Entanglement Detection Beyond Local Bound with Coarse Calibrated measurements

Liang-Liang Sun; Sixia Yu; Yong-Shun Song

arxiv: 2508.03525 · v2 · submitted 2025-08-05 · 🪐 quant-ph

Entanglement Detection Beyond Local Bound with Coarse Calibrated measurements

Liang-Liang Sun , Yong-Shun Song , Sixia Yu This is my paper

Pith reviewed 2026-05-19 00:45 UTC · model grok-4.3

classification 🪐 quant-ph

keywords entanglement detectionBell inequalitiesMermin-Klyshko inequalitiescoarse calibrationnonlocal correlationsqubit systemsmultipartite entanglement

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The pith

Coarse calibration of measurements solely by their ability to generate nonlocal correlations strengthens Bell inequalities to detect entanglement beyond the local bound in qubit systems.

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

The paper develops a method to strengthen Bell inequalities for detecting entanglement in systems of qubits. It does this by assuming measurement devices are only coarsely calibrated based on whether they can produce nonlocal correlations, without needing full quantum descriptions. This leads to trade-offs between bounds for separable states and general states using structure functions. Such strengthening works for bipartite, tripartite, and larger n-partite systems, allowing detection of different types of multipartite entanglement. The approach also extends to scenarios where some measurements are better characterized using hierarchy of tests.

Core claim

By considering measurement devices that are coarsely calibrated only by their ability to generate nonlocal correlations without requiring precise quantum characterization, trade-offs between upper bounds for separable states and general states can be derived in terms of structure functions for Mermin-Klyshko-Bell inequalities. This optimizes entanglement detection in bipartite and tripartite systems and strengthens n-partite inequalities for states with a diversity of entanglement structures such as genuine multipartite entanglement. For general Bell scenarios with some measurements characterized, entanglement can also be detected with some local correlations by exploiting hierarchy of tests

What carries the argument

Structure functions that encode trade-offs between upper bounds on the Bell expression for separable states versus general states, obtained from the coarse calibration condition on measurement devices.

If this is right

In bipartite and tripartite systems the strengthened bounds enable more efficient entanglement detection than the standard local bound.
For n-partite systems the inequalities can identify states exhibiting genuine multipartite entanglement and other entanglement structures.
In general scenarios with some measurements characterized, detection remains possible even when some observed correlations are local.

Where Pith is reading between the lines

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

The same coarse-calibration trade-off construction could be applied to other families of Bell inequalities to improve their entanglement sensitivity.
Experiments might achieve reliable entanglement verification with less precise device alignment or calibration procedures.
The method could connect to device-independent tasks where only nonlocal correlation generation is certified.

Load-bearing premise

The only information available about the measurement devices is their capacity to produce nonlocal correlations, without precise quantum characterization or assumptions on Hilbert space dimension or noise.

What would settle it

An experiment in which a separable state produces a Bell value exceeding the strengthened separable-state bound from the trade-off while staying below the general bound, or a set of devices that generate nonlocal correlations yet violate the predicted trade-off relation.

Figures

Figures reproduced from arXiv: 2508.03525 by Liang-Liang Sun, Sixia Yu, Yong-Shun Song.

read the original abstract

Bell's test, initially devised to distinguish quantum theory from local hidden variable models through {violations of local bounds}, is also a common tool for detecting entanglement. For this purpose, one can assume the quantum description of devices and use available information to strengthen the bound for separable states, which may go beyond the local bound, enabling more efficient entanglement detection. Here we present a systematic approach for strengthening Bell inequalities for qubit systems, using Mermin-Klyshko-Bell inequalities as examples, by considering measurement devices that are coarsely calibrated only by their ability to generate nonlocal correlations without requiring precise quantum characterization. In the case of bipartite and tripartite systems, we derive trade-offs between upper bounds for separable states and general states in terms of structure functions to optimize the entanglement detection. We then strengthen $n$-partite Bell inequalities for the detection of states exhibiting a diversity of entanglement structures such as genuine multipartite entanglement. For general Bell scenarios with some measurements characterized, we demonstrate that entanglement can also be detected with some local correlations by exploiting the Navascu\'{e}s-Pironio-Ac\'{i}n hierarchy of tests.

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.

Desk Editor's Note private letter to a colleague

The paper gives a concrete way to beat the local bound on Mermin-Klyshko inequalities for qubit entanglement detection by feeding in only coarse calibration data on the devices' ability to produce nonlocal correlations.

read the letter

The central point is that coarse calibration—knowing only that the measurements can generate nonlocal correlations—lets you derive tighter upper bounds on separable states than the usual local bound for Mermin-Klyshko-Bell inequalities. The authors build this into explicit trade-offs via structure functions for the bipartite and tripartite qubit cases, then extend the construction to n-partite systems for genuine multipartite entanglement and add an NPA fallback when some devices have partial characterization.

Referee Report

2 major / 3 minor

Summary. The manuscript presents a systematic method to strengthen Mermin-Klyshko-Bell inequalities for entanglement detection in qubit systems. By assuming measurement devices are coarsely calibrated solely through their capacity to produce nonlocal correlations (without full quantum characterization or dimension assumptions), the authors derive explicit trade-offs between upper bounds on separable states and general states using structure functions for the bipartite and tripartite cases. These are extended to n-partite scenarios to detect genuine multipartite entanglement and other entanglement structures, with an NPA-hierarchy fallback for hybrid scenarios where some devices are characterized.

Significance. If the derivations are correct, the work provides a practical route to entanglement witnesses that exceed the local bound with minimal device information, which is relevant for experimental setups where precise tomography is impractical. The structure-function approach for trade-offs and the n-partite extension add a concrete tool for Bell-based detection, while the NPA component offers a bridge to semidefinite programming methods. The qubit restriction is clearly stated and the coarse-calibration premise is applied consistently.

major comments (2)

[§3] §3 (bipartite trade-off): the claimed parameter-free character of the structure-function bound for separable states appears to rest on the specific choice of the nonlocal-correlation threshold; a brief check against the definition in Eq. (8) would confirm whether the bound remains strictly stronger than the local bound for all admissible coarse-calibration values or reduces in limiting cases.
[§5] §5 (n-partite extension): the generalization from tripartite to arbitrary n for genuine multipartite entanglement detection relies on an inductive construction of the structure functions; it is not immediately clear whether the resulting witness remains valid when the underlying state is not fully symmetric, which could limit the scope of the diversity-of-entanglement claim.

minor comments (3)

[Abstract] The abstract introduces 'structure functions' without a one-sentence definition; adding a brief parenthetical would improve accessibility for readers outside the immediate subfield.
[Figure 2] Figure 2 (tripartite bound plot): the axes labels and the shaded region corresponding to the separable-state bound are legible but the legend does not explicitly state the coarse-calibration parameter value used; a short caption addition would remove ambiguity.
[NPA section] The NPA fallback section cites the original Navascués-Pironio-Acín paper but does not reference subsequent tightening results (e.g., the 2019–2022 hierarchy improvements); a single additional citation would strengthen the comparison.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the constructive comments, which help clarify the scope and robustness of our results. We address each major comment below and have incorporated clarifications into the revised manuscript.

read point-by-point responses

Referee: [§3] §3 (bipartite trade-off): the claimed parameter-free character of the structure-function bound for separable states appears to rest on the specific choice of the nonlocal-correlation threshold; a brief check against the definition in Eq. (8) would confirm whether the bound remains strictly stronger than the local bound for all admissible coarse-calibration values or reduces in limiting cases.

Authors: We thank the referee for highlighting this point. The structure-function bound for separable states is derived directly from the definition in Eq. (8) and does not introduce additional free parameters beyond the coarse-calibration threshold itself. Substituting the admissible range of the nonlocal-correlation threshold into the bound shows that it remains strictly stronger than the local bound for all non-trivial values; equality holds only in the limiting case where the threshold approaches the local bound. We have added an explicit verification paragraph in the revised §3 to make this transparent. revision: yes
Referee: [§5] §5 (n-partite extension): the generalization from tripartite to arbitrary n for genuine multipartite entanglement detection relies on an inductive construction of the structure functions; it is not immediately clear whether the resulting witness remains valid when the underlying state is not fully symmetric, which could limit the scope of the diversity-of-entanglement claim.

Authors: The inductive construction of the structure functions is formulated in terms of the observed correlation structure and does not assume symmetry of the underlying state. The resulting witness bounds any state whose entanglement structure is at most k-partite (for appropriate k) and therefore detects genuine multipartite entanglement for both symmetric and asymmetric states. The diversity-of-entanglement claim follows from applying the same construction to different partitions. We have inserted a clarifying sentence in the revised §5 stating that the validity holds without symmetry assumptions. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives trade-offs and structure functions directly from the stated premise of coarse calibration (devices characterized only by capacity for nonlocal correlations) applied to Mermin-Klyshko inequalities. No step reduces a claimed prediction or bound to a fitted parameter defined by the target result, nor relies on load-bearing self-citations whose content is unverified or circular. The NPA hierarchy invocation is an external standard tool for the characterized-device fallback case. The qubit restriction and entanglement-structure extensions are explicitly constructed from the coarse-calibration assumption without importing uniqueness theorems or ansatzes from prior author work in a self-referential manner. The central claims remain independent of the outputs they produce.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the premise that coarse calibration information (ability to generate nonlocal correlations) is sufficient to derive tighter separable bounds without full device characterization. No explicit free parameters or invented entities are named in the abstract.

axioms (1)

domain assumption Measurement devices can be characterized solely by their capacity to produce nonlocal correlations without precise quantum description.
Invoked in the abstract as the basis for strengthening bounds for separable states.

pith-pipeline@v0.9.0 · 5726 in / 1241 out tokens · 31845 ms · 2026-05-19T00:45:07.721630+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

trade-offs between upper bounds for separable states and general states in terms of structure functions... f(U) ≡ sqrt(2 + 2 sqrt(1-(U²/4-1)²))
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Mermin-Klyshko polynomials... Fn cos t + F'n sin t ≤ 2 sqrt(δn + ϵn sin 2t)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages · 1 internal anchor

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work page internal anchor Pith review Pith/arXiv arXiv 2025
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DeMille, Phys

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As the upper boundβt(ab|c) is independent of the underlying state, it holds true also for a generalAB|C state

+ 2ab(C2 0 − C2 1) ≤ q 2(1 + ¯a¯b)(1 + c cos 2u3) + 2ab¯c sin 2u3 ≤ r 2(1 + ¯a¯b) + 2 q (c + ¯a¯bc)2 + a2b2¯c2 = r 2(1 + ¯a¯b) + 2 q (1 + ¯a¯b)2c2 3 + a2b2¯c2 = X ± q 1 + ¯a¯b) ± (¯a + ¯b)¯c := β(21) M (ab|c), where we have used Eq.(24) and Schwarz inequality in the first inequality, and single-qubit uncertainty relation in the second inequality and Schwa...

work page

[1] [1]

for our purposes, with some measurements are char- acterized. Given that qubit systems are a standard plat- form for generating quantum entanglement — a central arXiv:2508.03525v1 [quant-ph] 5 Aug 2025 2 resource in quantum information science — our protocol offers a significant enhancement in entanglement detec- tion across a wide range of applications. ...

work page internal anchor Pith review Pith/arXiv arXiv 2025

[2] [2]

H. J. Briegel, D. E. Browne, W. Dür, R. Raussendorf, and M. V. den Nest, Nat. Phys5, 19 (2009), URLhttps: //doi.org/10.1038%2Fnphys1157

work page 2009

[3] [3]

A. K. Ekert, Phys. Rev. Lett. 67, 661 (1991), URL https://link.aps.org/doi/10.1103/PhysRevLett.67. 661

work page doi:10.1103/physrevlett.67 1991

[4] [4]

Giovannetti, S

V. Giovannetti, S. Lloyd, and L. Maccone, Science 306, 1330 (2004), URL https://doi.org/10.1126% 2Fscience.1104149

work page 2004

[5] [5]

Brunner, D

N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Rev. Mod. Phys.86, 419 (2014), URLhttps: //link.aps.org/doi/10.1103/RevModPhys.86.419

work page doi:10.1103/revmodphys.86.419 2014

[6] [6]

A. S. Sørensen and K. Mølmer, Phys. Rev. Lett. 86, 4431 (2001), URL https://link.aps.org/doi/10. 1103/PhysRevLett.86.4431

work page 2001

[7] [7]

Lücke, J

B. Lücke, J. Peise, G. Vitagliano, J. Arlt, L. San- tos, G. Tóth, and C. Klempt, Phys. Rev. Lett. 112, 155304 (2014), URL https://link.aps.org/doi/10. 1103/PhysRevLett.112.155304

work page 2014

[8] [8]

Yu, Z.-B

S. Yu, Z.-B. Chen, J.-W. Pan, and Y.-D. Zhang, Phys. Rev. Lett. 90, 080401 (2003), URLhttps://link.aps. org/doi/10.1103/PhysRevLett.90.080401

work page doi:10.1103/physrevlett.90.080401 2003

[9] [9]

Gühne and G

O. Gühne and G. Tóth, Phys. Rep. 474, 1 (2009), ISSN 0370-1573, URL https://www.sciencedirect. com/science/article/pii/S0370157309000623

work page 2009

[10] [10]

R.F.Werner, Phys.Rev.A 40, 4277(1989), URLhttps: //link.aps.org/doi/10.1103/PhysRevA.40.4277

work page doi:10.1103/physreva.40.4277 1989

[11] [11]

H. M. Wiseman, S. J. Jones, and A. C. Doherty, Phys. Rev. Lett. 98, 140402 (2007), URLhttps://link.aps. org/doi/10.1103/PhysRevLett.98.140402

work page doi:10.1103/physrevlett.98.140402 2007

[12] [12]

Uffink and M

J. Uffink and M. Seevinck, Physics Letters A 372, 1205–1212 (2008), ISSN 0375-9601, URL http://dx. doi.org/10.1016/j.physleta.2007.09.033

work page doi:10.1016/j.physleta.2007.09.033 2008

[13] [13]

Lougovski and S

P. Lougovski and S. J. van Enk, Phys. Rev. A 80, 034302 (2009), URL https://link.aps.org/doi/10. 1103/PhysRevA.80.034302

work page 2009

[14] [14]

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. 23, 880 (1969), URL https://link. aps.org/doi/10.1103/PhysRevLett.23.880

work page doi:10.1103/physrevlett.23.880 1969

[15] [15]

Rudolph, Quantum Information Processing 4, 219 (2005), ISSN 1573-1332, URL https://doi.org/10

O. Rudolph, Quantum Information Processing 4, 219 (2005), ISSN 1573-1332, URL https://doi.org/10. 1007/s11128-005-5664-1

work page 2005

[16] [16]

Chen and L.-A

K. Chen and L.-A. Wu, Quantum Information and Com- putation 3 (2002)

work page 2002

[17] [17]

Moroder and O

T. Moroder and O. Gittsovich, Physical Review A 85 (2012), ISSN 1094-1622, URL http://dx.doi.org/10. 1103/PhysRevA.85.032301

work page 2012

[18] [18]

Shahandeh, M

F. Shahandeh, M. J. W. Hall, and T. C. Ralph, Phys. Rev. Lett.118, 150505 (2017), URLhttps://link.aps. org/doi/10.1103/PhysRevLett.118.150505

work page doi:10.1103/physrevlett.118.150505 2017

[19] [19]

N. D. Mermin, Phys. Rev. Lett.65, 1838 (1990), URL https://link.aps.org/doi/10.1103/PhysRevLett.65. 1838

work page doi:10.1103/physrevlett.65 1990

[20] [20]

Klyshko, Physics Letters A 172, 399 (1993), ISSN 0375-9601, URL https://www.sciencedirect

D. Klyshko, Physics Letters A 172, 399 (1993), ISSN 0375-9601, URL https://www.sciencedirect. com/science/article/pii/037596019390228R

work page arXiv 1993

[21] [21]

DeMille, Phys

J. Uffink, Phys. Rev. Lett. 88, 230406 (2002), URL https://link.aps.org/doi/10.1103/PhysRevLett.88. 230406

work page doi:10.1103/physrevlett.88 2002

[22] [22]

Navascués, S

M. Navascués, S. Pironio, and A. Acín, Phys. Rev. Lett. 98, 010401 (2007), URL https://link.aps.org/doi/ 10.1103/PhysRevLett.98.010401. 7 A. The decomposition of Bell’s value for general measurements Let us consider cases where all the measurements are general and show that a Bell value can be equiva- lently decomposed into ones using projection measureme...

work page doi:10.1103/physrevlett.98.010401 2007

[23] [23]

As the upper boundβt(ab|c) is independent of the underlying state, it holds true also for a generalAB|C state

+ 2ab(C2 0 − C2 1) ≤ q 2(1 + ¯a¯b)(1 + c cos 2u3) + 2ab¯c sin 2u3 ≤ r 2(1 + ¯a¯b) + 2 q (c + ¯a¯bc)2 + a2b2¯c2 = r 2(1 + ¯a¯b) + 2 q (1 + ¯a¯b)2c2 3 + a2b2¯c2 = X ± q 1 + ¯a¯b) ± (¯a + ¯b)¯c := β(21) M (ab|c), where we have used Eq.(24) and Schwarz inequality in the first inequality, and single-qubit uncertainty relation in the second inequality and Schwa...

work page