Preskill, “Holographic quantum error- correcting codes: toy models for the bulk/boundary correspondence,” J. High Energy Phys. (2015), no. 6, 149

    2015

  58. [66]

    Entanglement Wedge Reconstruction and the Information Paradox,

    G. Penington, “Entanglement Wedge Reconstruction and the Information Paradox,” JHEP 09, 002 (2020)

    2020

  59. [67]

    Petz, Sufficient subalgebras and the relative entropy of states of a von Neumann algebra, Comm

    D. Petz, Sufficient subalgebras and the relative entropy of states of a von Neumann algebra, Comm. Math. Phys. 105 (1986), no. 1, 123-131

    1986

  60. [68]

    On ultrapowers of non commutative Lp-spaces,

    Y. Raynaud, “On ultrapowers of non commutative Lp-spaces,” J. Operator Theory 48 (2002), no. 1, 41-68

    2002

  61. [69]

    Methods of modern mathematical physics. I. Functional anal- ysis,

    M. Reed and B. Simon, “Methods of modern mathematical physics. I. Functional anal- ysis,” Academic Press, New York-London, 1972. xvii+325 pp

    1972

  62. [70]

    Algebraic holography,

    K. H. Rehren, “Algebraic holography,” Annales Henri Poincare 1, 607-623 (2000) doi:10.1007/PL00001009

    2000 doi

  63. [71]

    Structural and Dynamical Aspects of the AdS-CFT Correspondence: a Rigorous Approach,

    P. Ribeiro, “Structural and Dynamical Aspects of the AdS-CFT Correspondence: a Rigorous Approach,” Ph.D. Thesis, University of S˜ ao Paulo (2007)

    2007

  64. [72]

    Holographic derivation of entanglement entropy from AdS/CFT,

    S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,” Phys. Rev. Lett. 96 (2006), 181602

    2006

  65. [73]

    Positive functions on C*-algebras,

    W. F. Stinespring, “Positive functions on C*-algebras,” Proceedings of the American Mathematical Society, 6 (1955), no 2, pp. 211-216

    1955

  66. [74]

    On the local structure of the Klein-Gordon field on curved spacetimes,

    A. Strohmaier, “On the local structure of the Klein-Gordon field on curved spacetimes,” Lett. Math. Phys. 54 (2000), no.4, 249-261

    2000

  67. [75]

    Analytic states in quantum field theory on curved space- times,

    A. Strohmaier and E. Witten, “Analytic states in quantum field theory on curved space- times,” Ann. Henri Poincar´ e25 (2024), no. 10, 4543-4590

    2024

  68. [76]

    Conditional Expectations in von Neumann Algebras,

    M. Takesaki, “Conditional Expectations in von Neumann Algebras,” J. Funct. Anal. 9 (1972), 306-321

    1972

  69. [77]

    Takesaki, Theory of Operator Algebras II , Encyclopedia of Mathematical Sciences 125, Springer-Verlag Berlin–Heidelberg–New York, 2003

    M. Takesaki, Theory of Operator Algebras II , Encyclopedia of Mathematical Sciences 125, Springer-Verlag Berlin–Heidelberg–New York, 2003

    2003

  70. [78]

    Treves, Topological vector spaces, distributions and kernels , Academic press, 1967

    F. Treves, Topological vector spaces, distributions and kernels , Academic press, 1967

    1967

  71. [79]

    The wave equation on asymptotically anti-de Sitter spaces,

    A. Vasy, “The wave equation on asymptotically anti-de Sitter spaces,” Anal. PDE 5 (2012), 81-144

    2012

  72. [80]

    The Massive Wave Equation in Asymptotically AdS Spacetimes,

    C. M. Warnick, “The Massive Wave Equation in Asymptotically AdS Spacetimes,” Comm. Math. Phys. 321, no. 1 (2013), 85-111. 35

    2013

  73. [81]

    Willard, General topology, Dover Publications, Inc., Mineola, NY, 2004

    S. Willard, General topology, Dover Publications, Inc., Mineola, NY, 2004. xii+369 pp

    2004

  74. [82]

    Anti-de Sitter space and holography,

    E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. 2 (1998), 253-291

    1998

  75. [83]

    The holographic Hadamard condition on asymptotically anti-de Sitter spacetimes,

    M. Wrochna, “The holographic Hadamard condition on asymptotically anti-de Sitter spacetimes,” Lett. Math. Phys. 107 (2017), 2291-331

    2017

  76. [84]

    Adding the algebraic Ryu-Takayanagi formula to the algebraic reconstruction theorem,

    M. Xu and H. Zhong “Adding the algebraic Ryu-Takayanagi formula to the algebraic reconstruction theorem,” arXiv:2411.06361v2

  77. [85]

    The Klein-Gordon equation in Anti-de Sitter spacetime,

    K. Yagdjian and A. Galstian, “The Klein-Gordon equation in Anti-de Sitter spacetime,” Rend. Sem. Mat. Univ. Pol. Torino, 67:291-292, (2009)

    2009

  78. [86]

    The Role of type III factors in quantum field theory,

    J. Yngvason, “The Role of type III factors in quantum field theory,” Rept. Math. Phys. 55, 135-147 (2005). 36

    2005

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