How post-selection affects device-independent claims under the fair sampling assumption

Davide Orsucci; Jean-Daniel Bancal; Nicolas Sangouard; Pavel Sekatski

How post-selection affects device-independent claims under the fair sampling assumption

T0 review · 0 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Under the fair sampling assumption, post-selected Bell statistics are exactly those of an ideal experiment measuring a locally filtered state \(\Psi_{\checkmark}\), so device-independent certifications target \(\Psi_{\checkmark}\) rather…

desk verdict A rigorous and useful framework for what post-selected Bell data actually certifies: the locally filtered state, not the original—worth a serious referee. read the letter →

arxiv 1908.11123 v2 pith:BI2HCJ3Q submitted 2019-08-29 quant-ph

Davide Orsucci , Jean-Daniel Bancal , Nicolas Sangouard , Pavel Sekatski This is my paper · ORCID

classification quant-ph MSC 81P4081P15 PACS 03.65.Ud03.67.Dd

keywords fairsamplingpost-selectiondetectionloopholedevice-independentcertificationBellinequalitiesfilteredquantumstatetotalvariationdistancekeydistribution

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

The paper asks what remains of device-independent certification when Bell experiments rely on post-selected data under the fair sampling assumption. It shows that if a lossy detector's decision to click factorizes into an independent classical part and an independent quantum part, then the post-selected statistics are exactly reproduced by an ideal lossless experiment measuring a locally filtered state \(\Psi_{\checkmark}\). Trusted conclusions about Bell violation, self-testing, randomness, and key distribution therefore apply to \(\Psi_{\checkmark}\), and not automatically to the original state \(\Psi\). Under strong fair sampling, where the quantum filter is proportional to the identity, the conclusions do apply to \(\Psi\). The paper also proves that small deviations from fair sampling shift the post-selected distribution by at most \(\epsilon/(1-\epsilon)\) in total variation distance.

What carries the argument

The load-bearing object is the filter: a probabilistic preprocessing stage placed before an ideal detector, which either accepts (\(\checkmark\)) or rejects (\(\varnothing\)) each round. Fair sampling is defined by demanding that the filter factorizes as \(F=\wedge(F_C\otimes F_Q)\), meaning the acceptance probability splits into a classical part depending only on the setting \(x\) and a quantum part depending only on the input state \(\rho\); equivalently, the detection efficiency factorizes as \(E(x,\rho)=E_C(x)E_Q(\rho)\). This factorization is what allows the rejected rounds to be discarded and the surviving statistics to be re-interpreted as ideal measurements on the locally filtered state \(\Psi_{\checkmark}\). The argument works because the quantum branch of the filter commutes through the Bell experiment, so post-selection becomes a heralded state preparation.

What would settle it

Measure, for a single lossy detector, the detection efficiency \(E(x,\rho)\) for two settings \(x\) and two probe states \(\rho\); if \(E(x,\rho_1)/E(y,\rho_1)\neq E(x,\rho_2)/E(y,\rho_2)\), the factorization \(E(x,\rho)=E_C(x)E_Q(\rho)\) fails and the paper's Proposition 2 cannot apply. Alternatively, exhibit a local hidden-variable model with state-dependent detection efficiency whose post-selected data reach the CHSH algebraic bound, which would show that post-selected statistics alone cannot certify the claims the paper makes under fair sampling.

Watch

Extended reading notes

Core claim

The central discovery is that fair sampling, formalized as a factorization of the filter or of the detection efficiency \(E(x,\rho)=E_C(x)E_Q(\rho)\), turns post-selection into a benign local operation. For any Bell experiment with lossy detectors satisfying this condition, the post-selected distribution equals the distribution obtained from lossless detectors acting on the normalized filtered state \(\Psi_{\checkmark}\), which is produced from \(\Psi\) by local probabilistic maps (Proposition 2). Consequently a Bell violation in the post-selected data certifies that \(\Psi_{\checkmark}\) is Bell-correlated, implying that \(\Psi\) itself possesses hidden nonlocality, and any self-testing or cryptographic statement drawn from the data refers to \(\Psi_{\checkmark}\). Strong fair sampling, in which the quantum filter is proportional to the identity, makes \(\Psi_{\checkmark}=\Psi\) and restores the usual device-independent meaning. Under approximate fair sampling the post-selected and ideal distributions are within \(\epsilon/(1-\epsilon)\) in total variation distance (Proposition 4).

Load-bearing premise

The whole result rests on the hypothesis that a detector's acceptance probability factorizes into a part depending only on the setting and a part depending only on the quantum state, \(E(x,\rho)=E_C(x)E_Q(\rho)\), a property that cannot be verified from the observed statistics alone.

Editorial extensions

If this is right

A Bell violation obtained from post-selected data under fair sampling proves that the filtered state \(\Psi_{\checkmark}\) is Bell-correlated, not that the original prepared state is.
Self-testing, certified randomness, and DI-QKD protocols run on post-selected data certify properties of \(\Psi_{\checkmark}\); a separate argument is needed to relate them to \(\Psi\).
If strong fair sampling holds, the post-selected statistics coincide with ideal lossless statistics on the original state \(\Psi\), so all usual device-independent conclusions are restored unchanged.
Small deviations from exact fair sampling are controlled: the total variation distance between post-selected and ideal filtered-state distributions is at most \(\epsilon/(1-\epsilon)\), and Bell-operator expectation values shift by at most \(2\epsilon_{\rm tot}\) in units of the algebraic bound.
State-dependent fair sampling extends the result to devices that fail fair sampling globally but satisfy the factorization on the actual experimental state, reproducing the same filtered-state conclusion.

Reading between the lines

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

One consequence the authors leave implicit is that post-selected Bell tests are effectively entanglement-distillation procedures: the certified object is the state after local filtering, so the amount of certified randomness or key should be quoted together with the filter success probability.
The paper's recommendation to publish the operator \(M_{\checkmark}\) (or bounds on it) rather than the full POVM suggests a practical calibration standard for post-selected DI experiments, a protocol step that goes beyond the theorems proved here.
A quantitative prediction of Section 8 is that deliberately detuning the two detectors of a polarization analyser by a relative amount \(\delta\) should move the post-selected distribution by no more than \(\delta(1-\eta)/\eta\) in total variation distance; an experimental check of this bound would be a direct test of the approximate-fair-sampling framework.

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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

This paper is worth your time: it gives a clean, general answer to what survives of device-independent certification when you post-select under fair sampling. The short version is that post-selected statistics are exactly those of an ideal lossless experiment measuring a locally filtered state Ψ✓, so DI claims (Bell correlation, self-testing, randomness) apply to Ψ✓, not to the original Ψ. That's the right way to frame the detection loophole: fair sampling doesn't let you ignore loss; it tells you what you are actually certifying.

The main theorem (Prop 2) is solid. The filter formalism is intuitive, the proof in Appendix C checks out, and the equivalence with Berry et al.'s efficiency factorization (Prop 1) is explicitly acknowledged. The genuinely new pieces are the approximate fair sampling bound (Prop 4), the state-preparation-error bound (Prop 5), and the concrete polarization-analyser analysis. Prop 4's ε/(1−ε) total-variation bound is practical and constructive—they even give the explicit lossless device. The paper is honest about the central limitation: fair sampling is not verifiable from the observed statistics alone, and the local hidden-variable models that fake Bell violations violate the factorization. That's not a flaw in the paper; it's the nature of the assumption, and they say so clearly.

Soft spots, in proportion. The bound in Prop 4 is not tight, and they admit it; for low detector efficiencies ε/(1−ε) can get large, so the 'robustness' guarantee is meaningful only for small deviations. The cryptographic section (Sec 5) is more of a threat-model discussion than a formal security proof—it argues composability, but I'd want a fuller proof if this were a crypto paper. The non-i.i.d. extension is sketched rather than proven in detail; the definition is plausible but the treatment is lighter than the i.i.d. case. None of this undercuts the central argument.

Who it's for: anyone running or interpreting Bell tests with post-selection—experimentalists in quantum optics, DI-QKD/QRNG people, and theorists who want to know what fair sampling actually buys. It deserves a serious referee. I'd accept it at a good journal, and I'd cite it in my own work when discussing detection loopholes.

Recommendation: send to peer review. My verdict: accept, with minor revisions at most; the authors could tighten the non-i.i.d. section and add a note about the regimes where the ε/(1−ε) bound is loose.

Referee Report

0 major / 6 minor

Summary. This paper analyses what can be concluded in device-independent protocols when Bell tests are performed with lossy detectors and the data are post-selected under a fair sampling assumption. The authors represent a lossy measurement as a filter followed by a lossless measurement, and define weak fair sampling as the factorization of the filter into classical and quantum parts (Eq. (11)). Their central result (Proposition 2, Eq. (17)) states that, under this assumption, the post-selected statistics of any Bell experiment are exactly reproduced by an ideal experiment in which lossless devices measure the locally filtered state Psi_check, which is obtained from the actual state by local probabilistic operations. They then define strong fair sampling (Section 4), for which Psi_check = Psi; discuss cryptographic consequences and an explicit attack (Section 5); prove an approximate version (Proposition 4) with a total-variation bound epsilon/(1-epsilon); prove robustness to state-preparation imperfections (Proposition 5); and apply the formalism to polarization analysers (Sections 6 and 8) and to state-dependent fair sampling (Section 9).

Significance. If the results hold — and the proofs in the appendices are sound — the paper provides a clean and useful clarification of an assumption that is ubiquitous in experimental Bell tests. The contribution is not merely terminological: Proposition 2 gives an explicit, falsifiable identity, Proposition 4 gives a quantitative robustness guarantee with a constructive proof, and the optical examples show how to certify the needed POVM element M_check in practice. The authors also honestly state the main limitation, namely that fair sampling cannot be verified from the observed statistics alone (Section 3), and they carefully phrase the conclusions as applying to the filtered state rather than to the original state. The equivalence to the earlier definition of Berry et al. is disclosed, and the new claims do not rely on circular reasoning or fitted parameters.

minor comments (6)

[Section 2.2 and Appendix D, Eq. (78)] The "lossless" POVM elements sum to the projector Pi_check rather than to the identity; since the relevant state rho_check has support in Pi_check this is harmless, but the completion to a full POVM on the orthogonal complement should be stated explicitly.
[Appendix C, after Eq. (67)] The sentence introducing the symbols defines F_{C,check} twice; the second occurrence should refer to F_{Q,check}.
[Section 8.1, Eqs. (42) and (44)] The no-click operators are typeset with superscripts in a way that obscures the intended expression R^{\hat N_theta}(1+delta)^{\hat N_theta}R^{\hat N_{theta_perp}}; please fix the typesetting.
[Appendix G, Eq. (109)] The inequalities in this equation contain garbled subscripts and superscripts (for example, the terms involving R^{n-1} and R_2) that should be cleaned up; the final result is correct.
[Sections 8.2 and 8.3] The computed approximation error epsilon = (1-eta)delta/eta is used in Proposition 4, which requires epsilon < 1; please state the corresponding restriction delta < eta/(1-eta) explicitly.
[Sections 2.2, 3, and 4] The notation F_{Q,check} is used both for a Kraus operator (e.g., Eq. (16)) and for the corresponding CP map (e.g., Eq. (20)); defining this distinction once would avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Proposition 2 is a direct theorem from the explicitly stated filter-factorization assumption, with self-contained proofs and no fitted inputs.

full rationale

The central claim, Eq. (17), is proved from Definition 1 by an explicit construction of the filtered state Ψ✓ and lossless devices M✓_k, with the verification carried out in Appendix C. The fair sampling definition (Eq. 11) is a condition on the device filter: it requires factorization into a classical part and a quantum part. It does not define fair sampling in terms of the post-selected distribution or in terms of the lossless filtered-state distribution that Proposition 2 derives, so the result is not self-definitional. Proposition 1's equivalence to the Berry et al. formulation is established inside the paper through Eqs. (14)-(16), not imported as an unverified black box; the paper openly states the equivalence, so this is disclosed benchmarking rather than a renamed result. No parameter is fitted to data, and no quantity is relabeled as a prediction: Ψ✓ and M✓_k are constructed from the assumed factorization, and the probability identity is checked by a direct trace computation. The approximate fair sampling result (Proposition 4) is an explicit epsilon-delta bound proven in Appendix D from the operator-norm condition (39); the bound ε/(1-ε) is derived, not assumed. The self-citations in the paper (e.g., Refs. [24], [25], [38], [41]) appear as contextual applications or examples and are not load-bearing for the main derivation. The paper's own admission that fair sampling cannot be verified device-independently is an honest premise-level limitation, not a circular step. Overall, the derivation is self-contained conditional on the stated assumption, and no circularity was found.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted. The central premise is the fair sampling factorization, an untestable device assumption explicitly identified by the authors. No new physical entities are introduced.

assumptions (6)

domain assumption Quantum theory provides a valid description of the source and measurement devices (Section 1).
All propositions are stated within the POVM and CPTP formalism of quantum mechanics.
domain assumption Outcomes in a sequence of measurement rounds are independent and identically distributed (Section 1).
The main theorems assume i.i.d. rounds; Section 5.2 provides a non-i.i.d. extension under a stricter definition of fair sampling.
domain assumption Fair sampling factorization holds: F = ∧(F_C ⊗ F_Q), equivalently E(x,ρ) = E_C(x) E_Q(ρ) (Deﬁnition 1 and Proposition 1, Eqs. 11-12).
This is the central unverifiable device premise; if it fails, Proposition 2 breaks and local hidden-variable models can fake Bell violations via post-selection.
domain assumption For cryptographic scenarios, fair sampling must hold also according to the adversary's POVM description (Section 5).
Security is proven only against adversaries whose description of the device also satisfies fair sampling; the Makarov attack shows what happens when this fails.
domain assumption The POVM elements M_x✓ (or their estimates) are known or calibrated for the approximate results (Section 7).
Proposition 4 defines ε from the actual POVM; applying the bound in practice requires knowledge of these operators.
domain assumption No-leakage assumption in cryptographic protocols (Section 5).
Standard device-independent security assumes no information about the measurement leaks to the adversary; the paper extends this to post-selected settings.

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Cite this review

Pith. "Pith review of How post-selection affects device-independent claims under the fair sampling assumption." pith.science (2026). https://pith.science/paper/BI2HCJ3Q

@misc{pith2026190811123,
  author       = {Pith},
  title        = {Pith review of: How post-selection affects device-independent claims under the fair sampling assumption},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BI2HCJ3Q}},
  note         = {Machine review of arXiv:1908.11123}
}

read the original abstract

Device-independent certifications employ Bell tests to guarantee the proper functioning of an apparatus from the sole knowledge of observed measurement statistics, i.e. without assumptions on the internal functioning of the devices. When these Bell tests are implemented with devices having too low efficiency, one has to post-select the events that lead to successful detections and thus rely on a fair sampling assumption. The question that we address in this paper is what remains of a device-independent certification under fair sampling. We provide an intuitive description of post-selections in terms of filters and define the fair sampling assumption as a property of these filters, equivalent to the definition introduced in [Berry et. al., PRA 81(1), 012109 (2010)]. When this assumption is fulfilled, the post-selected data is reproduced by an ideal experiment where lossless devices measure a filtered state which can be obtained from the actual state via local probabilistic maps. Trusted conclusions can thus be obtained on the quantum properties of this filtered state and the corresponding measurement statistics can reliably be used, e.g., for randomness generation or quantum key distribution. We also explore a stronger notion of fair sampling leading to the conclusion that the post-selected data is a fair representation of the data that would be obtained with lossless detections. Furthermore, we show that our conclusions hold in cases of small deviations from exact fair sampling. Finally, we describe setups previously or potentially used in Bell-type experiments under fair sampling and identify the underlying device-specific assumptions.

Figures

Figures reproduced from arXiv: 1908.11123 by the authors.

**Figure 2.** Example of a two-party experiment (e.g., a test of a Bell inequality). The picture schematically [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** A polarization analyser. The polarization of the incoming photons is transformed by a variable [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

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