Pith. sign in

REVIEW 5 major objections 5 minor 5 cited by

More on the upper bound of holographic n-partite information

T0 review · 5 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper shows that by tuning the number and shape of a region E, holographic conditional mutual information saturates its quantum upper bound $2\,\mathrm{EoSP}(A:B)$, so distant small regions are fully tripartite entangled.

desk verdict Creative, technically rich paper; the disconnectivity proof in Sec. 4.1 is the real weak spot, and the no-Bell-pairs claim needs softening. read the letter →

arxiv 2411.19207 v2 pith:CYXNSGMB submitted 2024-11-28 hep-th

Xin-Xiang Ju , Wen-Bin Pan , Ya-Wen Sun , Yuan-Tai Wang , Yang Zhao This is my paper · ORCID
classification hep-th
keywords holographicentanglemententropyconditionalmutualinformationn-partitemultipartiteRyu-TakayanagisurfaceofpurificationAdS3/CFT2Araki-Liebinequality
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 claims that holographic n-partite information is much larger than previously appreciated: for fixed regions A and B, tuning the third region E can drive the conditional mutual information $I(A:B|E)$ all the way to its quantum-information-theoretic ceiling, and in asymptotically AdS3 this ceiling is $2\min(S_A,S_B)$. The central identity is $\sup_E I(A:B|E) = 2\,\mathrm{EoSP}(A:B)$, where EoSP is the entanglement of state-constrained purification, the minimal area of a surface that separates A from B while the purifying regions are constrained to be boundary subregions. Because the required E is a collection of intervals whose number m is allowed to grow without bound, $-I_3$ can diverge as $m\to\infty$ even though A and B are distant and have zero mutual information. A sympathetic reading of the argument is that any two small distant boundary regions are fully tripartite entangled with some third region, and that all bipartite entanglement in holography emerges from tripartite entanglement.

What carries the argument

The argument is carried by the multi-entanglement phase transition (MPT) rule and its diagrammatic bookkeeping, together with the disconnectivity condition; RT here refers to the Ryu-Takayanagi surfaces, the bulk minimal surfaces whose areas give boundary entanglement entropies. The MPT rule says that, for a fixed number m of intervals in E, the CMI maximum occurs when 2m entanglement phase transitions of RT surfaces happen simultaneously, which fixes the 2m endpoints of E; the 'zigzag' MPT diagram is the one that wins for every m tested (m up to 5). The disconnectivity condition states that at the maximum, $I(A:E)=I(B:E)=0$, and a weaker version is proved by splitting intervals of E and enlarging gaps: any connected configuration can be replaced by a disconnected one with no smaller CMI. With that condition, the CMI collapses to $S_A+S_B+\sum_i S(E_i)-\sum_i S(\mathrm{Gap}_i)$, and requiring the RT surface of AE to stay disconnected yields inequalities whose tight form is $I(A:B|E) \le 2S_D$ for any D separating A and B; taking the minimal such $S_D$ defines $\mathrm{EoSP}(A:B)=\min S(AA')$, the entanglement of state-constrained purification. Generating polynomials for the maximal $\exp[I/2]$, such as $(1+\mathrm{CR})\,x(x-1)^m - \mathrm{CR}(x+1)^{m+1}$ in the one-gap case, give the exact divergence rates in m.

What would settle it

Take AdS3/CFT2 with two fixed intervals A and B at a fixed cross ratio (say 1/3), and for m = 6, 7, 8 solve the full system of quadratic phase-transition equations for every MPT diagram allowed by the disconnectivity constraints, comparing $\exp[I(A:B|E)/2]$ with the root of the generating polynomial $x(x-1)^m - (\mathrm{CR}/4)[(\sqrt{x}+1)^{m+1}-(\sqrt{x}-1)^{m+1}]^2$ (plus the $(x-1)^m$ term when m is odd). If any non-zigzag diagram yields a larger value, the claimed divergence behavior fails. A complementary check in AdS4/CFT3 would measure $I_4$ for three small disks and a strip region E with m strips approaching all three gap regions; if $I_4$ fails to approach $2\min(S_A,S_B,S_C)$ as m grows, the higher-dimensional saturation claim is false.

Watch

Extended reading notes

Core claim

The discovery is that the upper bound of conditional mutual information is saturated by holographic configurations, and that the saturating configuration is the limit of an m-interval region at a multi-entanglement phase transition. In AdS3/CFT2 the bound is $I(A:B|E) \le 2\min(S_A,S_B)$, and with E chosen as the 'zigzag' MPT diagram with m intervals living in the gaps around A and B, $\exp[I(A:B|E)/2]$ grows linearly, quadratically, or quartically in m depending on how many gap regions E occupies, so $-I_3$ diverges as $m\to\infty$ and the CMI approaches $2\min(S_A,S_B)$. At the saturating configuration the disconnectivity conditions $I(A:E)=I(B:E)=0$ hold while $I(A:BE)=2S_A$, so A shares no bipartite correlation with B or E individually but is maximally entangled with their union; the paper takes this as a signature that $-I_3$ measures genuine tripartite global entanglement rather than classical correlations. The same method gives $I_4 \le 2\min(S_A,S_B,S_C)$ in higher dimensions with divergence as the number of strips grows, while in AdS3/CFT2 the upper bound of $I_4$ with three fixed regions is finite; for $-I_5$ the higher-dimensional upper bound reaches the information-theoretic value $2S_A$. From these the paper concludes that every bipartite entanglement emerges from tripartite entanglement and that no Bell pairs exist in holographic states.

Load-bearing premise

The load-bearing step is the claim that for every number m of intervals, the global maximum of CMI sits at the zigzag multi-entanglement phase transition diagram; this is verified explicitly only up to m = 5, and the hill-climbing argument used beyond that is not a rigorous proof of global maximality.

Editorial extensions

If this is right

  • Two fixed distant small regions in a holographic CFT can be fully tripartite entangled with a third region: CMI reaches $2\min(S_A,S_B)$ as the number of intervals in E goes to infinity.
  • The saturating configuration has $I(A:E)=I(A:B)=0$ but $I(A:BE)=2S_A$, so A is purely quantum entangled with the union BE, making $-I_3$ a faithful measure of tripartite global entanglement for that state.
  • The general upper bound is $\sup_E I(A:B|E) = 2\,\mathrm{EoSP}(A:B)$, where EoSP is the area of the minimal surface dividing the entanglement wedge of A and B; in the two-sided black hole this is twice the throat area, so the throat encodes tripartite rather than bipartite entanglement.
  • In AdS3/CFT2, $I_4$ with three fixed regions is finite no matter how complex E is, while in higher dimensions it diverges and can reach $2\min(S_A,S_B,S_C)$; $-I_5$ likewise reaches $2S_A$ in higher dimensions, so the multipartite entanglement structure is qualitatively different in three bulk dimensions.
  • All bipartite entanglement between arbitrary intervals emerges from tripartite global entanglement of their subregions, so pure Bell-pair-like bipartite entanglement is absent in holographic states.

Reading between the lines

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

  • The proof that the zigzag MPT diagram is globally maximal is verified explicitly for m up to 5 and extrapolated to all m; if a different diagram overtook it at some large m, the claimed polynomial divergence rates would likely change while the saturation bound $2\,\mathrm{EoSP}(A:B)$ might still hold.
  • Identifying the large-m cutoff with the UV scale maps the linear, quadratic, and quartic growth of $\exp[I/2]$ into logarithmic divergences of $-I_3$ that match twice the entanglement-entropy divergence, suggesting the O(1) IR contributions also saturate, a point the paper argues directly via EoSP.
  • The same construction should apply to other holographic backgrounds (for example, higher-genus or multi-boundary wormholes), predicting that $\sup_E I(A:B|E)=2\,\mathrm{EoSP}(A:B)$ holds whenever a separating minimal surface exists, with the throat replaced by the minimal cross-section of the entanglement wedge.
  • For n ≥ 6 the paper does not prove the upper bound but constructs configurations reaching $2\min(S_A,\dots)$; extending the disconnectivity proof to $I_n$ would either confirm the hierarchy that any n-1 distant regions are highly n-partite entangled or reveal a failure of the pattern.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

Summary. This paper studies the maximum of holographic n-partite information when n-1 boundary regions are fixed and the remaining region E is varied. For the tripartite case it proposes a multi-entanglement phase transition (MPT) rule, introduces MPT diagrams, derives a generating polynomial for the maximum CMI as a function of the number m of intervals in E, and identifies a divergence of -I3 as m tends to infinity. It then proves, in a weaker form, a disconnectivity condition and claims sup_E I(A:B|E) = 2 EoSP(A:B), with EoSP the entanglement of state-constrained purification; in AdS3/CFT2 for distant intervals this is 2 min(S_A,S_B), saturating the Araki-Lieb upper bound. For I4 and I5 the paper obtains finite upper bounds in AdS3/CFT2 and divergent bounds in higher dimensions, and interprets these results as evidence that all bipartite entanglement in holography emerges from tripartite global entanglement.

Significance. If correct, the results are striking: any two distant small regions in a holographic state can be fully tripartite-entangled with a third region, and the conditional mutual information can saturate its information-theoretic upper bound. The paper's strengths include the absence of free parameters fitted to the target quantities, the extensive explicit evaluations reported in Figures 14-15 and Tables 1-3, and the concrete, falsifiable prediction for the m-dependence of the maximal CMI encoded in the generating polynomial (3.5). The central theorems, however, are not proven with full rigor: the MPT rule and the disconnectivity condition are argued through geometric phase-transition and monotonicity pictures, and several limits are asserted rather than derived. The conclusions are plausible and interesting, but the load-bearing inequalities need to be established more carefully before the claims can be regarded as secure.

major comments (5)
  1. [Sec. 2.2] The text states that the multi-entanglement phase transition rule is 'proved' by an iterative hill-climbing argument, but the proof paragraph concludes that the rule 'naturally holds in practice.' No theorem shows that a local maximum of CMI as a function of the 2m endpoints must occur at a configuration where 2m phase transitions occur simultaneously, nor that the coordinate-ascent iteration converges to the global maximum. Because the finite-m maximum values in Section 3 and the generating polynomials in Appendix A rely on this rule and on the zigzag diagram being the global maximizer, this gap is load-bearing. Please either provide a rigorous proof or explicitly state the rule as a conjecture whose verification is limited to the small-m cases.
  2. [Sec. 4.1, Fig. 17] The proof of the weaker disconnectivity condition rests on the assertion that 'when we enlarge the gap, CMI increases,' justified only by the sign pattern of the RT surfaces in Figure 17. No explicit inequality is given, and the proof does not track whether the entanglement wedge of ABE remains fully connected and whether E remains fully disconnected during the splitting. These properties are required for formula (4.1) and the bound (4.15). If the monotonicity claim fails in some phase, a connected E could yield CMI larger than 2 EoSP(A:B), invalidating Eq. (4.17). Please supply a quantitative proof of the monotonicity, or state the disconnectivity condition as an assumption and test it numerically in each phase used.
  3. [Sec. 4.2, Eqs. (4.3)-(4.6)] The derivation of the upper bound I(A:B|E) ≤ 2 S_B relies on the limits in Eq. (4.4), which are asserted rather than derived. In particular, the claim that lim_{m→∞} S_{Gap_{m+1} B Gap_{m+2}} = S_B requires specifying how the intervals E_i are chosen and how the UV cutoff ε is taken relative to m; the paper elsewhere (Eq. (3.14)) uses m ∝ 1/ε. The limiting procedure must be made precise before the saturation statement can be accepted.
  4. [Sec. 3.1, Table 1 and Figs. 12-13] The claim that the zigzag MPT diagram gives the global maximum for every m is verified only for m ≤ 5, and Table 1 presents the m=4 comparison at a single cross ratio (CR=1/3). The generating polynomial (3.5) and the linear/quadratic/quartic divergence rates in Section 3.2 assume this pattern for all m. If a different diagram overtakes the zigzag one at large m, the finite-m maximum values and the specific rates would change (although the existence of divergence may still follow from the explicit construction in Section 4.2). Please either prove the zigzag maximality or explicitly restrict the divergence-rate claims to the zigzag family.
  5. [Sec. 5.2, Fig. 20] The disconnectivity condition for I4 is argued through Figure 20, but at a crucial step the text states that ruling out diagram (1.1) 'seems difficult' and instead uses the purification argument with F. This is an acknowledgment that the proof is incomplete. Because the upper bound (5.11) and the finiteness/divergence dichotomy in Section 5.3 depend on this condition, the argument needs to be completed or the condition must be stated as a conjecture.
minor comments (5)
  1. [Sec. 3.2, Figs. 14-15] The notation is confusing: in the text I denotes exp(CMI/2), while the CMI itself appears in Eq. (1.1); please define the plotted quantity explicitly in each figure caption and use distinct markers for even and odd m in Figure 15, since they obey different polynomials (A.22) and (A.27).
  2. [Sec. 4.3, Eq. (4.16)] The definition of EoSP as min(S_AA') is imprecise: please state precisely what is minimized over (which boundary subregions A' and B', under which homology constraints) and how the condition that AA'BB' is the entire boundary is formalized in higher dimensions.
  3. [Appendix A] The induction of the polynomials from values at l=1,2,3,4 is described clearly, but the statement that the pattern 'has been checked to apply to any non-integer l' is vague; please state the check explicitly (e.g., for which values of l and m it was performed).
  4. [Acknowledgement] There are typos in the Acknowledgement: 'Bart lomiej Czech' and 'theirhis' should be corrected.
  5. [Sec. 1 and Table 2] The divergence table uses notation such as ∫ ∂G_AB log n_E which is not explained; please define ∂G_AB and G_ABC in the text or caption before the table is used in Section 6.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: EoSP is defined independently as a constrained purification entropy, and the bound/saturation argument does not reduce to the target equality.

full rationale

The central claim sup_E I(A:B|E) = 2EoSP(A:B) is not circular. EoSP is defined in Eq. (4.16) as min(S_AA') over boundary-constrained purifications AA'BB', which is an independent information-theoretic quantity; it is not defined as half the CMI upper bound. The paper then proves that the geometric minimization over separating boundary regions D equals this quantity, and Eq. (4.15) establishes I(A:B|E) <= 2S_D by the disconnectivity condition, with saturation argued through explicit m-to-infinity configurations (Sec. 4.2). Thus Eq. (4.17) is a derived statement, not a definitional restatement. No free parameters are fitted to the target results: the cross-ratio formulas and RT-surface calculations are self-contained, and the generating polynomials in Appendix A are pattern extrapolations from small m, which is a correctness/completeness concern rather than circularity. Self-citations [6,7,10] are used as background context and are not load-bearing for the main derivation. The diagrammatic proof of the weaker disconnectivity condition and the informal hill-climbing argument for the MPT rule are non-rigorous in places, but they do not assume the conclusion; they are open mathematical gaps, not circular reasoning.

Assumptions & free parameters 0 free parameters · 7 assumptions · 1 invented entities

No free parameters are fitted to data. The load-bearing assumptions are the RT formula, the disconnectivity conditions, the conjectured patterns for MPT diagrams and generating polynomials, and the correlated large-m/small-epsilon limit. These are clearly stated in the text, though several are not rigorously proven.

assumptions (7)
  • domain assumption Ryu-Takayanagi formula S_A = Area(gamma_A)/(4G_N) for entanglement entropy (Eq. 2.1)
    Basis of all entropy computations in the paper; standard holographic dictionary assumption, not derived here.
  • standard math Holographic mutual information is monogamous, and strong subadditivity holds for RT surfaces (used implicitly in Sections 2.2, 3.2, 5.4)
    Invoked to argue signs of I3 and bounds; standard property in holography proven in prior work [29].
  • domain assumption Only configurations satisfying the disconnectivity condition I(A:E)=I(B:E)=0 need be considered for the maximum of CMI (Section 4.1)
    A weaker form is proven geometrically, but the proof assumes monotonicity of CMI under splitting/enlarging gaps and simultaneous phase transitions; the stronger form used for the entanglement interpretation is assumed.
  • ad hoc to paper The maximum CMI configuration for m intervals is given by the 'zigzag' MPT diagram for every m (Section 3.1, Figures 12, 13)
    Established by enumeration only for m = 2, 3, 4, 5; extrapolated to all m.
  • ad hoc to paper Asymptotic ansatz x = a m + b + c/m for the large-m zero of the generating polynomial (Eq. 3.7)
    The form of the asymptotic expansion is assumed; coefficients are then determined by substitution into the polynomial.
  • ad hoc to paper The generating polynomial P_m(CR) is inferred from polynomial values at integer l = 1,2,3,4 via Pascal-triangle patterns (Appendix A)
    Pattern induction from finitely many exact results; asserted to hold for all m and non-integer l without a proof.
  • domain assumption In higher dimensions, E can be chosen as arbitrarily many thin strips with vanishing gap entropies as m->infinity and epsilon->0 (Section 4.2, Eq. 4.4)
    Assumes the continuum RT formula remains valid and the entropy of sub-cutoff gaps tends to zero; order of limits m ~ 1/epsilon is not rigorously established.
invented entities (1)
  • Entanglement of state-constrained purification (EoSP) independent evidence
    purpose: Defined as min S_AA' with A',B' constrained to be boundary subregions; gives the compact upper bound sup_E I(A:B|E) = 2 EoSP(A:B) (Eq. 4.16-4.17).
    It is defined operationally from boundary regions and RT areas, so it can be computed in any holographic state; it is a new measure rather than a speculative entity.

how reviews work

0 comments
Cite this review

Pith. "Pith review of More on the upper bound of holographic n-partite information." pith.science (2026). https://pith.science/paper/CYXNSGMB

@misc{pith2026241119207,
  author       = {Pith},
  title        = {Pith review of: More on the upper bound of holographic n-partite information},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CYXNSGMB}},
  note         = {Machine review of arXiv:2411.19207}
}
abstract

We show that there exists a huge amount of multipartite entanglement in holography by studying the upper bound for holographic $n$-partite information $I_n$ that $n-1$ fixed boundary subregions participate. We develop methods to find the $n$-th region $E$ that makes $I_n$ reach the upper bound. Through the explicit evaluation, it is shown that $I_n$, an IR term without UV divergence, could diverge when the number of intervals or strips in region $E$ approaches infinity. At this upper bound configuration, we could argue that $I_n$ fully comes from the $n-$partite global quantum entanglement. Our results indicate: fewer-partite entanglement in holography emerges from more-partite entanglement; $n-1$ distant local subregions are highly $n$-partite entangling. Moreover, the relationship between the convexity of a boundary subregion and the multipartite entanglement it participates, and the difference between multipartite entanglement structure in different dimensions are revealed as well.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The Holographic Multi-Entropy Cone

    hep-th 2026-06 accept novelty 7.0 of 10

    Holographic multi-entropy vectors form a rational polyhedral cone; its n=3,4 facets yield seven fundamental multi-entropy inequality orbits, with ordinary HEC facets arising as convex combinations of HMEC facets.

  2. Holographic multipartite entanglement dynamics in AdS$_3$-Vaidya

    hep-th 2026-08 conditional novelty 6.0 of 10

    During a holographic global quench, multipartite entanglement's spatial range first expands then contracts, with higher-party entanglement relaxing later, while some tripartite signals persist or return to vacuum values.

  3. Genuine multi-entropy and holography

    hep-th 2025-02 conditional novelty 6.0 of 10

    A new 'genuine multi-entropy' separates true q-party entanglement from lower-party pieces, and holographic systems are shown to carry O(1/G_N) genuine multipartite entanglement for connected regions.

  4. New insights on mutual information in the island approach to the Page curve

    hep-th 2026-07 conditional novelty 5.0 of 10

    At scrambling time I(B+:B−)=0 forces I(I:R)→∞, interpreted as conservation of geometric correlation, while I(I:R+:R−) is shown always negative via Cauchy-slice identities.

  5. Multipartite entanglement characterizing topological phase transitions in holographic nodal line semimetals

    hep-th 2026-02 conditional novelty 5.0 of 10

    Tripartite entanglement measures in holographic nodal line semimetals vanish at long distance but decay with phase-dependent power laws that jump at the quantum critical point.

Reference graph

Works this paper leans on

59 extracted references · 16 canonical work pages · cited by 5 Pith papers

  1. [1]

    J. M. Maldacena, The Large N limit of superconformal field theories and supergravity , Adv. Theor. Math. Phys. 2 (1998) 231–252, [ hep-th/9711200]

    arXiv 1998

  2. [2]

    Ryu and T

    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from the anti–de sitter space/conformal field theory correspondence, Physical Review Letters 96 (may, 2006)

    work page 2006

  3. [3]

    Maldacena and L

    J. Maldacena and L. Susskind, Cool horizons for entangled black holes , Fortschritte der Physik 61 (aug, 2013) 781–811

    work page 2013

  4. [4]

    Van Raamsdonk, Building up spacetime with quantum entanglement , Gen

    M. Van Raamsdonk, Building up spacetime with quantum entanglement , Gen. Rel. Grav. 42 (2010) 2323–2329, [ 1005.3035]

    arXiv 2010

  5. [5]

    Akers and P

    C. Akers and P. Rath, Entanglement Wedge Cross Sections Require Tripartite Entanglement , JHEP 04 (2020) 208, [ 1911.07852]

    arXiv 2020

  6. [6]

    Ju, W.-B

    X.-X. Ju, W.-B. Pan, Y.-W. Sun and Y. Zhao, Holographic multipartite entanglement from the upper bound of n-partite information, 2411.07790

    arXiv

  7. [7]

    Ju, B.-H

    X.-X. Ju, B.-H. Liu, W.-B. Pan, Y.-W. Sun and Y.-T. Wang, Squashed Entanglement from Generalized Rindler Wedge, 2310.09799

    arXiv

  8. [8]

    J. K. Basak, V. Malvimat and J. Yoon, A New Genuine Multipartite Entanglement Measure: from Qubits to Multiboundary Wormholes , 2411.11961

    arXiv

Show all 59 references
  1. [9]

    Bao and I

    N. Bao and I. F. Halpern, Holographic Inequalities and Entanglement of Purification , JHEP 03 (2018) 006, [ 1710.07643]

    2018 arXiv

  2. [10]

    Ju, T.-Z

    X.-X. Ju, T.-Z. Lai, B.-H. Liu, W.-B. Pan and Y.-W. Sun, Entanglement structures from modified IR geometry, JHEP 07 (2024) 181, [ 2404.02737]

    2024 arXiv

  3. [11]

    V. E. Hubeny, Covariant residual entropy, Journal of High Energy Physics 2014 (sep, 2014) . – 59 –

    2014

  4. [12]

    Czech, P

    B. Czech, P. Hayden, N. Lashkari and B. Swingle, The information theoretic interpretation of the length of a curve , Journal of High Energy Physics 2015 (jun, 2015)

    2015

  5. [13]

    Czech, X

    B. Czech, X. Dong and J. Sully, Holographic Reconstruction of General Bulk Surfaces , JHEP 11 (2014) 015, [ 1406.4889]

    2014 arXiv

  6. [14]

    R. C. Myers, J. Rao and S. Sugishita, Holographic Holes in Higher Dimensions , JHEP 06 (2014) 044, [ 1403.3416]

    2014 arXiv

  7. [15]

    Headrick, R

    M. Headrick, R. C. Myers and J. Wien, Holographic Holes and Differential Entropy , JHEP 10 (2014) 149, [ 1408.4770]

    2014 arXiv

  8. [16]

    Balasubramanian and C

    V. Balasubramanian and C. Rabideau, The dual of non-extremal area: differential entropy in higher dimensions , JHEP 09 (2020) 051, [ 1812.06985]

    2020 arXiv

  9. [17]

    Balasubramanian, B

    V. Balasubramanian, B. D. Chowdhury, B. Czech, J. de Boer and M. P. Heller, Bulk curves from boundary data in holography , Phys. Rev. D 89 (2014) 086004, [ 1310.4204]

    2014 arXiv

  10. [18]

    Czech and L

    B. Czech and L. Lamprou, Holographic definition of points and distances , Phys. Rev. D 90 (2014) 106005, [ 1409.4473]

    2014 arXiv

  11. [19]

    Balasubramanian, B

    V. Balasubramanian, B. Czech, B. D. Chowdhury and J. de Boer, The entropy of a hole in spacetime, JHEP 10 (2013) 220, [ 1305.0856]

    2013 arXiv

  12. [20]

    Ju, W.-B

    X.-X. Ju, W.-B. Pan, Y.-W. Sun and Y.-T. Wang, Generalized Rindler Wedge and Holographic Observer Concordance, 2302.03340

    arXiv

  13. [21]

    Vidal and Y

    G. Vidal and Y. Chen, Entanglement contour, J. Stat. Mech. 2014 (2014) P10011, [1406.1471]

    2014 arXiv

  14. [22]

    Wen, Fine structure in holographic entanglement and entanglement contour , Phys

    Q. Wen, Fine structure in holographic entanglement and entanglement contour , Phys. Rev. D 98 (2018) 106004, [ 1803.05552]

    2018 arXiv

  15. [23]

    Wen, Formulas for Partial Entanglement Entropy , Phys

    Q. Wen, Formulas for Partial Entanglement Entropy , Phys. Rev. Res. 2 (2020) 023170, [1910.10978]

    2020 arXiv

  16. [24]

    Nozaki, T

    M. Nozaki, T. Numasawa and T. Takayanagi, Holographic Local Quenches and Entanglement Density, JHEP 05 (2013) 080, [ 1302.5703]

    2013 arXiv

  17. [25]

    Bhattacharya, V

    J. Bhattacharya, V. E. Hubeny, M. Rangamani and T. Takayanagi, Entanglement density and gravitational thermodynamics , Phys. Rev. D 91 (2015) 106009, [ 1412.5472]

    2015 arXiv

  18. [26]

    Shimaji, T

    T. Shimaji, T. Takayanagi and Z. Wei, Holographic Quantum Circuits from Splitting/Joining Local Quenches, JHEP 03 (2019) 165, [ 1812.01176]

    2019 arXiv

  19. [27]

    Alishahiha, M

    M. Alishahiha, M. R. Mohammadi Mozaffar and M. R. Tanhayi, On the Time Evolution of Holographic n-partite Information, JHEP 09 (2015) 165, [ 1406.7677]

    2015 arXiv

  20. [28]

    Lo Monaco, L

    G. Lo Monaco, L. Innocenti, D. Cilluffo, D. A. Chisholm, S. Lorenzo and G. M. Palma, Quantum scrambling via accessible tripartite information , Quantum Sci. Technol. 8 (2023) 035006, [2305.19334]

    2023 arXiv

  21. [29]

    Hayden, M

    P. Hayden, M. Headrick and A. Maloney, Holographic mutual information is monogamous , Physical Review D 87 (feb, 2013)

    2013

  22. [30]

    Bengtsson and K

    I. Bengtsson and K. Zyczkowski, A brief introduction to multipartite entanglement , arXiv preprint arXiv:1612.07747 (2016)

    2016 arXiv

  23. [31]

    Dutta and T

    S. Dutta and T. Faulkner, A canonical purification for the entanglement wedge cross-section , JHEP 03 (2021) 178, [ 1905.00577]. – 60 –

    2021 arXiv

  24. [32]

    Calabrese, J

    P. Calabrese, J. Cardy and E. Tonni, Entanglement negativity in quantum field theory , Phys. Rev. Lett. 109 (2012) 130502, [ 1206.3092]

    2012 arXiv

  25. [33]

    Kusuki, J

    Y. Kusuki, J. Kudler-Flam and S. Ryu, Derivation of holographic negativity in AdS 3/CFT2, Phys. Rev. Lett. 123 (2019) 131603, [ 1907.07824]

    2019 arXiv

  26. [34]

    Gadde, V

    A. Gadde, V. Krishna and T. Sharma, New multipartite entanglement measure and its holographic dual, Phys. Rev. D 106 (2022) 126001, [ 2206.09723]

    2022 arXiv

  27. [35]

    Penington, M

    G. Penington, M. Walter and F. Witteveen, Fun with replicas: tripartitions in tensor networks and gravity , JHEP 05 (2023) 008, [ 2211.16045]

    2023 arXiv

  28. [36]

    M.-K. Yuan, M. Li and Y. Zhou, Reflected multi-entropy and its holographic dual , 2410.08546

  29. [37]

    N. Bao, S. Nezami, H. Ooguri, B. Stoica, J. Sully and M. Walter, The Holographic Entropy Cone, JHEP 09 (2015) 130, [ 1505.07839]

    2015 arXiv

  30. [38]

    V. E. Hubeny, M. Rangamani and M. Rota, Holographic entropy relations, Fortsch. Phys. 66 (2018) 1800067, [ 1808.07871]

    2018 arXiv

  31. [39]

    V. E. Hubeny, M. Rangamani and M. Rota, The holographic entropy arrangement , Fortsch. Phys. 67 (2019) 1900011, [ 1812.08133]

    2019 arXiv

  32. [40]

    T. He, M. Headrick and V. E. Hubeny, Holographic Entropy Relations Repackaged, JHEP 10 (2019) 118, [ 1905.06985]

    2019 arXiv

  33. [41]

    Hern´ andez Cuenca,Holographic entropy cone for five regions , Phys

    S. Hern´ andez Cuenca,Holographic entropy cone for five regions , Phys. Rev. D 100 (2019) 026004, [1903.09148]

    2019 arXiv

  34. [42]

    T. He, V. E. Hubeny and M. Rangamani, Superbalance of Holographic Entropy Inequalities, JHEP 07 (2020) 245, [ 2002.04558]

    2020 arXiv

  35. [43]

    Avis and S

    D. Avis and S. Hern´ andez-Cuenca,On the foundations and extremal structure of the holographic entropy cone, Discrete Appl. Math. 328 (2023) 16–39, [ 2102.07535]

    2023 arXiv

  36. [44]

    Fadel and S

    M. Fadel and S. Hern´ andez-Cuenca,Symmetrized holographic entropy cone, Phys. Rev. D 105 (2022) 086008, [ 2112.03862]

    2022 arXiv

  37. [45]

    N. Bao, K. Furuya and J. Naskar, Towards a complete classification of holographic entropy inequalities, 2409.17317

    arXiv

  38. [46]

    J. D. Brown and M. Henneaux, Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity , Commun. Math. Phys. 104 (1986) 207–226

    1986

  39. [47]

    Li and A

    K. Li and A. Winter, Squashed entanglement, $$\mathbf {k}$$ k -extendibility, quantum markov chains, and recovery maps , Foundations of Physics 48 (feb, 2018) 910–924

    2018

  40. [48]

    Li and A

    K. Li and A. Winter, Relative entropy and squashed entanglement , Communications in Mathematical Physics 326 (jan, 2014) 63–80

    2014

  41. [49]

    M. M. Wilde, Squashed entanglement and approximate private states , Quantum Information Processing 15 (sep, 2016) 4563–4580

    2016

  42. [50]

    D. Avis, P. Hayden and I. Savov, Distributed compression and multiparty squashed entanglement, Journal of Physics A: Mathematical and Theoretical 41 (mar, 2008) 115301

    2008

  43. [51]

    D. Yang, K. Horodecki, M. Horodecki, P. Horodecki, J. Oppenheim and W. Song, Squashed entanglement for multipartite states and entanglement measures based on the mixed convex – 61 – roof, IEEE Transactions on Information Theory 55 (jul, 2009) 3375–3387

    2009

  44. [52]

    squashed entanglement

    M. Christandl and A. Winter, “squashed entanglement”: An additive entanglement measure , Journal of Mathematical Physics 45 (mar, 2004) 829–840

    2004

  45. [53]

    F. G. S. L. Brand˜ ao, M. Christandl and J. Yard, Faithful squashed entanglement, Communications in Mathematical Physics 306 (aug, 2011) 805–830

    2011

  46. [54]

    Bhattacharjee and J

    A. Bhattacharjee and J. Naskar, Revisiting holographic codes with fractal-like boundary erasures, 2411.02825

    arXiv

  47. [55]

    Takayanagi and K

    T. Takayanagi and K. Umemoto, Entanglement of purification through holographic duality , Nature Phys. 14 (2018) 573–577, [ 1708.09393]

    2018 arXiv

  48. [56]

    Mori and B

    T. Mori and B. Yoshida, Does connected wedge imply distillable entanglement? , 2411.03426

  49. [57]

    P. Jain, N. Jokela, M. Jarvinen and S. Mahapatra, Bounding entanglement wedge cross sections, JHEP 03 (2023) 102, [ 2211.07671]

    2023 arXiv

  50. [58]

    Ju, W.-B

    X.-X. Ju, W.-B. Pan, Y.-W. Sun and Y. Zhao, In progress,

  51. [59]

    Erdmenger, D

    J. Erdmenger, D. Fernandez, M. Flory, E. Megias, A.-K. Straub and P. Witkowski, Time evolution of entanglement for holographic steady state formation , JHEP 10 (2017) 034, [1705.04696]. – 62 –

    2017 arXiv

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.