Bell violation with path-entangled number states under realistic detection

Christoph F. Wildfeuer

Bell violation with path-entangled number states under realistic detection

T0 review · 0 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read A single photon split between two paths can violate Bell's inequality in a detection-loophole-free way once the overall detection efficiency exceeds 82.58%, with simple on–off detectors beating parity measurement.

desk verdict Solid, honest extension that turns a thirty-year conceptual dispute into a concrete experimental budget; the central μ=pη collapse and 0.8258 threshold hold up, but the new numerics aren't independently reproducible from the text. read the letter →

arxiv 2607.29012 v1 pith:N3E5QDAE submitted 2026-07-31 quant-ph

Christoph F. Wildfeuer This is my paper

classification quant-ph MSC 81P4081P1581V80 PACS 03.65.Ud42.50.Xa

keywords Bellinequalitysingle-photonentanglementN00Nstatesdetectionloopholeon–offparityCHSHmodematching

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 argues that the decades-old question of whether a single photon split between two paths can violate a Bell inequality is settled in principle, and that what remains is a question about apparatus: how efficient detectors must be. The central result reduces the entire loss budget to one number, the overall probability μ that a heralded photon is detected, because loss and displacement commute up to a rescaling, so upstream loss and detector efficiency enter only through μ=pη. For a single delocalized photon, maximizing CHSH over the measurement settings gives μ_c=0.8258 with on–off detection and μ_c=0.9427 with parity detection, so the detector that needs no photon-number resolution is the better choice. The paper then argues the real obstacle is effective mode overlap, which carries the heralded photon's single-mode weight; at a buildable μ=0.862 the test needs overlap 0.92. If these numbers are right, a loophole-free Bell test on single-photon path entanglement is within reach of currently demonstrated components, and the limiting factor shifts from efficiency to source engineering.

What carries the argument

The one-parameter displaced detector family M̂(x,α)=D̂†(α)x^{n̂}D̂(α), with x=1−η for on–off and x=1−2η for parity, is the central object; it unifies the two schemes under loss and ties detector efficiency to quasiprobability ordering. The load-bearing identity is displacement covariance of the loss channel, Φ_η[D̂(α)ρD̂†(α)]=D̂(√η α)Φ_η[ρ]D̂†(√η α), which makes upstream loss p and detector efficiency η interchangeable and reduces a symmetric experiment to the single efficiency μ=pη. The correlator for N=1 then includes dark counts, phase noise and the effective overlap ξ, and the CHSH maximum is found by unrestricted numerical optimization over the four complex settings.

What would settle it

Compute the CHSH maximum for the single-photon state using the full Schmidt-mode decomposition of a realistic heralded source instead of a single overlap parameter and compare with Eq. (13); if the discrepancy exceeds the paper's stated numerical verification tolerance, the ξ-factorization is wrong. Alternatively, run the proposed experiment at measured μ=0.862 and effective overlap 0.92: a measured |S|≤2 falsifies the central claim.

Watch

Extended reading notes

Core claim

The paper's central claim is that the single-photon path-entangled state (|1,0⟩−|0,1⟩)/√2 is Bell-nonlocal once the two stations share a phase reference, and that a realistic, detection-loophole-free test reduces to a single number: the overall probability μ that a heralded photon is detected. Because photonic loss is displacement-covariant, all loss before the displacement and at the detector collapses into μ=pη for symmetric arms, and both detection schemes belong to one operator family x^n after displacement. Maximizing CHSH over the four complex displacement settings gives μ_c=0.8258 for on–off detection and μ_c=0.9427 for parity detection, so the detector that needs no photon-number res

Load-bearing premise

The load-bearing premise is that all mode mismatch—local-oscillator mode match and the heralded photon's single-mode weight alike—can be compressed into one multiplicative factor ξ²=ξ_LO² w appearing in front of the interference terms; if a real source's spectral structure is too complex for a single overlap parameter, the quoted thresholds and the 0.92 target could shift.

Editorial extensions

If this is right

A loophole-free (no fair-sampling) CHSH test on a single delocalized photon is feasible with on–off detectors once the probability that a heralded photon is detected exceeds 0.8258, a threshold marginally easier than the 0.8284 qubit limit.
Parity detection is not worth building: it demands 0.9427 overall efficiency and dies at 0.0282 dark counts per window, while on–off detection survives to 0.1395; number-resolving hardware read out as on–off is fine.
The practical bottleneck is source mode structure: the effective overlap ξ² equals ξ_LO² times the heralded photon's single-mode weight, so filtering for spectral purity reduces heralding efficiency; the target at μ=0.862 is ξ=0.92, and no source has reported coupling and overlap on one device.
An experiment must gate the local oscillator to heralded windows, generate settings on a free clock with spacelike separation from emission, and use memory-safe statistics; with realistic parameters a five-standard-deviation violation needs roughly 1.1×10^5 trials.
For N≥2, on–off detection still violates CHSH at unit efficiency for the tested N up to 5, but thresholds rise steeply (0.923 for N=2), so N=1 is the experimentally relevant case.

Reading between the lines

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

If these thresholds are correct, the single-photon path-entanglement platform is a standard-difficulty Bell test, and further progress should concentrate on sources engineered for spectral factorability; a source that reports both coupling h and overlap ξ on one device would be the enabling step.
The paper's optimization is restricted to the x^n detector family; a systematic search over arbitrary dichotomic functions of detected photon number might find a threshold below 0.8258, since the paper explicitly notes the family leaves room.
The mild favourability of unbalanced arms suggests the threshold depends on the geometric mean of the two per-arm efficiencies, so deliberately unbalancing the arms could serve as a controlled test of the μ-collapse prediction.

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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

Short take: this is a careful, honest paper that reduces a long conceptual debate to a concrete experimental specification, and the central number—the 0.8258 on–off efficiency threshold—agrees with prior literature and the paper's own brute-force checks. The genuinely new pieces, the μ=pη collapse and the 0.9427 parity threshold, are plausible and well-derived.

What's new and good: the closed correlator Eq. (9) for arbitrary N with efficiency, dark counts, mode mismatch, phase noise, and upstream loss; the displacement-covariance argument (Eq. 17) that collapses all loss to a single overall efficiency; the clean demonstration that on–off detection is more robust than parity; and the practical budget showing that mode overlap, not efficiency, is now the bottleneck. The paper also does a service by carefully separating the three questions (entangled? nonlocal? measurable?) and giving the historical debate its due without muddle. The self-citation to Ref. [43] is appropriate—it's the direct predecessor, and the new efficiency dependence is independently checked.

Soft spots, in proportion. The new numerical results—parity threshold, dark-count ceiling, multiphoton contamination curve—depend on numerical optimization, and no code or data are shipped. The claimed ~1e-16 checks against brute-force Fock-space computation are credible, but a referee can't reproduce them from the text alone. The claim that no source has reported high heralding efficiency and high overlap together rests on an AI-assisted literature search and should be independently verified; the paper discloses this honestly. The mode-mismatch model reduces all overlap effects to a single ξ; that's standard and likely fine, but if a real source has more complex spectral structure the quantitative trade-off could shift. These are verification-level concerns, not load-bearing flaws.

This deserves a serious referee. It's a useful, constructive paper that an experimental group could take directly to the lab. I'd cite it, and I'd bring it to a reading group if anyone works on single-photon nonlocality.

Referee Report

0 major / 5 minor

Summary. The paper analyzes Bell tests with a single photon delocalized over two modes (the N=1 N00N state) using displacement measurements enabled by a local oscillator, and argues that the longstanding conceptual debate is closed while the practical question is one of apparatus efficiency. The central technical results are: (i) closed correlators for N00N states at arbitrary efficiency, with dark counts, mode mismatch, phase noise, and upstream loss; (ii) a covariance argument showing that for symmetric arms all loss collapses into a single overall efficiency mu = p*eta; (iii) numerically optimized CHSH thresholds mu_c = 0.8258 for on-off detection and mu_c = 0.9427 for parity detection; (iv) an analysis of multiphoton contamination, statistics, and experimental layout concluding that mode overlap, not efficiency, is the main obstacle, and that on-off detection is preferable.

Significance. If correct, this is a substantial contribution. It gives a clean one-parameter budget for a long-disputed experiment, identifies a quantitative threshold that current technology can plausibly meet, and turns a decades-old conceptual debate into a concrete apparatus question. The paper is careful to separate exact results (Eqs. 10, 17, 19) from approximate fits (Eq. 23), it reproduces the known on-off threshold from prior literature, and it reports extensive internal numerical cross-checks against brute-force Fock-space computation (6-8e-16 in Secs. IV and V). The strengths are the closed-form correlator family, the loss-collapse theorem, and the practical guidance. The main new quantitative claims (especially the parity threshold) are numerical, so reproducibility of the optimization is the principal verification concern.

minor comments (5)

[Sec. VI C, Eq. (20)] The displayed formula for nu*_parity is inconsistent as printed. It reads nu* = (1/4) ln(max|S|_{mu=1/2}), but the parity value at mu=1/2 is 0.8881 (Sec. III), whose logarithm is negative, whereas the quoted number 0.0282 corresponds to (1/4) ln(max|S|_{mu=1}/2) = (1/4) ln(2.2387/2). Please correct the equation to the intended expression.
[Secs. VI B and VII D, Eq. (19), Eq. (23), Fig. 3] The central new numbers, in particular mu_c^parity = 0.9427, the on-off dark-count ceiling, and the multiphoton contamination curve, are obtained by numerical optimization, but no code, data, or tables of max|S|(mu) are provided. The analytic correlators are given, so the results are in principle reproducible, but the paper would be considerably stronger if the optimization code or a supplementary table of max|S| versus mu (and versus f_2, xi, sigma) were included. This is a reproducibility request, not a claim of error.
[Sec. VII A and Abstract] The statement that "no source has reported the two together" is a strong negative literature claim. Given the disclosed AI-assisted literature search, please provide the search protocol, specify the databases and date, or qualify the claim as "to our knowledge" with a manual verification before publication.
[Sec. IV B, Eq. (11)] The mode-mismatch factorization xi^2 = xi_LO^2 w is physically explained and numerically verified, but the derivation is compressed. A two-line expression for the displaced multi-mode state in the Schmidt basis would make it clearer why xi^2 multiplies only the p lambda^2 group and why the factorization is exact at N=1. This is a clarity suggestion.
[Fig. 2] The three panels use different horizontal scales and directions (mode overlap runs right to left), which is unconventional and can be misread. Please add explicit direction arrows or unify the axis orientation where possible.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the μ-collapse theorem and thresholds follow from explicit closed-form correlators and numerical optimization, with self-citations used only as baseline checks, not as load-bearing premises.

full rationale

The central derivation is self-contained rather than circular. The collapse of upstream loss and detector efficiency into the single product μ=pη is derived from the displacement covariance of the loss channel, Eq. (17), together with composition of loss channels, and the paper explicitly names the three hypotheses required (symmetric arms, unconstrained displacement amplitude, and commuting imperfections). This is a mathematical argument, not a fit. The closed correlators of Eq. (9) and Eq. (13) are given in explicit analytic form, and the thresholds in Eq. (19) are obtained by optimizing the full complex four-setting problem and bisecting the predicate max|S|>2. The on–off threshold reproduces previously published independent results (Lee et al., Alwehaibi et al.), and the parity threshold, although new, follows from the same explicit formulas rather than from an imported self-citation. The mode-mismatch factorization ξ²=ξ_LO² w in Eq. (11) is an explicitly stated modeling assumption for a pure single-mode local oscillator and mode-independent bucket detection; it is not a redefinition that assumes the conclusion, and the paper reports verification against direct Fock-space construction to 6×10⁻¹⁶. The ancillary coefficient 0.23 in Eq. (23) is a summary of a direct numerical computation, not a fitted input used to produce the threshold predictions. The only self-citation, Ref. [43], is used as an acknowledged ideal-detector baseline and as a consistency check; the efficiency-dependent results are derived independently, and the N≥2 parity remark in Sec. VIII is a side statement, not load-bearing for the paper's central claim. No self-citation is used to forbid alternatives or to import a uniqueness theorem. Remaining concerns—that no code or data are shipped, and that the literature-search claim about source reports is AI-assisted—are verification-level issues, not circularity.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new particles, forces, or entities. It relies on standard quantum optics and a small number of physical modeling assumptions about the source, detectors, and local oscillator. The only fitted quantity is an ancillary coefficient in an approximate scaling law.

free parameters (1)

Fit coefficient in Eq. (23) = ≈0.23 (for f2 ≤ 10^-2)
Empirical coefficient in the approximate linear relation Δμ_c ≈ 0.23 f2 for multiphoton contamination. Ancillary: the exact two-pair computation is also given, and the coefficient is a convenient approximation to that exact result.

assumptions (5)

standard math Fock-space quantum optics with beam-splitter loss channel (Kraus operators in Sec. III).
Standard formalism for lossy photodetection; the loss channel and its covariance are standard results in quantum optics.
standard math CHSH and CH inequalities with local hidden variables, |S|≤2.
The Bell bounds are taken as given; the paper uses them to define thresholds.
domain assumption Local oscillator is a strong classical field, so unbalanced homodyning implements an exact displacement D(α).
Invoked throughout Secs. III–IV. Corrections from a bounded reference are cited to [54] and stated to be negligible at the few-hundred-photon optimum.
domain assumption Heralded SPDC photon is described as a mixture over Schmidt modes with a single-mode weight w; bucket herald leaves this mixture.
Used to derive Eq. (11) and the mode-overlap trade-off. This is the standard model for pulsed SPDC with bucket heralding, but the factorization ξ²=ξ_LO² w is specific to this model.
domain assumption Dark counts are Poissonian in the detection window.
Used for the scaling of the parity and on-off dark-count effects (Sec. IV B). Real detector dark counts may have non-Poissonian components, but the paper argues they are orders of magnitude below concern.

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Cite this review

Pith. "Pith review of Bell violation with path-entangled number states under realistic detection." pith.science (2026). https://pith.science/paper/N3E5QDAE

@misc{pith2026260729012,
  author       = {Pith},
  title        = {Pith review of: Bell violation with path-entangled number states under realistic detection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N3E5QDAE}},
  note         = {Machine review of arXiv:2607.29012}
}

read the original abstract

A single photon delocalised over two modes is entangled, and whether that entanglement can violate a Bell inequality has been disputed for three decades. In principle it is settled: the surviving objection was the absence of a shared phase reference, and a local oscillator supplies one, acting as a shared frame rather than as a measurement setting. In practice it is open, and what remains is a question about the apparatus. Loss does not spoil a parity measurement so much as rescale it, carrying it into the same one-parameter family of detector operators that already contains on--off detection. Within that family we give closed correlators for path-entangled number states, or N00N states, at arbitrary efficiency, with dark counts, mode mismatch, phase noise and upstream loss. Because loss and displacement commute up to a rescaling of the displacement amplitude, loss before and after the displacement is interchangeable, and a symmetric experiment is governed by one overall efficiency: the probability that a heralded photon is detected. For a single photon the Clauser--Horne--Shimony--Holt threshold on that efficiency is 0.83 with on--off detection and 0.95 with parity, so the scheme that needs no photon-number resolution is the more robust, by twelve percentage points of loss and a factor of five in tolerable dark counts. Efficiency is no longer the obstacle; the mode overlap is. The best demonstrated telecom coupling, a buildable interferometer and today's detectors give an overall efficiency of 0.86, at which the required overlap is 0.92. We show that this overlap carries the heralded photon's single-mode weight as well as the local oscillator's mode match, so buying more of it costs heralding efficiency --- and no source has reported the two together.

Figures

Figures reproduced from arXiv: 2607.29012 by the authors.

**Figure 2.** FIG. 2. Threshold overall efficiency for a CHSH violation with [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** FIG. 3. The frontier the experiment has to cross. Shaded [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

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Works this paper leans on

93 extracted references · 57 canonical work pages

[43]

C. F. Wildfeuer, A. P. Lund, and J. P. Dowling, Phys. Rev. A76, 052101 (2007). doi:10.1103/PhysRevA.76.052101

work page doi:10.1103/physreva.76.052101 2007
[1]

B. C. Sanders, Phys. Rev. A40, 2417 (1989). doi:10.1103/PhysRevA.40.2417

work page doi:10.1103/physreva.40.2417 1989
[2]

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, Phys. Rev. Lett.85, 2733 (2000). doi:10.1103/PhysRevLett.85.2733

doi:10.1103/physrevlett.85.2733 2000
[3]

H. Lee, P. Kok, and J. P. Dowling, J. Mod. Opt.49, 2325 (2002). doi:10.1080/0950034021000011536

doi:10.1080/0950034021000011536 2002
[4]

K. T. Kapale, L. D. Didomenico, H. Lee, P. Kok, and J. P. Dowling, Concepts of PhysicsII, 225 (2005). doi:10.48550/arXiv.quant-ph/0507150

work page Pith review arXiv doi:10.48550/arxiv.quant-ph/0507150 2005
[5]

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

doi:10.1103/physrevlett.23.880 1969
[6]

J. F. Clauser and M. A. Horne, Phys. Rev. D10, 526 (1974). doi:10.1103/PhysRevD.10.526

doi:10.1103/physrevd.10.526 1974
[7]

R. F. Werner, Phys. Rev. A40, 4277 (1989). doi:10.1103/PhysRevA.40.4277

doi:10.1103/physreva.40.4277 1989

Show all 93 references

[8]

Barrett, Phys

J. Barrett, Phys. Rev. A65, 042302 (2002). doi:10.1103/PhysRevA.65.042302

2002 doi
[9]

Gisin, Phys

N. Gisin, Phys. Lett. A154, 201 (1991). doi:10.1016/0375-9601(91)90805-I

1991 doi
[10]

Gisin and A

N. Gisin and A. Peres, Phys. Lett. A162, 15 (1992). doi:10.1016/0375-9601(92)90949-M

1992 doi
[11]

Popescu and D

S. Popescu and D. Rohrlich, Phys. Lett. A166, 293 (1992). doi:10.1016/0375-9601(92)90711-T

1992 doi
[12]

H. M. Wiseman, S. J. Jones, and A. C. Do- herty, Phys. Rev. Lett.98, 140402 (2007). doi:10.1103/PhysRevLett.98.140402

2007 doi
[13]

M. O. Renou, D. Rosset, A. Martin, and N. Gisin, J. Phys. A50, 255301 (2017). doi:10.1088/1751- 8121/aa6f78

2017 doi
[14]

S. M. Tan, D. F. Walls, and M. J. Collett, Phys. Rev. Lett.66, 252 (1991). doi:10.1103/PhysRevLett.66.252

1991 doi
[15]

Santos, Phys

E. Santos, Phys. Rev. Lett.68, 894 (1992); S. M. Tan, D. F. Walls, and M. J. Collett,ibid.68, 895 (1992). doi:10.1103/PhysRevLett.68.894

1992 doi
[16]

Hardy, Phys

L. Hardy, Phys. Rev. Lett.73, 2279 (1994). doi:10.1103/PhysRevLett.73.2279

1994 doi
[17]

Peres, Phys

A. Peres, Phys. Rev. Lett.74, 4571 (1995);ibid.76, 2205(E) (1996). doi:10.1103/PhysRevLett.74.4571

1995 doi
[18]

Vaidman, Phys

L. Vaidman, Phys. Rev. Lett.75, 2063 (1995). doi:10.1103/PhysRevLett.75.2063

doi
[19]

D. M. Greenberger, M. A. Horne, and A. Zeilinger, Phys. Rev. Lett.75, 2064 (1995); L. Hardy,ibid.75, 2065 (1995). doi:10.1103/PhysRevLett.75.2064

doi
[20]

S. J. van Enk, Phys. Rev. A72, 064306 (2005). doi:10.1103/PhysRevA.72.064306

2005 doi
[21]

Drezet, Phys

A. Drezet, Phys. Rev. A74, 026301 (2006). doi:10.1103/PhysRevA.74.026301

2006 doi
[22]

S. J. van Enk, Phys. Rev. A74, 026302 (2006). doi:10.1103/PhysRevA.74.026302

2006 doi
[23]

T. Das, M. Karczewski, A. Mandarino, M. Markiewicz, B. Wo loncewicz, and M. ˙Zukowski, New J. Phys.23, 073042 (2021). doi:10.1088/1367-2630/ac0ffe

2021 doi
[24]

T. Das, M. Karczewski, A. Mandarino, M. Markiewicz, B. Wo loncewicz, and M. ˙Zukowski, arXiv:2102.03254. doi:10.48550/arXiv.2102.03254

doi arXiv
[25]

J. J. Cooper and J. A. Dunningham, New J. Phys.10, 113024 (2008). doi:10.1088/1367-2630/10/11/113024

2008 doi
[26]

T. Das, M. Karczewski, A. Mandarino, M. Markiewicz, B. Wo loncewicz, and M. ˙Zukowski, New J. Phys.24, 038001 (2022). doi:10.1088/1367-2630/ac55b1

2022 doi
[27]

Schlichtholz, B

K. Schlichtholz, B. Wo loncewicz, T. Das, M. Markiewicz, and M. ˙Zukowski, Quantum10, 2084 (2026). doi:10.22331/q-2026-04-28-2084

doi
[28]

Aharonov and L

Y. Aharonov and L. Susskind, Phys. Rev.155, 1428 (1967). doi:10.1103/PhysRev.155.1428

1967 doi
[29]

S. D. Bartlett, A. C. Doherty, R. W. Spekkens, and H. M. Wiseman, Phys. Rev. A73, 022311 (2006). doi:10.1103/PhysRevA.73.022311

2006 doi
[30]

S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Rev. Mod. Phys.79, 555 (2007). doi:10.1103/RevModPhys.79.555

2007 doi
[31]

J. A. Dunningham and V. Vedral, Phys. Rev. Lett.99, 180404 (2007). doi:10.1103/PhysRevLett.99.180404

2007 doi
[32]

Ashhab, K

S. Ashhab, K. Maruyama, ˇC. Brukner, and F. Nori, Phys. Rev. A80, 062106 (2009). doi:10.1103/PhysRevA.80.062106

2009 doi
[33]

J. B. Brask, R. Chaves, and N. Brunner, Phys. Rev. A 88, 012111 (2013). doi:10.1103/PhysRevA.88.012111

2013 doi
[34]

Abiuso, T

P. Abiuso, T. Kriv´ achy, E.-C. Boghiu, M.- O. Renou, A. Pozas-Kerstjens, and A. Ac ´ ın, Phys. Rev. Research4, L012041 (2022). doi:10.1103/PhysRevResearch.4.L012041

2022 doi
[35]

M. G. A. Paris, Phys. Lett. A217, 78 (1996). doi:10.1016/0375-9601(96)00339-8

1996 doi
[36]

K. E. Cahill and R. J. Glauber, Phys. Rev.177, 1882 (1969). doi:10.1103/PhysRev.177.1882

1969 doi
[37]

Banaszek and K

K. Banaszek and K. W´ odkiewicz, Phys. Rev. Lett.76, 4344 (1996). doi:10.1103/PhysRevLett.76.4344

1996 doi
[38]

Wallentowitz and W

S. Wallentowitz and W. Vogel, Phys. Rev. A53, 4528 (1996). doi:10.1103/PhysRevA.53.4528

1996 doi
[39]

Banaszek and K

K. Banaszek and K. W´ odkiewicz, Phys. Rev. Lett.82, 2009 (1999). doi:10.1103/PhysRevLett.82.2009

2009 doi
[40]

(8) of Ref

— the on–off branch iss= 1−2/(ηT d) — or into Eq. (8) of Ref. [44], and we claim no priority for it. It fixes the geometry of what follows. What is spent as the efficiency falls is not the detector’s information — the counting distribution is still there — but the useful- ness...

2016
[41]

Royer, Phys

A. Royer, Phys. Rev. A15, 449 (1977). doi:10.1103/PhysRevA.15.449

1977 doi
[42]

Banaszek, C

K. Banaszek, C. Radzewicz, K. W´ odkiewicz, and J. S. Krasi´ nski, Phys. Rev. A60, 674 (1999). doi:10.1103/PhysRevA.60.674

1999 doi
[44]

Moya-Cessa and P

H. Moya-Cessa and P. L. Knight, Phys. Rev. A48, 2479 (1993). doi:10.1103/PhysRevA.48.2479

1993 doi
[45]

S.-Y. Lee, J. Park, J. Kim, and C. Noh, Phys. Rev. A 95, 012134 (2017). doi:10.1103/PhysRevA.95.012134

2017 doi
[46]

S.-W. Lee, H. Jeong, and D. Jaksch, Phys. Rev. A80, 022104 (2009). doi:10.1103/PhysRevA.80.022104

2009 doi
[47]

Garg and N

A. Garg and N. D. Mermin, Phys. Rev. D35, 3831 (1987). doi:10.1103/PhysRevD.35.3831

1987 doi
[48]

Chaves and J

R. Chaves and J. B. Brask, Phys. Rev. A84, 062110 (2011). doi:10.1103/PhysRevA.84.062110

2011 doi
[49]

A. J. Bennet, D. A. Evans, D. J. Saunders, C. Branciard, E. G. Cavalcanti, H. M. Wiseman, and G. J. Pryde, Phys. Rev. X2, 031003 (2012). doi:10.1103/PhysRevX.2.031003

2012 doi
[50]

P. H. Eberhard, Phys. Rev. A47, R747 (1993). doi:10.1103/PhysRevA.47.R747

1993 doi
[51]

H. M. Wiseman and J. A. Vaccaro, Phys. Rev. Lett.91, 097902 (2003). doi:10.1103/PhysRevLett.91.097902

2003 doi
[52]

Paterek, P

T. Paterek, P. Kurzy´ nski, D. K. L. Oi, and D. Kaszlikowski, New J. Phys.13, 043027 (2011). 16 doi:10.1088/1367-2630/13/4/043027

2011 doi
[53]

S. J. van Enk and C. A. Fuchs, Phys. Rev. Lett.88, 027902 (2002). doi:10.1103/PhysRevLett.88.027902

2002 doi
[54]

Rudolph and B

T. Rudolph and B. C. Sanders, Phys. Rev. Lett.87, 077903 (2001). doi:10.1103/PhysRevLett.87.077903

2001 doi
[55]

Barrett, D

J. Barrett, D. Collins, L. Hardy, A. Kent, and S. Popescu, Phys. Rev. A66, 042111 (2002). doi:10.1103/PhysRevA.66.042111

2002 doi
[56]

S. D. Bartlett, T. Rudolph, R. W. Spekkens, and P. S. Turner, New J. Phys.11, 063013 (2009). doi:10.1088/1367-2630/11/6/063013

2009 doi
[57]

Bierhorst, J

P. Bierhorst, J. Phys. A48, 195302 (2015). doi:10.1088/1751-8113/48/19/195302

2015 doi
[58]

R. D. Gill, inFoundations of Probability and Physics — 2(V¨ axj¨ o University Press, 2003), p. 179; arXiv:quant- ph/0301059. doi:10.48550/arXiv.quant-ph/0301059

2003 doi
[59]

Larsson, J

J.- ˚A. Larsson, J. Phys. A47, 424003 (2014). doi:10.1088/1751-8113/47/42/424003

2014 doi
[60]

Larsson and R

J.- ˚A. Larsson and R. D. Gill, Europhys. Lett.67, 707 (2004). doi:10.1209/epl/i2004-10124-7

2004 doi
[61]

L. K. Shalmet al., Phys. Rev. Lett.115, 250402 (2015). doi:10.1103/PhysRevLett.115.250402

2015 doi
[62]

Hensenet al., Nature526, 682 (2015)

B. Hensenet al., Nature526, 682 (2015). doi:10.1038/nature15759

2015 doi
[63]

Scheidlet al., Proc

T. Scheidlet al., Proc. Natl. Acad. Sci. USA107, 19708 (2010). doi:10.1073/pnas.1002780107

2010 doi
[64]

Rosenfeld, D

W. Rosenfeld, D. Burchardt, R. Garthoff, K. Redeker, N. Ortegel, M. Rau, and H. Wein- furter, Phys. Rev. Lett.119, 010402 (2017). doi:10.1103/PhysRevLett.119.010402

2017 doi
[65]

D. Kun, T. Str¨ omberg, B. Daki´ c, P. Walther, and L. A. Rozema, Optica13, 745 (2026). doi:10.1364/OPTICA.586172

2026 doi
[66]

Caprara Vivoli, P

V. Caprara Vivoli, P. Sekatski, J.-D. Bancal, C. C. W. Lim, B. G. Christensen, A. Martin, R. T. Thew, H. Zbinden, N. Gisin, and N. Sangouard, Phys. Rev. A91, 012107 (2015). doi:10.1103/PhysRevA.91.012107

2015 doi
[67]

Y. K. Alwehaibi, E. Mer, G. J. Machado, S. Yu, I. A. Walmsley, and R. B. Patel, Phys. Rev. Research7, 043198 (2025). doi:10.1103/xw66-nqfs

2025 doi
[68]

J. B. Brask and R. Chaves, Phys. Rev. A86, 010103(R) (2012). doi:10.1103/PhysRevA.86.010103

2012 doi
[69]

M. E. Mycroft, T. McDermott, A. Buraczewski, and M. Stobi´ nska, Phys. Rev. A107, 012607 (2023). doi:10.1103/PhysRevA.107.012607

2023 doi
[70]

A. J. E. Bjerrum, J. B. Brask, J. S. Neergaard-Nielsen, and U. L. Andersen, Phys. Rev. A107, 052611 (2023). doi:10.1103/PhysRevA.107.052611

2023 doi
[71]

S. A. Babichev, J. Appel, and A. I. Lvovsky, Phys. Rev. Lett.92, 193601 (2004). doi:10.1103/PhysRevLett.92.193601

2004 doi
[72]

Hessmo, P

B. Hessmo, P. Usachev, H. Heydari, and G. Bj¨ ork, Phys. Rev. Lett.92, 180401 (2004). doi:10.1103/PhysRevLett.92.180401

2004 doi
[73]

M. Fuwa, S. Takeda, M. Zwierz, H. M. Wiseman, and A. Furusawa, Nat. Commun.6, 6665 (2015). doi:10.1038/ncomms7665

2015 doi
[74]

D’Angelo, A

M. D’Angelo, A. Zavatta, V. Parigi, and M. Bellini, Phys. Rev. A74, 052114 (2006). doi:10.1103/PhysRevA.74.052114

2006 doi
[75]

Liu, Y.-Z

W.-Z. Liu, Y.-Z. Zhang, Y.-Z. Zhen, M.-H. Li, Y. Liu, J. Fan, F. Xu, Q. Zhang, and J.- W. Pan, Phys. Rev. Lett.129, 050502 (2022). doi:10.1103/PhysRevLett.129.050502

2022 doi
[76]

Monteiro, V

F. Monteiro, V. Caprara Vivoli, T. Guerreiro, A. Martin, J.-D. Bancal, H. Zbinden, R. T. Thew, and N. Sangouard, Phys. Rev. Lett.114, 170504 (2015). doi:10.1103/PhysRevLett.114.170504

2015 doi
[77]

is plotted at its own reported efficiency and overlap. C. Statistics and systematics The violation is not hard to accumulate. All num- bers in this subsection are for the realistic caseξ= 0.97. Atµ= 0.86, where|S|= 2.067, the four optimal correla- tors are (−0.56,−0.56,−0.56,+...
[78]

Meyer-Scott, N

E. Meyer-Scott, N. Montaut, J. Tiedau, L. Sansoni, H. Herrmann, T. J. Bartley, and C. Silberhorn, Phys. Rev. A 95, 061803(R) (2017). doi:10.1103/PhysRevA.95.061803

2017 doi
[79]

Guerreiroet al., Phys

T. Guerreiroet al., Phys. Rev. Lett.117, 070404 (2016). doi:10.1103/PhysRevLett.117.070404

2016 doi
[80]

Caspar, E

P. Caspar, E. Oudot, P. Sekatski, N. Maring, A. Martin, N. Sangouard, H. Zbinden, and R. T. Thew, Quantum 6, 671 (2022). doi:10.22331/q-2022-03-22-671

2022 doi
[81]

Giustinaet al., Phys

M. Giustinaet al., Phys. Rev. Lett.115, 250401 (2015). doi:10.1103/PhysRevLett.115.250401

2015 doi
[82]

D. V. Reddy, R. R. Nerem, S. W. Nam, R. P. Mirin, and V. B. Verma, Optica7, 1649 (2020). doi:10.1364/OPTICA.400751

2020 doi
[83]

Esmaeil Zadehet al., ACS Photonics7, 1780 (2020)

I. Esmaeil Zadehet al., ACS Photonics7, 1780 (2020). doi:10.1021/acsphotonics.0c00433

2020 doi
[84]

Changet al., APL Photonics6, 036114 (2021)

J. Changet al., APL Photonics6, 036114 (2021). doi:10.1063/5.0039772

2021 doi
[85]

Korzhet al., Nat

B. Korzhet al., Nat. Photonics14, 250 (2020). doi:10.1038/s41566-020-0589-x

2020 doi
[86]

Fukudaet al., Opt

D. Fukudaet al., Opt. Express19, 870 (2011). doi:10.1364/OE.19.000870

2011 doi
[87]

J. W. N. Los, M. Sidorovaet al., APL Photonics9, 066101 (2024). doi:10.1063/5.0204340

2024 doi
[88]

da Cunha Pereira, F

M. da Cunha Pereira, F. E. Becerra, B. L. Glebov, J. Fan, S. W. Nam, and A. Migdall, Opt. Lett.38, 1609 (2013). doi:10.1364/OL.38.001609

2013 doi
[89]

Dinget al., Nat

X. Dinget al., Nat. Photonics19, 387 (2025). doi:10.1038/s41566-025-01639-8

2025 doi
[90]

Margariaet al., Nat

N. Margariaet al., Nat. Commun.16, 7553 (2025). doi:10.1038/s41467-025-62712-y

2025 doi
[91]

Lamet al., Quantum Sci

H. Lamet al., Quantum Sci. Technol.10, 045061 (2025). doi:10.1088/2058-9565/ae0a7a

2025 doi
[92]

ˇSvarc, M

V. ˇSvarc, M. Nov´ akov´ a, M. Dudka, and M. Jeˇ zek, Opt. Express31, 12562 (2023). doi:10.1364/OE.480569

2023 doi
[93]

C. Ding, X. Zhang, J. Xiong, Y. Xiao, T. Zhang, J. Huang, H. Xu, X. Liu, L. You, Z. Wang, and H. Li, ACS Photonics12, 4924 (2025). doi:10.1021/acsphotonics.5c00508

2025 doi

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [43]

C. F. Wildfeuer, A. P. Lund, and J. P. Dowling, Phys. Rev. A76, 052101 (2007). doi:10.1103/PhysRevA.76.052101

work page doi:10.1103/physreva.76.052101 2007

[2] [1]

B. C. Sanders, Phys. Rev. A40, 2417 (1989). doi:10.1103/PhysRevA.40.2417

work page doi:10.1103/physreva.40.2417 1989

[3] [2]

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, Phys. Rev. Lett.85, 2733 (2000). doi:10.1103/PhysRevLett.85.2733

doi:10.1103/physrevlett.85.2733 2000

[4] [3]

H. Lee, P. Kok, and J. P. Dowling, J. Mod. Opt.49, 2325 (2002). doi:10.1080/0950034021000011536

doi:10.1080/0950034021000011536 2002

[5] [4]

K. T. Kapale, L. D. Didomenico, H. Lee, P. Kok, and J. P. Dowling, Concepts of PhysicsII, 225 (2005). doi:10.48550/arXiv.quant-ph/0507150

work page Pith review arXiv doi:10.48550/arxiv.quant-ph/0507150 2005

[6] [5]

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

doi:10.1103/physrevlett.23.880 1969

[7] [6]

J. F. Clauser and M. A. Horne, Phys. Rev. D10, 526 (1974). doi:10.1103/PhysRevD.10.526

doi:10.1103/physrevd.10.526 1974

[8] [7]

R. F. Werner, Phys. Rev. A40, 4277 (1989). doi:10.1103/PhysRevA.40.4277

doi:10.1103/physreva.40.4277 1989