Nonlocality distillation can outperform entanglement distillation

Jibran Rashid; Peter H{\o}yer; Razeen ud Din

REVIEW 4 major objections 5 minor 28 references

Nonlocality distillation can outperform entanglement distillation

T0 review · 4 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read The paper proves that for two or three copies of a weakly entangled state, directly distilling nonlocal correlations yields a higher CHSH violation than first distilling a Bell state, even though the latter uses communication.

desk verdict Plausible finite-copy comparison, but the proof has load-bearing gaps—especially the unproven n=3 attainability claim. read the letter →

arxiv 2603.00940 v1 pith:X6ZDDVXD submitted 2026-03-01 quant-ph

Peter H{o}yer , Jibran Rashid , Razeen ud Din This is my paper · ORCID

classification quant-ph

keywords nonlocalitydistillationentanglementCHSHinequalitycollectivemeasurementsquantumresourceestimationsparsestatepreparationdevice-independentQKDsmall-copyregime

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 compares two ways to amplify the nonlocality of a small number of shared partially entangled states. It establishes that for two or three copies of a specific two-qubit pure state, a protocol that directly maximizes the CHSH violation via collective measurements can achieve a higher CHSH value than the optimal entanglement-distillation protocol, which first concentrates a Bell state and then measures it—even though entanglement distillation is allowed one-way communication. The advantage is shown for explicit parameter ranges and disappears by four copies, where entanglement distillation is always better. The comparison extends to a family of mixed states, and a resource estimate indicates the direct nonlocality approach also needs fewer qubits, gates, and runtime.

What carries the argument

The comparison is driven by two quantities: V_ED, the CHSH value after optimal entanglement distillation, which equals 2√2 times the Bell-state success probability plus 2 times the failure probability; and V_ND, the best known CHSH value from collective measurements on n copies, computed from the coefficients λ_i of the n-copy bipartite decomposition via the formula V_ND = 2 Σ_{n=1}^{⌊d/2⌋} √( (λ_{2n−1}^2 + λ_{2n}^2)^2 + 4λ_{2n}^2 λ_{2n−1}^2 ). For mixed states, the proof bounds the operator norm of the effective measurement N by writing N^2 = 4 I⊗K̃ + (2A)⊗L̃ and bounding K̃ and L̃, yielding ∥N∥ ≤ 2√(1+(1−2p)^2) for n=1,2 and a higher-degree polynomial bound for n=3.

What would settle it

Explicitly construct or find the measurements that achieve the n=3 collective CHSH bound and demonstrate a CHSH value above V_ED at some p in [0.746,0.904]. If no such measurements exist—that is, the bound is not attainable—the n=3 claim fails. A direct experiment on two copies at p=0.7 would test the n=2 claim: the measured CHSH should exceed V_ED for p in [0.5,0.85].

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Extended reading notes

Core claim

For the state |Ψ⟩ = √p|ψ⟩ + √(1−p)|ϕ⟩ (a superposition of two maximally entangled states), the paper proves that the best known nonlocality-distillation value for n=2 and n=3 copies exceeds the CHSH value of optimal entanglement distillation in the intervals p∈[0.5,0.85] and p∈[0.746,0.904]. The entanglement value is a weighted average: with success probability p_succ the protocol yields a Bell state with CHSH 2√2, otherwise a product state with CHSH 2. The nonlocality value comes from a collective-measurement bound expressed in terms of the coefficients of the n-copy bipartite decomposition. For the mixed state ρ = p|ψ⟩⟨ψ| + (1−p)|ϕ⟩⟨ϕ|, the paper proves the optimal nonlocality-distillation

Load-bearing premise

The n=3 pure-state advantage assumes the collective-measurement CHSH bound in Eq. (3) is attainable, but the paper states the corresponding measurements have only been verified numerically; if the bound is merely an upper bound, the n=3 branch of Theorem 1 is not established.

Editorial extensions

If this is right

For n=2 and n=3 copies of the pure state, nonlocality distillation achieves a higher CHSH value than optimal entanglement distillation for concrete parameter ranges (p∈[0.5,0.85] and p∈[0.746,0.904] respectively).
At n=4 copies, the comparison flips: entanglement distillation is always better, so the advantage is a small-copy phenomenon.
For mixed states, the optimal nonlocality distillation protocol for two copies attains the same CHSH value as for the corresponding pure state, so the pure-state advantage carries over to mixed states with the same form.
Nonlocality distillation requires fewer quantum resources than entanglement distillation in the n=2 pure-state case—about 3 fewer logical qubits, roughly 8 times less logical depth, and about 3 times fewer T states according to the resource estimate.
The advantage exists even though entanglement distillation uses one-way communication; the communicated bit carries no input information, so the comparison isolates the value of the distillation strategy itself.

Reading between the lines

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

If the n=3 nonlocality-distillation bound is proven tight with explicit measurements (currently only numerical), the advantage window might extend or shift; until then, the n=3 branch of the theorem rests on an unverified attainment assumption.
For device-independent QKD, higher CHSH values from nonlocality distillation could improve key rates or tolerable noise, but only if the distilled correlations have the right secrecy structure—nonlocal correlations do not automatically yield secrecy.
The resource estimate suggests a concrete experimental test: with two copies of |Ψ⟩, directly performing the collective CHSH measurements should outperform the full distillation-plus-measurement circuit on near-term hardware, provided a fair accounting of communication overhead.
The result points to a general principle: when the target is a Bell-inequality violation, the optimal distillation strategy should be optimized for that observable, not for Bell-state fidelity; entanglement concentration is a different goal.

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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

Colleague, here's my read. The paper asks the right question: for a few copies of a bipartite pure state, is it better to distill entanglement first or just perform collective CHSH measurements? The claim that nonlocality distillation wins for n=2 and 3 is plausible and worth knowing. I think the n=2 case may survive contact with a careful proof, and the mixed-state bound in Eq. (6) is a useful addition. The resource comparison is rough but honest. The soft spots are real. The n=3 branch of Theorem 1 depends on Eq. (3) being an attainable CHSH value, but the authors themselves say the measurements have only been verified numerically. That is an upper bound, not an attainment claim, so the 'can attain' is unsupported. This is the main problem. Theorem 2 says 'optimal' but the proof only bounds the norm of a specific operator; it does not show that the bound is achieved or optimal. The proof details also have inconsistencies: the n=3 interval in the proof does not match the theorem statement, and a careful reader will want to check the displayed inequality for n=2 at p=0.5—I suspect a typo, but as printed it gives an impossible CHSH value. The reader's note about Eq. (2) at p=1/2 is a misreading: that state is product, not a Bell state, so zero there is not necessarily wrong. But the formula itself has issues at p=1, where it gives a success probability above one. So the technical core needs serious cleanup. If the n=2 case is correct after cleanup, the paper is a solid finite-copy comparison. The n=3 claim needs either a proof of attainability or a softening to 'the best known nonlocality-distillation protocol achieves...' which would still be new. I would send it to peer review, because the question is significant and a referee can help separate typos from actual contributions. As written, I wouldn't cite it yet. Bring it to reading group if people want to untangle the proof.

Referee Report

4 major / 5 minor

Summary. The paper compares nonlocality distillation with entanglement distillation for the task of maximizing CHSH violation from n copies of a bipartite state. For pure states of the form |Ψ⟩ = √p|ψ⟩ + √(1−p)|ϕ⟩, Theorem 1 claims that for n=2 and n=3 nonlocality distillation achieves a higher CHSH value than the optimal entanglement distillation protocol. Theorem 2 makes a similar claim for a mixed-state family ρ = p|ψ⟩⟨ψ| + (1−p)|ϕ⟩⟨ϕ| for n=2, and provides a (non-tight) bound for n=3. The paper also includes a resource-estimation comparison for the n=2 pure-state case, finding that nonlocality distillation requires fewer logical qubits, lower logical depth, and fewer T states than entanglement distillation.

Significance. The central question — whether nonlocality distillation can beat entanglement distillation when only a few copies are available — is interesting and conceptually relevant to the distinction between entanglement and nonlocality as resources. If the n=2 pure-state claim holds, it provides a concrete small-copy regime where directly measuring collective nonlocality outperforms first concentrating a Bell state, even though the latter uses communication. The paper also makes an explicit attempt at resource estimation, which is a useful addition for practical relevance. However, the manuscript as written contains several load-bearing algebraic and logical gaps that prevent the claims from being accepted as proven; the n=3 branch relies on an unproven attainability statement, and the comparison is made against a protocol that is optimal for Bell-state probability, not necessarily for CHSH.

major comments (4)

[Section III, proof of Theorem 1] The displayed formulas for V_ED are algebraically incorrect and produce unphysical values. For n=2, the formula V_ED = 2 + 4(2√2−2)t + 2((√p+√(1−p))/√2)^4 gives V_ED=4 at p=0.5, while the state is then product and the true CHSH value is 2. For n=3, the analogous formula gives values above 2√2 in the stated intervals (e.g., at p=0.9 it exceeds 3.7). These errors invalidate the proof of Theorem 1 as written. The authors must re-derive the expressions for p_succ and V_ED and re-check the claimed inequalities.
[Section II-B and Theorem 1 n=3 branch] The n=3 branch of Theorem 1 uses V_ND from Eq. (3) as an attained value, but the paper states in Sec. II-B that for n=3 the corresponding measurements 'have only been verified numerically.' If Eq. (3) is only an upper bound, the inequality V_ED ≤ V_ND only shows that the optimal entanglement-distillation value does not exceed a possibly unreachable upper bound; it does not prove that nonlocality distillation attains the higher value. To support the claim, the authors must either prove attainability of Eq. (3) for n=3 or rephrase the theorem as a comparison of upper bounds.
[Section II-A and Theorem 1] The comparison is made against the Lo-Popescu protocol, which is optimal for the probability of distilling a Bell state, but the task here is to maximize CHSH. It is not shown that the Lo-Popescu protocol is optimal for CHSH, nor is it justified that the failure state is product (which is assumed in the V_ED formula). Without an optimality proof for CHSH, the claim that nonlocality distillation 'outperforms the optimal entanglement distillation protocol' is not established. A counterexample or a proof that any successful Bell-distillation protocol has failure product and is CHSH-optimal is needed.
[Section III, proof of Theorem 2] The proof bounds ||N^2||, which yields an upper bound on the CHSH value, but the theorem claims that the optimal nonlocality distillation protocol 'can attain' a higher CHSH value for n=2. The proof does not explicitly show that the bound is attained for n=2; it only observes that the bound equals the n=1 value. For n=3, the text admits the bound is not known to be tight, so the theorem's statement for n=3 is only an upper bound. Additionally, the comparison to 'optimal entanglement distillation' for mixed states is missing entirely: no mixed-state V_ED is defined or computed. The n=2 L̃ bound also contains a duplicated sum (the same four C terms appear twice with different coefficients), which appears to be a typo and obscures the derivation.

minor comments (5)

[Eq. (2)] The notation p_succ = min_{1≤r≤2} (2/r)(λ_{2−r+1}+...+λ_8) is unclear for general n; the upper limit of the sum should depend on the dimension of the state. Please clarify the formula and specify the Schmidt dimension for each n.
[Section II-C] There is a punctuation error: 'where, The CHSH value is then given by' should be a new sentence. Also, the derivation of the CHSH value for ρ would benefit from explicitly writing the four correlators and the resulting expression.
[Section III, Theorem 2 proof] The bound ||L1|| ≤ 8|a−b| + 24|c−b| + 8|d−b| is stated without derivation. Please provide the eigenvalue or norm argument, or cite a standard result, so the reader can verify the step.
[Figure 1 and proof of Theorem 1] The intervals stated in the text are inconsistent: for n=3 the proof says the inequality holds for p∈[0.5,0.746] and then concludes it proves the result for p∈[0.746,0.904]. The figure labels should match the text.
[Section III-A] The resource comparison is based on a specific circuit implementation and a particular sparse-state-preparation algorithm. Please state clearly that these are implementation-dependent estimates, not general complexity-theoretic claims.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central comparisons use external Lo-Popescu and Liang-Doherty results; the stated n=3 attainability gaps are missing proofs, not circular reductions.

full rationale

The paper's core derivations are not circular. Theorem 1 compares the Lo-Popescu entanglement-distillation value V_ED (Eq. 2, from Ref. [5]) with the Liang-Doherty collective-measurement bound V_ND (Eq. 3, from Ref. [8]). Both are external, parameter-free expressions in the Schmidt coefficients of the same state, and the proof is an explicit algebraic inequality. Theorem 2 is a direct operator-norm estimate on the operator N defined in Eq. 7; no fitted constants appear and the argument does not assume the theorem it claims to prove. The only self-references are [11] (background on box-world nonlocality distillation) and [23] (a GitHub implementation used for resource estimation); neither is load-bearing for the main results. The manuscript itself flags real limitations: for n=3, "the corresponding measurements have only been verified numerically" (Sec. II-B); for the mixed-state n=3 bound, "it is not known to be tight" (Sec. II-C) and "It is open whether the bound for n=3 is tight" (Sec. III). These weaken the attainability of the n=3 branch of Theorem 1 and the n=3 expression in Theorem 2, but they are omitted proofs or unverified attainability claims, not circular redefinitions or fitted-parameter predictions. Since no prediction reduces to its input by construction and no uniqueness theorem is imported from the authors' own prior work, the circularity score is low.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted. The argument rests on standard CHSH/Tsirelson bounds and on prior distillation results adopted as input. No new physical entities are introduced.

assumptions (4)

standard math Tsirelson bound: CHSH of any quantum state is at most 2√2, and classically correlated states give at most 2.
Used throughout as the benchmark for Bell-state and product-state CHSH values (Sections II-A and III).
domain assumption Lo-Popescu is the optimal entanglement-distillation protocol and Eq. (2) gives its optimal success probability.
Borrowed from Ref. [5]; the paper does not re-derive optimality for the CHSH objective rather than the distillation fidelity objective.
domain assumption The Liang-Doherty bound, Eq. (3), is an attainable CHSH value for collective measurements on n copies.
Load-bearing for Theorem 1; for n=3 the paper admits only numerical verification (Sec. II-B).
domain assumption A failed entanglement-distillation attempt leaves a product state with CHSH value 2.
Used to write V_ED = 2√2 p_succ + 2(1 - p_succ) in Section II-A.

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Pith. "Pith review of Nonlocality distillation can outperform entanglement distillation." pith.science (2026). https://pith.science/paper/X6ZDDVXD

@misc{pith2026260300940,
  author       = {Pith},
  title        = {Pith review of: Nonlocality distillation can outperform entanglement distillation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X6ZDDVXD}},
  note         = {Machine review of arXiv:2603.00940}
}

read the original abstract

Given the goal of maximizing CHSH violation, we compare the optimal strategies of entanglement and nonlocality distillation. In the limit of the number of copies of the shared state, entanglement distillation is guaranteed to work by generating a Bell state. For a small number of copies of the state, we show that nonlocality distillation can achieve a higher CHSH value, even though optimal entanglement distillation requires communication. Nonlocality distillation not only outperforms entanglement distillation but also demonstrates superior resource efficiency across multiple metrics for quantum resource estimation.

Figures

Figures reproduced from arXiv: 2603.00940 by the authors.

**Figure 2.** Nonlocality distillation bounds for one, two and three copies of the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 4.** The optimal measurement circuit for entanglement distillation. The [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: The circuit for optimal nonlocality distillation measurements given [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 3.** Figure 3: Circuit for preparing two copies of the initial quantum state [PITH_FULL_IMAGE:figures/full_fig_p005_3.png]

**Figure 6.** Figure 6: The entanglement distillation circuit that generates a Bell state [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [1]

Entanglement and nonlocality are different resources,

N. Brunner, N. Gisin, and V . Scarani, “Entanglement and nonlocality are different resources,”NJP, vol. 7, p. 88, 2005

2005

[2] [2]

Bell nonlocality,

N. Brunner, D. Cavalcanti, S. Pironio, V . Scarani, and S. Wehner, “Bell nonlocality,”RMP, vol. 86, pp. 419–478, 2014

2014

[3] [3]

Concentrat- ing partial entanglement by local operations,

C. Bennett, H. Bernstein, S. Popescu, and B. Schumacher, “Concentrat- ing partial entanglement by local operations,”PRA, vol. 53, p. 2046, 1996

[4] [4]

Purification of noisy entanglement and faithful teleportation via noisy channels,

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,”PRL, vol. 76, p. 722, 1996

1996

[5] [5]

Concentrating entanglement by local actions: Beyond mean values,

H. Lo and S. Popescu, “Concentrating entanglement by local actions: Beyond mean values,”PRA, vol. 63, p. 022301, 2001

2001

[6] [6]

Collective tests for quantum nonlocality,

A. Peres, “Collective tests for quantum nonlocality,”PRA, vol. 54, p. 4, 1996

1996

[7] [7]

Proposed experiment to test local hidden–variable theories,

J. Clauser, M. Horne, A. Shimony, and R. Holt, “Proposed experiment to test local hidden–variable theories,”PRL, vol. 23, p. 880, 1969

1969

[8] [8]

Better Bell-inequality violation by collective measurements,

Y . Liang and A. Doherty, “Better Bell-inequality violation by collective measurements,”PRA, vol. 73, p. 052116, 2006

2006