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arxiv: 2502.04437 · v3 · pith:MD5SE767new · submitted 2025-02-06 · 🪐 quant-ph · hep-th

Tripartite Haar random state has no bipartite entanglement

Zhi Li , Takato Mori , Beni Yoshida This is my paper

Pith reviewed 2026-05-23 03:30 UTC · model grok-4.3

classification 🪐 quant-ph hep-th
keywords Haar random statetripartite entanglementbipartite entanglement distillationquantum error-correcting codeAdS/CFTconcentration of measurevolume argumentlocal operations
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The pith

Tripartite Haar random states have no distillable EPR-like bipartite entanglement when each subsystem is smaller than half the total system.

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

The paper establishes that local unitaries and local operations cannot distill EPR-like pairs from any two subsystems of a tripartite Haar random state when each subsystem holds fewer than half the qubits. The probability of sampling a state that allows such distillation at a fixed fidelity threshold is bounded above by a doubly exponential function of the qubit number. A reader would care because this shows that typical random multipartite states support only intricate forms of entanglement that resist reduction to simple pairwise links. The same volume-plus-concentration argument also rules out distillation of W or GHZ states and the presence of nontrivial global symmetries.

Core claim

We show that no EPR-like bipartite entanglement can be distilled from a tripartite Haar random state |Ψ⟩_ABC by local unitaries or local operations when each subsystem A, B, or C has fewer than half of the total qubits. We derive an upper bound on the probability of sampling a state with EPR-like entanglement at a given EPR fidelity tolerance, showing a doubly-exponential suppression in the number of qubits. The proof relies on a simple volume argument supplemented by an ε-net argument and concentration of measure. Viewing |Ψ⟩_ABC as a bipartite quantum error-correcting code C→AB implies that neither output subsystem A nor B supports any non-trivial logical operator. We also establish that W

What carries the argument

The volume argument with ε-net and concentration of measure that bounds the set of states possessing high EPR fidelity.

If this is right

  • Neither subsystem A nor B supports non-trivial logical operators when the state is viewed as a bipartite code C→AB.
  • W-like or GHZ-like entanglement cannot be distilled by local operations.
  • Tripartite Haar random states admit no nontrivial global symmetries.
  • In the AdS/CFT setting a connected entanglement wedge does not imply the existence of distillable bipartite entanglement.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same suppression may extend to other multipartite-to-bipartite distillation tasks beyond EPR pairs.
  • Small-system numerical checks could verify whether the doubly exponential bound already appears at modest qubit numbers.
  • The result suggests that random entanglement resources for quantum networks will typically require multipartite decoding protocols rather than pairwise extraction.

Load-bearing premise

The volume argument supplemented by an ε-net argument and concentration of measure applies to bound the probability for states with high EPR fidelity when each subsystem has fewer than half the total qubits.

What would settle it

Numerical sampling of Haar-random tripartite states on 6-12 qubits with each subsystem smaller than half the total, followed by an exhaustive search over local unitaries to check whether the fraction of states yielding EPR fidelity above a fixed threshold exceeds the doubly exponential upper bound.

Figures

Figures reproduced from arXiv: 2502.04437 by Beni Yoshida, Takato Mori, Zhi Li.

Figure 1
Figure 1. Figure 1: a) Bipartite Haar random states with |A| < |B|. EPR pairs can be distilled by applying a local unitary UB. b) Tripartite Haar random states. Can EPR pairs be distilled by local unitary rotations or local operations? as minimal toy models of a quantum black hole [4–6], and Haar random tensor networks serve as toy models obeying the Ryu-Takayanagi formula at the AdS scale at the leading order [7, 8]. Also, t… view at source ↗
Figure 2
Figure 2. Figure 2: A spherical cap and upper/lower bounds on its surface area on a unit sphere [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A schematic picture of an ϵ-net. Any point in the Hilbert space has some ϵ-neighbor point in the ϵ-net. constant α > 0 and an ϵ-net Mϵ(R) of R such that |Mϵ(R)| ≤ α p log p Area(R+2ϵ ) Area(Bϵ) . (18) Here, Bϵ ⊆ S p−1 is the ϵ spherical ball (in the Euclidean distance), R+2ϵ = {x ∈ S p−1 | dist(x, R) ≤ 2ϵ} and Mϵ(R) ⊆ S p−1 does not need to be contained in R. Proof. It is proven in [33] that S p−1 can be c… view at source ↗
Figure 4
Figure 4. Figure 4: Connected entanglement wedge. Here, the minimal area surface of [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Entanglement wedge reconstruction. a) When [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
read the original abstract

We show that no EPR-like bipartite entanglement can be distilled from a tripartite Haar random state $|\Psi\rangle_{ABC}$ by local unitaries or local operations when each subsystem $A$, $B$, or $C$ has fewer than half of the total qubits. Specifically, we derive an upper bound on the probability of sampling a state with EPR-like entanglement at a given EPR fidelity tolerance, showing a doubly-exponential suppression in the number of qubits. Our proof relies on a simple volume argument supplemented by an $\epsilon$-net argument and concentration of measure. Viewing $|\Psi\rangle_{ABC}$ as a bipartite quantum error-correcting code $C\to AB$, this implies that neither output subsystem $A$ nor $B$ supports any non-trivial logical operator. We also establish general constraints on the structure of tripartite entanglement in Haar random states, showing that W- or GHZ-like entanglement cannot be distilled and that nontrivial global symmetries are absent. Finally, we discuss a physical interpretation in the AdS/CFT correspondence, indicating that a connected entanglement wedge does not necessarily imply bipartite entanglement, contrary to a previous belief.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper claims that a Haar-random tripartite pure state |Ψ⟩_ABC admits no distillable EPR-like bipartite entanglement (at fixed fidelity) via local unitaries or LOCC whenever each subsystem dimension is strictly less than half the total number of qubits; the probability of sampling such a state is bounded above by a doubly-exponentially small quantity in the total qubit number. The argument proceeds by a volume estimate on the set of states that are close (after local unitaries) to an EPR ⊗ junk form, supplemented by an ε-net covering and concentration-of-measure tail bounds. Additional claims include the absence of distillable W- or GHZ-type tripartite entanglement, the lack of nontrivial global symmetries, and the consequence that the state viewed as a quantum error-correcting code from C to AB supports no nontrivial logical operators supported on A or on B. A brief discussion of implications for the AdS/CFT entanglement wedge is included.

Significance. If the central bound holds, the result supplies a clean, quantitative demonstration that typical tripartite entanglement is incompatible with bipartite distillability under the stated subsystem-size condition. The proof relies on standard, parameter-free tools (volume ratios, ε-nets, and Lévy concentration) that have been successfully applied to similar atypicality statements in quantum information; the doubly-exponential suppression is a strong, falsifiable prediction. The QEC reformulation and the holographic remark are natural corollaries that may stimulate further work on the structure of random states.

minor comments (2)
  1. The precise scaling (e.g., whether the bound holds for subsystem dimension ≤ 2^{n/2-1} or a slightly weaker threshold) should be stated explicitly in the first paragraph of the introduction and in the statement of the main theorem.
  2. Notation for the total Hilbert-space dimension (2^n) and the individual subsystem dimensions should be fixed once at the beginning rather than re-introduced in each section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment and recommendation to accept the manuscript. Their summary correctly reflects the central claims regarding the absence of distillable bipartite EPR entanglement in typical tripartite Haar-random states under the stated dimension condition, along with the supporting volume and concentration arguments, the QEC implications, and the holographic remarks.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central claim is established via a volume argument on the Haar measure, combined with an ε-net covering and standard concentration-of-measure bounds to show doubly-exponential suppression of states admitting local-unitary or LOCC distillation of an EPR pair when each subsystem is smaller than half the total dimension. These are external, parameter-free mathematical tools whose validity does not depend on any fitted quantities, self-definitions, or prior results by the same authors; the subsystem-size condition simply places the relevant submanifolds inside the regime where the bounds apply. No load-bearing step reduces by construction to the paper's own inputs or to a self-citation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard assumptions in quantum information and geometric probability; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Haar measure on the unitary group for the tripartite Hilbert space
    The states are sampled from the Haar distribution as stated in the abstract.
  • standard math Concentration of measure phenomenon in high-dimensional spaces
    Used to bound the probability as per the proof description in the abstract.

pith-pipeline@v0.9.0 · 5719 in / 1438 out tokens · 47072 ms · 2026-05-23T03:30:51.881401+00:00 · methodology

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Reference graph

Works this paper leans on

43 extracted references · 43 canonical work pages · 2 internal anchors

  1. [1]

    Srednicki, Chaos and Quantum Thermalization , Phys

    M. Srednicki, Chaos and Quantum Thermalization , Phys. Rev. E 50 (1994)

    work page 1994

  2. [2]

    Hayden and J

    P. Hayden and J. Preskill, Black holes as mirrors: Quantum information in random subsystems, JHEP 09 (2007) 120

    work page 2007

  3. [3]

    Mehta, Random matrices, vol

    M.L. Mehta, Random matrices, vol. 142, Elsevier (2004). 25

    work page 2004

  4. [4]

    Page, Average entropy of a subsystem , Phys

    D.N. Page, Average entropy of a subsystem , Phys. Rev. Lett. 71 (1993) 1291

    work page 1993

  5. [5]

    Hosur, X.-L

    P. Hosur, X.-L. Qi, D.A. Roberts and B. Yoshida, Chaos in quantum channels , JHEP 02 (2016) 004

    work page 2016

  6. [6]

    Cotler, G

    J.S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Saad, S.H. Shenker et al., Black Holes and Random Matrices, JHEP 05 (2017) 118

    work page 2017

  7. [7]

    Pastawski, B

    F. Pastawski, B. Yoshida, D. Harlow and J. Preskill, Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence , JHEP 06 (2015) 149

    work page 2015

  8. [8]

    Hayden, S

    P. Hayden, S. Nezami, X.-L. Qi, N. Thomas, M. Walter and Z. Yang, Holographic duality from random tensor networks , JHEP 11 (2016) 009

    work page 2016

  9. [9]

    Ambainis and J

    A. Ambainis and J. Emerson, Quantum t-designs: t-wise independence in the quantum world , in Twenty-Second Annual IEEE Conference on Computational Complexity (CCC’07) , p. 129, (2007)

    work page 2007

  10. [10]

    Lubkin, Entropy of an n-system from its correlation with a k-reservoir , J

    E. Lubkin, Entropy of an n-system from its correlation with a k-reservoir , J. Math. Phys. 19 (1978) 1028

    work page 1978

  11. [11]

    Lloyd and H

    S. Lloyd and H. Pagels, Complexity as thermodynamic depth , Ann. Phys. 188 (1988) 186

    work page 1988

  12. [12]

    Efficient decoding for the Hayden-Preskill protocol

    B. Yoshida and A. Kitaev, Efficient decoding for the Hayden-Preskill protocol , arXiv:1710.03363

    work page internal anchor Pith review Pith/arXiv arXiv

  13. [13]

    Lu and T

    T.-C. Lu and T. Grover, Entanglement transitions as a probe of quasiparticles and quantum thermalization, Phys. Rev. B 102 (2020) 235110

    work page 2020

  14. [14]

    Shapourian, S

    H. Shapourian, S. Liu, J. Kudler-Flam and A. Vishwanath, Entanglement Negativity Spectrum of Random Mixed States: A Diagrammatic Approach , PRX Quantum 2 (2021) 030347

    work page 2021

  15. [15]

    Yoshida and I.L

    B. Yoshida and I.L. Chuang, Framework for classifying logical operators in stabilizer codes , Phys. Rev. A 81 (2010) 052302

    work page 2010

  16. [16]

    Nezami and M

    S. Nezami and M. Walter, Multipartite Entanglement in Stabilizer Tensor Networks , Phys. Rev. Lett. 125 (2020) 241602

    work page 2020

  17. [17]

    Smith and D

    G. Smith and D. Leung, Typical entanglement of stabilizer states , Physical Review A—Atomic, Molecular, and Optical Physics 74 (2006) 062314

    work page 2006

  18. [18]

    Freedman and M

    M. Freedman and M. Headrick, Bit threads and holographic entanglement , Comm. Math. Phys. 352 (2017) 407

    work page 2017

  19. [19]

    S.X. Cui, P. Hayden, T. He, M. Headrick, B. Stoica and M. Walter, Bit Threads and Holographic Monogamy, Comm. Math. Phys. 376 (2019) 609

    work page 2019

  20. [20]

    Ag´ on, J

    C.A. Ag´ on, J. De Boer and J.F. Pedraza, Geometric Aspects of Holographic Bit Threads , JHEP 05 (2019) 075. 26

    work page 2019

  21. [21]

    Harper and M

    J. Harper and M. Headrick, Bit threads and holographic entanglement of purification , JHEP 08 (2019) 101

    work page 2019

  22. [22]

    Bao and G

    N. Bao and G. Suer, Holographic entanglement distillation from the surface state correspondence, JHEP 01 (2024) 091

    work page 2024

  23. [23]

    Akers and P

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

    work page 2020

  24. [24]

    Akers, T

    C. Akers, T. Faulkner, S. Lin and P. Rath, The Page curve for reflected entropy , JHEP 06 (2022) 089

    work page 2022

  25. [25]

    Akers, T

    C. Akers, T. Faulkner, S. Lin and P. Rath, Reflected entropy in random tensor networks. Part II. A topological index from canonical purification , JHEP 01 (2023) 067

    work page 2023

  26. [26]

    Akers, T

    C. Akers, T. Faulkner, S. Lin and P. Rath, Reflected entropy in random tensor networks. Part III. Triway cuts , JHEP 12 (2024) 209

    work page 2024

  27. [27]

    Bravyi and B

    S. Bravyi and B. Terhal, A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes , New J. Phys. 11 (2009) 043029

    work page 2009

  28. [28]

    Mori and B

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

    work page arXiv

  29. [29]

    Bennett, D.P

    C.H. Bennett, D.P. DiVincenzo, J.A. Smolin and W.K. Wootters, Mixed state entanglement and quantum error correction, Phys. Rev. A 54 (1996) 3824

    work page 1996

  30. [30]

    Devetak and A

    I. Devetak and A. Winter, Distillation of secret key and entanglement from quantum states , Proceedings of the Royal Society A: Mathematical, Physical and engineering sciences 461 (2005) 207

    work page 2005

  31. [31]

    Hayden, D.W

    P. Hayden, D.W. Leung and A. Winter, Aspects of generic entanglement , Comm. Math. Phys. 265 (2006) 95

    work page 2006

  32. [32]

    Hayden, D

    P. Hayden, D. Leung, P.W. Shor and A. Winter, Randomizing quantum states: Constructions and applications, Comm. Math. Phys. 250 (2004) 371

    work page 2004

  33. [33]

    B¨ or¨ oczky and G

    K. B¨ or¨ oczky and G. Wintsche,Covering the sphere by equal spherical balls , in Discrete and Computational Geometry: The Goodman-Pollack Festschrift , B. Aronov, S. Basu, J. Pach and M. Sharir, eds., (Berlin, Heidelberg), pp. 235–251, Springer Berlin Heidelberg (2003), DOI

    work page 2003

  34. [34]

    Marchal and J

    O. Marchal and J. Arbel, On the sub-gaussianity of the beta and dirichlet distributions , Electronic Communications in Probability 22 (2017)

    work page 2017

  35. [35]

    Nielsen and I.L

    M.A. Nielsen and I.L. Chuang, Quantum computation and quantum information , Cambridge university press (2010)

    work page 2010

  36. [36]

    Ledoux, The Concentration of Measure Phenomenon , no

    M. Ledoux, The Concentration of Measure Phenomenon , no. 89 in Mathematical surveys and monographs, American Mathematical Society (2001). 27

    work page 2001

  37. [37]

    Aubrun, J

    G. Aubrun, J. Jenkinson and S.J. Szarek, Optimal constants in concentration inequalities on the sphere and in the gauss space , arXiv preprint arXiv:2406.13581 (2024)

    work page arXiv 2024

  38. [38]

    Metric Entropy of Homogeneous Spaces

    S.J. Szarek, Metric entropy of homogeneous spaces , arXiv preprint math/9701213 (1997)

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  39. [39]

    Susskind, ER=EPR, GHZ, and the consistency of quantum measurements , Fortsch

    L. Susskind, ER=EPR, GHZ, and the consistency of quantum measurements , Fortsch. Phys. 64 (2016) 72

    work page 2016

  40. [40]

    Dong, X.-L

    X. Dong, X.-L. Qi and M. Walter, Holographic entanglement negativity and replica symmetry breaking, JHEP 06 (2021) 024

    work page 2021

  41. [41]

    Almheiri, X

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

    work page 2015

  42. [42]

    Barnum and E

    H. Barnum and E. Knill, Reversing quantum dynamics with near-optimal quantum and classical fidelity, J. Math. Phys. 43 (2002) 2097

    work page 2002

  43. [43]

    Akers and G

    C. Akers and G. Penington, Leading order corrections to the quantum extremal surface prescription, JHEP 04 (2021) 062 [ 2008.03319]. 28

    work page arXiv 2021