Open Access

Tripartite Haar Random State Has No Bipartite Entanglement

Zhi Li^1,2,*,†, Takato Mori ^3,2,4,‡, and Beni Yoshida ^2,§

¹National Research Council Canada, Waterloo, Ontario N2L 3W8, Canada
²Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 3W8, Canada
³Department of Physics, Rikkyo University, 3-34-1 Nishi-Ikebukuro, Toshima-ku, Tokyo 171-8501, Japan
⁴Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan

^*Present address: IBM Quantum.
^†Contact author: zli@ibm.com
^‡Contact author: takato.mori@yukawa.kyoto-u.ac.jp
^§Contact author:byoshida@perimeterinstitute.ca

Phys. Rev. X 16, 041013 – Published 13 October, 2026

DOI: https://doi.org/10.1103/d7b9-ctw2

Abstract

We show that no Einstein-Podolsky-Rosen (EPR)–like bipartite entanglement can be distilled from a tripartite Haar random state $| Ψ ⟩_{A B C}$ 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 $ε$ -net argument and concentration of measure. Viewing $| Ψ ⟩_{A B C}$ as a bipartite quantum error-correcting code $C \to A B$ , this implies that neither output subsystem $A$ nor $B$ supports any nontrivial logical operator. We also establish general constraints on the structure of tripartite entanglement in Haar random states, showing that W- or Greenberger-Horne-Zeilinger–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.

Physics Subject Headings (PhySH)

Popular Summary

Haar random quantum states play a fundamental role across modern physics, from statistical physics and quantum information to quantum gravity. A hallmark result is that a bipartite Haar random state is nearly maximally entangled, and its entanglement can essentially be viewed as a collection of Einstein-Podolsky-Rosen pairs, up to local changes of basis. Yet, surprisingly little has been understood about how entanglement is shared among three parts of such a state. Here, we show rigorously that, in a broad regime, no pair of parties can distill entanglement from a tripartite Haar random state using only local operations, contrary to the natural expectation that abundant global entanglement should leave useful entanglement between pairs. We apply this result to the AdS/CFT correspondence and quantum error correction, where random states provide models of complex quantum systems. More broadly, our work reveals that multipartite quantum entanglement can be far more intrinsically complex than previously appreciated.

Article Text

References (58)

M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50, 888 (1994).
P. Hayden and J. Preskill, Black holes as mirrors: Quantum information in random subsystems, J. High Energy Phys. 09 (2007) 120.
M. L. Mehta, Random Matrices (Elsevier, New York, 2004), Vol. 142.
D. N. Page, Average entropy of a subsystem, Phys. Rev. Lett. 71, 1291 (1993).
P. Hosur, X.-L. Qi, D. A. Roberts, and B. Yoshida, Chaos in quantum channels, J. High Energy Phys. 02 (2016) 004.
J. S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Saad, S. H. Shenker et al., Black holes and random matrices, J. High Energy Phys. 05 (2017) 118.
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. 06 (2015) 149.
P. Hayden, S. Nezami, X.-L. Qi, N. Thomas, M. Walter, and Z. Yang, Holographic duality from random tensor networks, J. High Energy Phys. 11 (2016) 009.
A. Ambainis and J. Emerson, Quantum t-designs: t-wise independence in the quantum world, in Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity (CCC’07) (IEEE, New York, 2007), p. 129.
E. Lubkin, Entropy of an n-system from its correlation with a k-reservoir, J. Math. Phys. (N.Y.) 19, 1028 (1978).
S. Lloyd and H. Pagels, Complexity as thermodynamic depth, Ann. Phys. (N.Y.) 188, 186 (1988).
Choosing a Haar random state means that one picks a quantum state uniformly at random from the set of all the $n$ -qubit pure states. In particular, a Haar measure can be characterized as a unique probability distribution that is left and right invariant under any unitary operators.
B. Yoshida and A. Kitaev, Efficient decoding for the Hayden-Preskill protocol, arXiv:1710.03363.
T.-C. Lu and T. Grover, Entanglement transitions as a probe of quasiparticles and quantum thermalization, Phys. Rev. B 102, 235110 (2020).
H. Shapourian, S. Liu, J. Kudler-Flam, and A. Vishwanath, Entanglement negativity spectrum of random mixed states: A diagrammatic approach, PRX Quantum 2, 030347 (2021).
B. Yoshida and I. L. Chuang, Framework for classifying logical operators in stabilizer codes, Phys. Rev. A 81, 052302 (2010).
S. Nezami and M. Walter, Multipartite entanglement in stabilizer tensor networks, Phys. Rev. Lett. 125, 241602 (2020).
G. Smith and D. Leung, Typical entanglement of stabilizer states, Phys. Rev. A 74, 062314 (2006).
M. Freedman and M. Headrick, Bit threads and holographic entanglement, Commun. Math. Phys. 352, 407 (2017).
S. X. Cui, P. Hayden, T. He, M. Headrick, B. Stoica, and M. Walter, Bit threads and holographic monogamy, Commun. Math. Phys. 376, 609 (2019).
C. A. Agón, J. De Boer, and J. F. Pedraza, Geometric aspects of holographic bit threads, J. High Energy Phys. 05 (2019) 075.
J. Harper and M. Headrick, Bit threads and holographic entanglement of purification, J. High Energy Phys. 08 (2019) 101.
N. Bao and G. Suer, Holographic entanglement distillation from the surface state correspondence, J. High Energy Phys. 01 (2024) 091.
C. Akers and P. Rath, Entanglement wedge cross sections require tripartite entanglement, J. High Energy Phys. 04 (2020) 208.
C. Akers, T. Faulkner, S. Lin, and P. Rath, The page curve for reflected entropy, J. High Energy Phys. 06 (2022) 089.
C. Akers, T. Faulkner, S. Lin, and P. Rath, Reflected entropy in random tensor networks. Part II. A topological index from canonical purification, J. High Energy Phys. 01 (2023) 067.
C. Akers, T. Faulkner, S. Lin, and P. Rath, Reflected entropy in random tensor networks. Part III. Triway cuts, J. High Energy Phys. 12 (2024) 209.
The inequalities here are informal, serving as the counterpart of the $n_{R} < n / 2$ ( $R = A$ , $B$ , $C$ ) condition in the results for tripartite Haar random states. Precise conditions may be obtained via Theorem 4 following the discussions below Theorems 1 and 2.
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, 043029 (2009).
C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Mixed state entanglement and quantum error correction, Phys. Rev. A 54, 3824 (1996).
I. Devetak and A. Winter, Distillation of secret key and entanglement from quantum states, Proc. R. Soc. A 461, 207 (2005).
T. Mori and B. Yoshida, Does connected wedge imply distillable entanglement?, J. High Energy Phys. 01 (2026) 125.
P. Hayden, D. W. Leung, and A. Winter, Aspects of generic entanglement, Commun. Math. Phys. 265, 95 (2006).
P. Hayden, D. Leung, P. W. Shor, and A. Winter, Randomizing quantum states: Constructions and applications, Commun. Math. Phys. 250, 371 (2004).
Here and throughout this paper, we consider pure states with phases, so that we work with the sphere $S^{2 d - 1}$ rather than the complex projective space $C P^{d - 1}$ . Two formulations are equivalent due to the global phase-rotation symmetry.
K. Böröczky and G. Wintsche, Covering the sphere by equal spherical balls, in Discrete and Computational Geometry: The Goodman-Pollack Festschrift, edited by B. Aronov, S. Basu, J. Pach, and M. Sharir (Springer, Berlin, Heidelberg, 2003), pp. 235–251.
O. Marchal and J. Arbel, On the sub-Gaussianity of the beta and dirichlet distributions, Electron. Commun. Probab. 22 (2017).
Here, we omit the $O (\log d)$ term, which is unfortunately unavoidable in this proof due to the $p \log p$ term in Sec. 2b. However, it can be avoided if we prove Theorem 1 via the argument used for Theorem 2. Moreover, in the regimes of interest—when $δ = Θ (1)$ and/or $δ ≫ (d_{A} + d_{B}) \sqrt{[(\log d) / d]}$ —the additional $O (\log d)$ term does not affect the asymptotic behavior.
M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, England, 2010).
When constructing $V_{A}$ , we have $d_{1} = d_{A}$ and $d_{2} = 2^{m}$ , and we denote $d_{A}^{'} = d_{2} d_{3}$ . Similar comments apply to $B$ .
M. Ledoux, The Concentration of Measure Phenomenon, Mathematical Surveys and Monographs Vol. 89 (American Mathematical Society, Providence, 2001).
G. Aubrun, J. Jenkinson, and S. J. Szarek, Optimal constants in concentration inequalities on the sphere and in the Gauss space, in High Dimensional Probability X (Birkhäuser, Cham, 2026).
S. J. Szarek, Metric entropy of homogeneous spaces, arXiv:math/9701213.
We may simply consider $ε$ -nets for $U (d^{'})$ , since an isometry can be (nonuniquely) extended to a unitary. Then, Theorem 2 still holds with $2^{2 m}$ replaced by $2^{4 m}$ in the subleading term. To obtain a tighter bound on the subleading term, we may also quotient out another $U (\sqrt{d d^{'}})$ , where $d d^{'}$ is the dimension of the subsystem being traced out. This would give us the dimension $d d^{'} - d^{2}$ , which merely improves a constant factor before $2^{2 m}$ .
S. Bravyi, Upper bounds on entangling rates of bipartite Hamiltonians, Phys. Rev. A 76, 052319 (2007).
While this result concerns LU distillation of pure states supported on smaller tripartite subsystems, it also has implications for the LU distillation of certain target states supported on the original tripartite systems. For example, it implies that the probability for a tripartite state to be LU equivalent to a GHZ state vanishes, since a GHZ state over $A B C$ is the tensor product of two GHZ states over $A_{1} B_{1} C_{1}$ and $A_{2} B_{2} C_{2}$ (the same conclusion also follows from a direct counting argument).
I. H. Kim, B. Shi, K. Kato, and V. V. Albert, Chiral central charge from a single bulk wave function, Phys. Rev. Lett. 128, 176402 (2022).
S. Vardhan, B. Shi, I. H. Kim, and Y. Zou, Chirality, magic, and quantum correlations in multipartite quantum states, SciPost Phys. 20, 066 (2026).
At the leading order in $G_{N}$ , the entanglement wedge of a boundary subregion $R$ is defined as a bulk subregion enclosed by the minimal surface $γ_{R}$ together with the AdS boundary. See Fig. 5 for its illustration.
L. Susskind, ER=EPR, GHZ, and the consistency of quantum measurements, Fortschr. Phys. 64, 72 (2016).
X. Dong, X.-L. Qi, and M. Walter, Holographic entanglement negativity and replica symmetry breaking, J. High Energy Phys. 06 (2021) 024.
A. Almheiri, X. Dong, and D. Harlow, Bulk locality and quantum error correction in AdS/CFT, J. High Energy Phys. 04 (2015) 163.
This can be done by applying the Petz recovery map [54].
H. Barnum and E. Knill, Reversing quantum dynamics with near-optimal quantum and classical fidelity, J. Math. Phys. (N.Y.) 43, 2097 (2002).
C. Akers and G. Penington, Quantum minimal surfaces from quantum error correction, SciPost Phys. 12, 157 (2022).
C. Akers and G. Penington, Leading order corrections to the quantum extremal surface prescription, J. High Energy Phys. 04 (2021) 062.
The reason why we consider a partition into four segments is that it leaves a sufficiently large bulk region (AdS size) where a bulk d.o.f. with $1 / G_{N}$ entropy can be placed without worrying about backreaction. One may replace $C$ with a small (sub-AdS size) black hole or a conical singularity in order to explicitly account for backreaction. We also emphasize that this setup does not require fine-tuning of the sizes of $A$ and $B$ as long as the condition on the two geodesic lengths is met.
T. Mori and B. Yoshida, Baby universe as logical qubits: Information recovery in random encoding, arXiv:2511.20747.

Phys. Rev. X 16, 041013– Published 13 October, 2026

Vol. 16, Iss. 4 — October - December 2026

Received 11 September 2025
Revised 20 May 2026
Accepted 28 July 2026

DOI: https://doi.org/10.1103/d7b9-ctw2

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physical Review X

Reuse & Permissions

Export citation

Export citation

Tripartite Haar Random State Has No Bipartite Entanglement

Abstract

Physics Subject Headings (PhySH)

Popular Summary

Article Text

References (58)

Outline

Information

Filter

Filter

Article Lookup

Enter a citation