Relative entropy of entanglement of Haar random states

Beni Yoshida; Takato Mori

REVIEW 2 major objections 3 minor 31 references

Relative entropy of entanglement of Haar random states

T0 review · 2 major / 3 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read This paper proves that for a reduced state of a tripartite Haar-random pure state, the relative entropy of entanglement is $\log(d_A d_B / \max(d_A,d_B,d_C)) + O(1)$ with probability exponentially close to one, nearly saturating…

desk verdict Likely right answer, but the lower-tail proof relies on a false identity (Eq. 29) equating bipartite and tripartite product overlaps; worth a serious referee, but as written it needs a major revision. read the letter →

arxiv 2608.05274 v1 pith:BBYLZKLT submitted 2026-08-05 quant-ph hep-th

Takato Mori , Beni Yoshida This is my paper · ORCID

classification quant-phhep-th

keywords relativeentropyofentanglementHaarrandomstatesmixed-stateone-sidedclassicalizationSchmidtdephasinggeometricmeasureinducedconcentration

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

This paper asks how entangled a typical mixed state is when it is obtained by tracing out one subsystem of a large pure state drawn uniformly at random (a Haar-random state). The answer it establishes is that the relative entropy of entanglement—the information-theoretic distance to the nearest separable state—is fixed by subsystem dimensions alone: $E_R(\rho_{AB}) = \log \tilde d + O(1)$ with $\tilde d = d_A d_B/\max(d_A,d_B,d_C)$, and this holds with probability exponentially close to one. Since the entanglement of formation and the mutual information are already known to take the same value up to constants, the result shows that the general upper bound $E_R \le \min(E_F, I(A:B))$ is nearly tight for Haar-random states. The paper also exhibits the approximately closest separable state: dephase one side in its Schmidt basis before tracing out the purifier. The significance is that a notoriously hard optimization over all separable states is resolved, up to a constant, by a simple dimension ratio in the random setting.

What carries the argument

The upper bound rests on the one-sided classicalized state $\sigma_{AB}^{A\to cl}=\sum_k q_k |u_k\rangle\langle u_k|_A\otimes\beta_k$, formed by dephasing the smaller side $A$ in the Schmidt basis of $|\psi\rangle$ across $A:BC$ and then tracing out $C$; Lemma 2 computes its relative entropy to $\rho_{AB}$ as $-S(\rho_{AB})+S(\rho_A)+\sum_k q_k S(\beta_k)$, and Haar entropy concentration puts this at $\log\tilde d+O(1)$. The lower bound rests on the maximal product overlap $\Lambda(\rho_{AB})=\max_{\|a\|=\|b\|=1}\langle a,b|\rho_{AB}|a,b\rangle$, which Lemma 3 converts into $E_R\ge -S(\rho_{AB})-\log\Lambda$, combined with Lemma 4's epsilon-net bound $\Lambda(\rho_{AB})\le C(d_A+d_B+d_C)/(d_Ad_Bd_C)$ with high probability. These two mechanisms meet at the dimensionless combination $\tilde d$, which automatically interpolates between the two regimes $d_C\ge d_+$ and $d_C\le d_+$.

What would settle it

Evaluate $E_R$ for a sequence of Haar-random tensors with $d_A=d_B=d_C=d$ by solving the defining minimization over separable states numerically for moderate $d$; Theorem 1 predicts all values lie within an absolute constant of $\log d$ for every $d$, so any spread or offset that grows with $d$ would disprove the $O(1)$ claim.

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Extended reading notes

Core claim

Let $|\psi\rangle_{ABC}$ be Haar-random with $1\le d_C\le d_A d_B$ and set $\rho_{AB}=\mathrm{Tr}_C|\psi\rangle\langle\psi|_{ABC}$. The paper's main theorem states that $E_R(\rho_{AB}) = \log\tilde d + O(1)$ except with probability at most $C e^{-c(d_A+d_B+d_C)}$, where $\tilde d = d_Ad_B/\max(d_A,d_B,d_C)$ and $C,c$ are absolute constants. Equivalently, among the two universal upper bounds $E_R\le E_F$ and $E_R\le I(A:B)$, the relative entropy of entanglement nearly saturates the smaller one throughout the regime $d_C\le D$. The upper tail is proven by constructing an explicit separable state, the one-sided classicalization $\sigma_{AB}^{A\to cl}$, and showing its relative entropy to $\rho_{AB}$ is $\log\tilde d+O(1)$ with high probability; the lower tail combines a general inequality $E_R\ge -S(\rho_{AB})-\log\Lambda(\rho_{AB})$ with an epsilon-net estimate showing the maximal product overlap $\Lambda$ is small for Haar tensors.

Load-bearing premise

The load-bearing premise is that the von Neumann entropy of the reduced state concentrates within an additive constant of $\log d_C$ with probability exponentially close to one; if that error grew like a logarithm of the dimensions, the main formula would degrade by the same amount.

Editorial extensions

If this is right

For any tripartite Haar-random state with $d_C\le d_Ad_B$, the relative entropy of entanglement is determined up to an additive constant by the three subsystem dimensions, so the usual optimization over separable states is unnecessary in this setting.
The general upper bound $E_R\le\min(E_F,I(A:B))$ is saturated to within $O(1)$ for Haar-random mixed states, making the smaller of entanglement of formation and mutual information the correct scale of $E_R$.
In the regime $d_A,d_B\ll d_C$, $E_R\approx\log(d_Ad_B/d_C)$ remains macroscopically large even though one-shot one-way distillable entanglement vanishes asymptotically; these random states contain entanglement inaccessible to such distillation protocols.
The one-sided Schmidt classicalized state is an approximately closest separable state, giving an explicit and simply described optimizer for the relative entropy of entanglement.
As a corollary, the geometric measure of tripartite entanglement—minus the logarithm of the maximum overlap with a product state—obeys $G(\psi)=\log(d_-d_C)+O(1)$ for $d_C\le D$, with no logarithmic correction.

Reading between the lines

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

If the same saturation survives regularization, the optimal asymptotic error exponent for distinguishing $\rho_{AB}^{\otimes n}$ from separable states would also be $\log\tilde d$; the paper leaves open the more likely possibility that the regularized quantity $E_R^\infty$ differs, and pinning down that difference is a natural next target.
The near-optimality of one-sided classicalization is a feature of full Haar randomness; in structured systems such as conformal field theories, random tensor networks, or short random circuits, dephasing in the modular Hamiltonian eigenbasis need not be near-optimal, so $E_R$ may obey a different, state-dependent formula.
The epsilon-net estimate on maximal product overlap is really a statement about random tensors: a random $d_A\times d_B\times d_C$ complex tensor has injective norm $O((d_A+d_B+d_C)/(d_Ad_Bd_C))$ with high probability, which could be reused in other LOCC discrimination or tensor-norm problems.
A direct test outside the Haar measure: sample random states from a finite-depth random circuit or a random tensor network with the same dimensions; if $E_R$ still equals $\log\tilde d+O(1)$, the formula is a symptom of generic entanglement spreading rather than full Haar randomness.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's the thing you should know: the paper's headline result is likely true, but the proof as written has a load-bearing gap in the lower tail. Eq. (29) asserts max_{a,b} ⟨a,b|ρ_AB|a,b⟩ = max_{a,b,c} |⟨a,b,c|ψ⟩|². That is false. The left side is a sum over c: ⟨a,b|ρ_AB|a,b⟩ = Σ_c |⟨a,b,c|ψ⟩|². For |ψ⟩ = d_C^{-1/2} Σ_c |00c⟩, the left side is 1 and the right side is 1/d_C. So the epsilon-net proof in Lemma 4 bounds only the tripartite product overlap, not the bipartite overlap that Lemma 3 needs. An upper bound on each term does not control the sum; in the worst case the sum is d_C times larger. Since Lemma 4 is the only input controlling Λ(ρ_AB), E_R ≥ log d̃ − O(1) is not established as written.

That's a shame, because the rest is good. The upper bound via one-sided classicalization (Lemma 2) is clean, completely general, and correctly gives D(ρ∥σ) = −S(ρ)+S(ρ_A)+Σ q_k S(β_k). The entropy-concentration input from Hayden-Leung-Winter is standard; the untracked constants are a minor annoyance, not a substantive issue. The explicit separable state is a nice contribution, and the discussion of the gap between E_R and distillable entanglement is worth having.

The fix is probably within reach. A direct tail bound on the sum Σ_c |⟨a,b,c|ψ⟩|² for fixed product a,b, followed by a net over a,b, should yield the same O((d_A+d_B+d_C)/(d_A d_B d_C)) upper bound on Λ(ρ_AB). The mean of that sum is 1/(d_A d_B), and fluctuations are smaller. But that argument is absent. The d_C=1 edge case is also treated casually, but that is minor relative to Eq. (29).

Who is this for? People working on random states, entanglement measures, and holographic toy models. The question is natural and the answer matters as a benchmark. The citation pattern looks honest; the self-citations are to directly relevant prior work on random-state entanglement. I'd send it to a serious referee, but my own position is that the lower tail needs repair before the theorem can be trusted. The reader's ACCEPT with high confidence doesn't match the state of the proof; I'd call it a major revision.

Referee Report

2 major / 3 minor

Summary. The paper studies the relative entropy of entanglement E_R of the bipartite reduced state ρ_AB obtained by tracing out C from a tripartite Haar-random pure state |ψ⟩_ABC. Theorem 1 claims that E_R(ρ_AB) = log(D/max(d_C,d_+)) + O(1) with probability exponentially close to one, where D=d_A d_B and d_+=max(d_A,d_B). The upper bound is proved by constructing an explicit separable state via one-sided Schmidt dephasing and using entropy concentration for Haar random states. The lower bound is approached by bounding the maximal overlap of ρ_AB with product states, using an epsilon-net argument, and then invoking E_R ≥ -S(ρ_AB) - log Λ(ρ_AB).

Significance. If the proof is completed, this is a valuable and clean result: E_R is generally very hard to compute, and the paper gives a simple universal formula for Haar-induced mixed states together with an explicit approximately closest separable state. The upper-bound strategy via one-sided classicalization is elegant, and Lemma 2 is a useful general statement. The main theorem would also imply a macroscopic separation between E_R and distillable entanglement in certain regimes, which is of conceptual interest. However, the lower-bound proof as written contains a load-bearing gap in Lemma 4, so the central claim is not yet established by the arguments given.

major comments (2)

[Section 3, Eq. (29)] The identity Λ(ρ_AB) = max_{a,b,c} |⟨a,b,c|ψ⟩|^2 is false. Since ρ_AB = Tr_C |ψ⟩⟨ψ|, one has Λ(ρ_AB) = max_{a,b} ∑_c |⟨a,b,c|ψ⟩|^2, and the maximum of the sum over c is not the maximum of the individual terms. For example, with |ψ⟩ = d_C^{-1/2} ∑_{c=1}^{d_C} |00c⟩, the left-hand side equals 1 while the right-hand side equals 1/d_C. Consequently, the epsilon-net argument in Lemma 4 bounds only the largest single overlap max_{a,b,c}|⟨a,b,c|ψ⟩|^2, and does not control Λ(ρ_AB). Since Lemma 4 is the only input that controls Λ(ρ_AB) in the lower-tail proof, the derivation of Eq. (27) and hence Theorem 1 has a genuine gap. The theorem may still be true, but it requires a different argument, for example concentration of the rank-d_C quadratic form Tr[(|a,b⟩⟨a,b|⊗I_C)|ψ⟩⟨ψ|] followed by a union bound over a,b nets. The same incorrect identification also affects the claimed corollary for the geometric measure in Eq. (49).
[Section 2, Eq. (26)] The upper-tail proof relies on the Hayden-Leung-Winter entropy concentration bound, but the statement in Eq. (26) is presented as a citation with an ad hoc condition d_A^{-1}+d_B^{-1}+d_C^{-1} ≲ (log d_C)^{-2}. This condition is not derived and is not satisfied for small d_C, such as d_C=2, nor for d_A=1 with large d_B. For bounded d_C the needed O(1) statement is trivial, and for d_A=1 the required concentration is still plausible, so the issue is likely repairable. However, as written the proof of the upper tail is not fully supported; please either state the precise HLW theorem being used or provide a self-contained derivation of the needed bound in the regimes covered by Theorem 1.

minor comments (3)

[Section 3, Eq. (42)-(46)] The epsilon-net comparison step assumes the maximum in Eq. (28) is attained by some unit vectors a,b,c. If the maximum is not attained, the same argument works with an ε-approximate maximizer; this should be stated for completeness.
[Section 2, Eq. (14)] The probability bound in Eq. (14) is stated with an exponent d_A d_B + d_A d_C + d_B d_C, which is stronger than needed for Theorem 1. The proof of Eq. (26) appears to give at least an exponent proportional to d_A d_B; since d_C ≤ D, this is already sufficient for the theorem, so the stronger statement should be justified or weakened.
[Section 4, Eq. (49)] The geometric-measure corollary should be revisited after correcting Eq. (29). In particular, the claimed formula G(ψ) = log(d_- d_C)+O(1) does not follow from the true definition of Λ(ρ_AB), and the stated O(1) can fail when d_C is much larger than d_+.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation of E_R for Haar-induced states relies on standard external concentration results and general lemmas; author self-citations are discussion-level only.

full rationale

The argument is not circular. The upper tail is derived by constructing an explicit separable state sigma_A_to_cl and evaluating D(rho_AB || sigma_A_to_cl) via general Lemma 2; the only probabilistic input is the Hayden-Leung-Winter entropy concentration bound (Eq. (26)), an external result, together with the standard upper bound E_R <= min(E_F, I) (Eq. (7), cited to [17,2]) and standard Haar formulas for E_F and I (Eq. (13), cited to [1,18,19]). The lower tail uses a general product-overlap bound (Lemma 3) and an epsilon-net/sub-Gaussian estimate (Lemma 4); no parameter is fitted to the target quantity, and no normalization defines log d_tilde in terms of E_R. Author self-citations [6,7,26] appear only in the Discussion, not as proof ingredients for Theorem 1. The manuscript does, however, contain a genuine mathematical gap that is unrelated to circularity: Eq. (29) asserts Lambda(rho_AB) = max_{a,b,c} |<a,b,c|psi>|^2, but since rho_AB = Tr_C |psi><psi|, Lambda(rho_AB) = max_{a,b} sum_c |<a,b,c|psi>|^2; the sum over c can exceed the largest single term (e.g., |psi> = d_C^{-1/2} sum_c |00c> gives left side 1 and right side 1/d_C). This invalidates the epsilon-net proof of Lemma 4 as written and is a correctness risk, but it is not an input-output equivalence, a fitted prediction, or a self-citation chain, so it does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard concentration-of-measure results for Haar random states and known inequalities for entanglement measures, all cited from the literature. No free parameters are fitted to data, and no new physical entities are introduced. The proof is essentially an original combination of existing tools.

assumptions (5)

standard math Hayden-Leung-Winter entropy concentration: P[S(ρ_AB) < log d_C − K] ≤ C exp(−c(d_A d_B + d_A d_C + d_B d_C)) for d_C ≤ d_A d_B
Cited as Eq. (26) from [1]; used to lower-bound S(ρ_AB) in the upper-tail argument. This is a standard concentration-of-measure result for random induced states.
domain assumption Sub-Gaussian tail for fixed product overlap with a Haar random state: P[|<a,b,c|ψ>| > t/√(d_A d_B d_C)] ≤ e^{−c0 t^2}
Stated in Lemma 4 proof as a standard property of Haar measure. Used to bound the maximum product overlap via an epsilon net.
standard math Epsilon-net covering bound: the unit sphere in C^{d_X} admits an ε-net of size at most (C_0/ε)^{2d_X}
Used in Lemma 4 to discretize the set of product states. Cited to [22]; standard result in high-dimensional geometry.
domain assumption Known asymptotics for Haar random states: E_F(ρ_AB) = log d_− + O(1) and I(A:B) = log(d_A d_B/d_C) + O(1) when d_C ≤ d_A d_B
Cited as Eq. (13) from [1,18,19]. Used to interpret the main theorem as nearly saturating min(E_F,I), but not load-bearing for the proof of Theorem 1.
standard math Upper bound E_R ≤ min(E_F, I(A:B))
Known inequality from [17], stated in Eq. (7). Used for interpretation and consistency, but not used to derive the new lower bound.

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Cite this review

Pith. "Pith review of Relative entropy of entanglement of Haar random states." pith.science (2026). https://pith.science/paper/BBYLZKLT

@misc{pith2026260805274,
  author       = {Pith},
  title        = {Pith review of: Relative entropy of entanglement of Haar random states},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BBYLZKLT}},
  note         = {Machine review of arXiv:2608.05274}
}

abstract

We determine the relative entropy of entanglement of a bipartite mixed state $\rho_{AB}$ obtained by tracing out one subsystem of a tripartite Haar-random pure state $|\psi\rangle_{ABC}$, finding $E_R(\rho_{AB})=\log\frac{d_Ad_B}{\max(d_A,d_B,d_C)}+O(1)$. Equivalently, the relative entropy of entanglement nearly saturates the smaller of the entanglement of formation $E_F(\rho_{AB})$ and the mutual information $I(A:B)$. The upper bound is achieved by an explicit separable state obtained through one-sided Schmidt dephasing, which is therefore approximately closest.

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [1]

Hayden, D

P. Hayden, D. W. Leung, and A. Winter,Aspects of Generic Entanglement,Commun. Math. Phys.265(2006), no. 1 95–117, [quant-ph/0407049]

arXiv 2006

[2] [2]

Horodecki, P

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki,Quantum Entanglement, Rev. Mod. Phys.81(2009) 865–942, [quant-ph/0702225]

arXiv 2009

[3] [3]

Entanglement thresholds for random induced states

G. Aubrun, S. J. Szarek, and D. Ye,Entanglement Thresholds for Random Induced States, Commun. Pure Appl. Math.67(2014), no. 1 129–171, [arXiv:1106.2264]

work page Pith review arXiv 2014

[4] [4]

Lu and T

T.-C. Lu and T. Grover,Entanglement transitions as a probe of quasiparticles and quantum thermalization,Phys. Rev. B102(2020), no. 23 235110, [arXiv:2008.11727]

arXiv 2020

[5] [5]

Shapourian, S

H. Shapourian, S. Liu, J. Kudler-Flam, and A. Vishwanath,Entanglement Negativity Spectrum of Random Mixed States: A Diagrammatic Approach,PRXQuantum2(2021), no. 3 030347, [arXiv:2011.01277]

arXiv 2021

[6] [6]

Mori and B

T. Mori and B. Yoshida,Does connected wedge imply distillable entanglement?,JHEP01 (2026) 125, [arXiv:2411.03426]

arXiv 2026

[7] [7]

Z. Li, T. Mori, and B. Yoshida,Tripartite Haar random state has no bipartite entanglement,arXiv:2502.04437

arXiv

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Vedral, M

V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight,Quantifying entanglement,Phys. Rev. Lett.78(1997) 2275–2279, [quant-ph/9702027]

arXiv 1997