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arxiv: 2204.11726 · v4 · submitted 2022-04-25 · 🪐 quant-ph

Exponentially decreasing critical detection efficiency for any Bell inequality

Nikolai Miklin , Anubhav Chaturvedi , Mohamed Bourennane , Marcin Paw{\l}owski , Ad\'an Cabello This is my paper

Pith reviewed 2026-05-24 12:19 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Bell inequalitydetection loopholecritical detection efficiencypenalized N-productCHSH inequalityquantum nonlocalityentanglementlocal hidden variables
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The pith

Penalized N-product Bell inequalities reduce the critical detection efficiency exponentially with the number of parallel tests N for any original inequality.

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

The paper proposes entangling particles in N orthogonal subspaces to run N Bell tests in parallel. It introduces penalized N-product Bell inequalities whose local hidden variable bound is the Nth power of the original and which close the simultaneous measurement loophole. For these inequalities the critical detection efficiency needed to close the detection loophole falls exponentially as N increases. This applies to any starting Bell inequality and is demonstrated explicitly for the CHSH case. A reader cares because it offers a route to loophole-free Bell tests with detectors of much lower efficiency than previously required.

Core claim

For the penalized N-product Bell inequalities, obtained by taking the Nth power of any Bell inequality while adding penalty terms to close the simultaneous measurement loophole, the critical detection efficiency decays exponentially with N. This is shown by calculating the threshold efficiency required for the quantum value to exceed the local bound in the presence of detection losses.

What carries the argument

Penalized N-product (PNP) Bell inequalities, which enforce the Nth power of the original local bound while penalizing simultaneous undetected events to close the measurement loophole.

If this is right

  • For any Bell inequality, sufficiently large N makes the critical detection efficiency arbitrarily small.
  • The maximum value for local hidden-variable theories of the PNP inequality is exactly the Nth power of the original.
  • The method closes both the detection efficiency loophole and the simultaneous measurement loophole.
  • Explicit calculations for the CHSH inequality confirm the exponential decay.

Where Pith is reading between the lines

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

  • This approach could enable loophole-free experiments using detectors with efficiencies below 50 percent by choosing large enough N.
  • Implementation would require creating entanglement across multiple degrees of freedom such as polarization and path.
  • Similar product constructions might reduce thresholds for other Bell-related tasks like randomness certification.

Load-bearing premise

That the parallel tests in orthogonal subspaces remain independent and that the penalty terms do not allow new local hidden variable strategies to evade the bound.

What would settle it

An experiment measuring the violation of a PNP inequality for N=2 or N=3 and finding that the observed critical efficiency does not follow the predicted exponential reduction compared to the single-test case.

Figures

Figures reproduced from arXiv: 2204.11726 by Ad\'an Cabello, Anubhav Chaturvedi, Marcin Paw{\l}owski, Mohamed Bourennane, Nikolai Miklin.

Figure 1
Figure 1. Figure 1: FIG. 1. (Solid line) Critical detection efficiency [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
read the original abstract

We address the problem of closing the detection efficiency loophole in Bell experiments, which is crucial for real-world applications. Every Bell inequality has a critical detection efficiency $\eta$ that must be surpassed to avoid the detection loophole. Here, we propose a general method for reducing the critical detection efficiency of any Bell inequality to arbitrary low values. This is accomplished by entangling two particles in $N$ orthogonal subspaces (e.g., $N$ degrees of freedom) and conducting $N$ Bell tests in parallel. Furthermore, the proposed method is based on the introduction of penalized $N$-product (PNP) Bell inequalities, for which the so-called simultaneous measurement loophole is closed, and the maximum value for local hidden-variable theories is simply the $N$th power of the one of the Bell inequality initially considered. We show that, for the PNP Bell inequalities, the critical detection efficiency decays exponentially with $N$. The strength of our method is illustrated with a detailed study of the PNP Bell inequalities resulting from the Clauser-Horne-Shimony-Holt inequality.

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

2 major / 2 minor

Summary. The paper claims a general construction, based on entangling two particles across N orthogonal subspaces and performing N parallel Bell tests, that produces penalized N-product (PNP) Bell inequalities whose local-hidden-variable (LHV) bound is exactly the Nth power of the original inequality's bound. The penalty term is asserted to close the simultaneous-measurement loophole, so that the critical detection efficiency η required to rule out LHV models decays exponentially with N. The construction is illustrated in detail for the CHSH inequality.

Significance. If the central claims hold, the result supplies a systematic, parameter-free route to arbitrarily low critical detection efficiencies for arbitrary Bell inequalities. This would be a substantial technical contribution to the detection-loophole literature and directly relevant to device-independent protocols.

major comments (2)
  1. [Definition of PNP inequalities and LHV bound proof] The assertion that the LHV value of the PNP inequality equals exactly the Nth power of the single-copy bound (and therefore that the effective violation ratio improves exponentially) is load-bearing for the exponential-decay claim. The derivation must explicitly rule out LHV strategies that correlate non-detection events across the N subspaces; the penalty term's sufficiency under such cross-subspace correlations is not demonstrated in the provided text.
  2. [Closure of simultaneous measurement loophole] The treatment of the simultaneous-measurement loophole assumes that the N subspace tests remain independent once the penalty is applied. Under the detection-efficiency loophole, an LHV model can in principle use a joint non-detection strategy across subspaces that evades the penalty while exceeding the claimed Nth-power bound; no explicit bound or counter-example analysis addressing this possibility appears in the manuscript.
minor comments (2)
  1. Notation for the penalty term and the orthogonal-subspace projectors should be introduced with explicit definitions before the main theorems.
  2. The numerical study of the CHSH-derived PNP inequalities would benefit from an explicit table listing the computed critical η versus N together with the corresponding violation ratios.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting two points that require clarification. We address each major comment below and will incorporate the requested explicit analyses into a revised version.

read point-by-point responses

  1. Referee: [Definition of PNP inequalities and LHV bound proof] The assertion that the LHV value of the PNP inequality equals exactly the Nth power of the single-copy bound (and therefore that the effective violation ratio improves exponentially) is load-bearing for the exponential-decay claim. The derivation must explicitly rule out LHV strategies that correlate non-detection events across the N subspaces; the penalty term's sufficiency under such cross-subspace correlations is not demonstrated in the provided text.

    Authors: We agree that an explicit demonstration ruling out cross-subspace correlations in non-detection events would strengthen the presentation. The PNP construction defines the penalty term over the joint detection pattern across all N subspaces; any LHV strategy that correlates non-detections must still respect the per-subspace local bounds while incurring the full penalty whenever at least one subspace fails to detect. This structure prevents the overall value from exceeding the Nth power of the single-copy bound. In the revision we will add a dedicated lemma that enumerates the possible joint non-detection assignments and shows that none can produce a super-multiplicative violation. revision: yes

  2. Referee: [Closure of simultaneous measurement loophole] The treatment of the simultaneous-measurement loophole assumes that the N subspace tests remain independent once the penalty is applied. Under the detection-efficiency loophole, an LHV model can in principle use a joint non-detection strategy across subspaces that evades the penalty while exceeding the claimed Nth-power bound; no explicit bound or counter-example analysis addressing this possibility appears in the manuscript.

    Authors: The manuscript asserts closure via the penalty term, but we acknowledge that an explicit counter-example analysis or bound under joint non-detection strategies is not spelled out. Because the penalty is triggered by any incomplete detection pattern across the N subspaces, a joint strategy that leaves one or more subspaces undetected necessarily reduces the effective contribution from those subspaces below the single-copy quantum value, keeping the total below the Nth-power threshold. We will add a short subsection that considers the most general joint non-detection map and derives the resulting upper bound, confirming it cannot surpass the claimed LHV value. revision: yes

Circularity Check

0 steps flagged

No circularity; exponential decay follows algebraically from PNP product construction

full rationale

The paper constructs PNP inequalities by taking the product of N copies of a base Bell inequality plus a penalization term that enforces simultaneous measurement. It then shows (via direct expansion) that the LHV maximum is exactly the Nth power of the base value. The critical detection efficiency η_crit is obtained by solving for the efficiency at which the quantum value exceeds this LHV bound; because the violation ratio grows as (violation)^N while the LHV bound grows only as (bound)^N, η_crit decays exponentially in N. This reduction is a direct algebraic consequence of the product definition and does not rely on fitting, renaming, or load-bearing self-citations. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction assumes that orthogonal subspaces allow independent parallel tests and that the product structure plus penalty term closes the simultaneous measurement loophole without new loopholes; no free parameters or invented entities are mentioned in the abstract.

axioms (2)
  • domain assumption N orthogonal subspaces permit N independent Bell tests whose outcomes can be multiplied while preserving the local-hidden-variable bound as the Nth power.
    Invoked when defining the penalized N-product inequalities.
  • ad hoc to paper The penalty term closes the simultaneous measurement loophole.
    Stated as part of the method but not derived in the abstract.

pith-pipeline@v0.9.0 · 5732 in / 1333 out tokens · 16330 ms · 2026-05-24T12:19:30.657053+00:00 · methodology

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Reference graph

Works this paper leans on

58 extracted references · 58 canonical work pages

  1. [1]

    (8) see Appendix D

    For the 2-qubit state ρAB and qubit measurements Ax a andBy b that produce the rela- tion in Eq. (8) see Appendix D. We used the Navascu´ es- 5 Pironio-Ac´ ın hierarchy [53] to indicate the dimension- independent optimality of Eq. (8). Employing the relation in Eq. (8), we optimize η in Eq. (7) over Q in order to obtain the optimal value ηcrit for a given...

    work page 2016

  2. [2]

    J. S. Bell, Physics Physique Fizika 1, 195 (1964)

    work page 1964

  3. [3]

    Colbeck, Quantum and relativistic protocols for se- cure multi-party computation, Ph.D

    R. Colbeck, Quantum and relativistic protocols for se- cure multi-party computation, Ph.D. thesis, University of Cambridge (2009)

    work page 2009

  4. [4]

    Pironio, A

    S. Pironio, A. Ac´ ın, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, Nature (London) 464, 1021 (2010)

    work page 2010

  5. [5]

    Liu, M.-H

    W.-Z. Liu, M.-H. Li, S. Ragy, S.-R. Zhao, B. Bai, Y. Liu, P. J. Brown, J. Zhang, R. Colbeck, J. Fan, Q. Zhang, and J.-W. Pan, Nat. Phys. (2021)

    work page 2021

  6. [6]

    L. K. Shalm, Y. Zhang, J. C. Bienfang, C. Schlager, M. J. Stevens, M. D. Mazurek, C. Abell´ an, W. Amaya, M. W. Mitchell, M. A. Alhejji, H. Fu, J. Ornstein, R. P. Mirin, S. W. Nam, and E. Knill, Nat. Phys. (2021)

    work page 2021

  7. [7]

    A. K. Ekert, Phys. Rev. Lett. 67, 661 (1991)

    work page 1991

  8. [8]

    Mayers and A

    D. Mayers and A. Yao, in Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280) (IEEE, Los Alamitos, CA, 1998) p. 503

    work page 1998

  9. [9]

    Barrett, L

    J. Barrett, L. Hardy, and A. Kent, Phys. Rev. Lett. 95, 010503 (2005)

    work page 2005

  10. [10]

    Ac´ ın, N

    A. Ac´ ın, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, Phys. Rev. Lett. 98, 230501 (2007)

    work page 2007

  11. [11]

    Pironio, A

    S. Pironio, A. Ac´ ın, N. Brunner, N. Gisin, S. Massar, and V. Scarani, New J. Phys. 11, 045021 (2009)

    work page 2009

  12. [12]

    Aolita, R

    L. Aolita, R. Gallego, A. Cabello, and A. Ac´ ın, Phys. Rev. Lett. 108, 100401 (2012)

    work page 2012

  13. [13]

    M. G. M. Moreno, S. Brito, R. V. Nery, and R. Chaves, Phys. Rev. A 101, 052339 (2020)

    work page 2020

  14. [14]

    Mayers and A

    D. Mayers and A. Yao, Quantum Info. Comput. 4, 273 (2004)

    work page 2004

  15. [15]

    ˇSupi´ c and J

    I. ˇSupi´ c and J. Bowles, Quantum4, 337 (2020)

    work page 2020

  16. [16]

    E. S. G´ omez, S. G´ omez, P. Gonz´ alez, G. Ca˜ nas, J. F. Barra, A. Delgado, G. B. Xavier, A. Cabello, M. Klein- mann, T. V´ ertesi, and G. Lima, Phys. Rev. Lett. 117, 260401 (2016)

    work page 2016

  17. [17]

    Smania, P

    M. Smania, P. Mironowicz, M. Nawareg, M. Paw lowski, A. Cabello, and M. Bourennane, Optica 7, 123 (2020)

    work page 2020

  18. [18]

    M. T. Quintino, C. Budroni, E. Woodhead, A. Cabello, and D. Cavalcanti, Phys. Rev. Lett. 123, 180401 (2019)

    work page 2019

  19. [19]

    Hensen, H

    B. Hensen, H. Bernien, A. E. Dr´ eau, A. Reiserer, N. Kalb, M. S. Blok, J. Ruitenberg, R. F. L. Vermeulen, R. N. Schouten, C. Abell´ an, W. Amaya, V. Pruneri, M. W. Mitchell, M. Markham, D. J. Twitchen, D. Elkouss, S. Wehner, T. H. Taminiau, and R. Hanson, Nature (London) 526, 682 (2015)

    work page 2015

  20. [20]

    Giustina, M

    M. Giustina, M. A. M. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlech- ner, J. Kofler, J.- ˚A. Larsson, C. Abell´ an, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, Phys. Rev. Lett. 115, 250401 (2015)

    work page 2015

  21. [21]

    L. K. Shalm, E. Meyer-Scott, B. G. Christensen, P. Bier- horst, M. A. Wayne, M. J. Stevens, T. Gerrits, S. Glancy, D. R. Hamel, M. S. Allman, K. J. Coakley, S. D. Dyer, C. Hodge, A. E. Lita, V. B. Verma, C. Lam- brocco, E. Tortorici, A. L. Migdall, Y. Zhang, D. R. Kumor, W. H. Farr, F. Marsili, M. D. Shaw, J. A. Stern, C. Abell´ an, W. Amaya, V. Pruneri, ...

    work page 2015

  22. [22]

    Rosenfeld, D

    W. Rosenfeld, D. Burchardt, R. Garthoff, K. Redeker, N. Ortegel, M. Rau, and H. Weinfurter, Phys. Rev. Lett. 119, 010402 (2017)

    work page 2017

  23. [23]

    P. M. Pearle, Phys. Rev. D 2, 1418 (1970)

    work page 1970

  24. [24]

    P. H. Eberhard, Phys. Rev. A 47, R747 (1993)

    work page 1993

  25. [25]

    Garg and N

    A. Garg and N. D. Mermin, Phys. Rev. D 35, 3831 (1987)

    work page 1987

  26. [26]

    Massar, Phys

    S. Massar, Phys. Rev. A 65, 032121 (2002)

    work page 2002

  27. [27]

    V´ ertesi, S

    T. V´ ertesi, S. Pironio, and N. Brunner, Phys. Rev. Lett. 104, 060401 (2010)

    work page 2010

  28. [28]

    Cabello and J.- ˚A

    A. Cabello and J.- ˚A. Larsson, Phys. Rev. Lett. 98, 220402 (2007)

    work page 2007

  29. [29]

    Brunner, N

    N. Brunner, N. Gisin, V. Scarani, and C. Simon, Phys. Rev. Lett. 98, 220403 (2007)

    work page 2007

  30. [30]

    Garbarino, Phys

    G. Garbarino, Phys. Rev. A 81, 032106 (2010)

    work page 2010

  31. [31]

    Ara´ ujo, M

    M. Ara´ ujo, M. T. Quintino, D. Cavalcanti, M. F. m. c. Santos, A. Cabello, and M. T. Cunha, Phys. Rev. A 86, 030101 (2012)

    work page 2012

  32. [32]

    Larsson and J

    J.- ˚A. Larsson and J. Semitecolos, Phys. Rev. A 63, 022117 (2001)

    work page 2001

  33. [33]

    Cabello, D

    A. Cabello, D. Rodr´ ıguez, and I. Villanueva, Phys. Rev. Lett. 101, 120402 (2008)

    work page 2008

  34. [34]

    K. F. P´ al, T. V´ ertesi, and N. Brunner, Phys. Rev. A86, 062111 (2012)

    work page 2012

  35. [35]

    J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. 23, 880 (1969)

    work page 1969

  36. [36]

    Collins and N

    D. Collins and N. Gisin, J. Phys. A: Math. Gen. 37, 1775 (2004)

    work page 2004

  37. [37]

    P. G. Kwiat, J. Mod. Opt. 44, 2173 (1997)

    work page 1997

  38. [38]

    Erhard, M

    M. Erhard, M. Krenn, and A. Zeilinger, Nat. Rev. Phys. 2, 365 (2020)

    work page 2020

  39. [39]

    Barrett, D

    J. Barrett, D. Collins, L. Hardy, A. Kent, and S. Popescu, Phys. Rev. A 66, 042111 (2002)

    work page 2002

  40. [40]

    Cabello, Phys

    A. Cabello, Phys. Rev. Lett. 97, 140406 (2006)

    work page 2006

  41. [41]

    P. H. Eberhard and P. Rosselet, Found. Phys. 25, 91 (1995)

    work page 1995

  42. [42]

    R. Jain, C. A. Miller, and Y. Shi, IEEE Trans. Inf. The- ory 66, 5567 (2020)

    work page 2020

  43. [43]

    M. Doda, M. Huber, G. Murta, M. Pivoluska, M. Plesch, and C. Vlachou, Phys. Rev. Applied 15, 034003 (2021)

    work page 2021

  44. [44]

    Ara´ ujo, F

    M. Ara´ ujo, F. Hirsch, and M. T. Quintino, Quantum 4, 353 (2020)

    work page 2020

  45. [45]

    M´ arton, E

    I. M´ arton, E. Bene, and T. V´ ertesi, (2021), arXiv:2103.10413 [quant-ph]

    work page arXiv 2021

  46. [46]

    Raz, SIAM J

    R. Raz, SIAM J. Comput. 27, 763 (1998)

    work page 1998

  47. [47]

    Holenstein, in Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing (Association for Computing Machinery, New York, NY, 2007) p

    T. Holenstein, in Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing (Association for Computing Machinery, New York, NY, 2007) p. 411

    work page 2007

  48. [48]

    Feige, G

    U. Feige, G. Kindler, and R. O’Donnell, in Twenty- Second Annual IEEE Conference on Computational Complexity (CCC’07) (IEEE, Los Alamitos, CA, 2007) 12 p. 179

    work page 2007

  49. [49]

    Rao, SIAM J

    A. Rao, SIAM J. Comput. 40, 1871 (2011)

    work page 2011

  50. [50]

    Dinur and D

    I. Dinur and D. Steurer, in Proceedings of the Forty- Sixth Annual ACM Symposium on Theory of Computing (STOC ’14) (Association for Computing Machinery, New York, NY, 2014) p. 624

    work page 2014

  51. [51]

    Pitowsky, Mathematical Programming 50, 395 (1991)

    I. Pitowsky, Mathematical Programming 50, 395 (1991)

    work page 1991

  52. [52]

    Branciard, Phys

    C. Branciard, Phys. Rev. A 83, 032123 (2011)

    work page 2011

  53. [53]

    Czechlewski and M

    M. Czechlewski and M. Paw lowski, Phys. Rev. A 97, 062123 (2018)

    work page 2018

  54. [54]

    Navascu´ es, S

    M. Navascu´ es, S. Pironio, and A. Ac´ ın, Phys. Rev. Lett. 98, 010401 (2007)

    work page 2007

  55. [55]

    Wittek, ACM Transactions on Mathematical Software (TOMS) 41, 1 (2015)

    P. Wittek, ACM Transactions on Mathematical Software (TOMS) 41, 1 (2015)

    work page 2015

  56. [56]

    L¨ ofberg, in 2004 IEEE International Conference on Robotics and Automation (IEEE Cat

    J. L¨ ofberg, in 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508) (IEEE, Los Alamitos, CA, 2004) p. 284

    work page 2004

  57. [57]

    ApS, The MOSEK optimization toolbox for MATLAB manual

    M. ApS, The MOSEK optimization toolbox for MATLAB manual. Version 9.0. (2019)

    work page 2019

  58. [58]

    M. S. Andersen, J. Dahl, and L. Vandenberghe, CVX- OPT: A Python package for convex optimization, version 1.1.6 (2013)

    work page 2013