Generalized measurements for Bell tests in different probability spaces

Alfredo Luis

REVIEW 2 major objections 5 minor 1 cited by

Generalized measurements for Bell tests in different probability spaces

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read One generalized photon measurement implements two Bell-test probability spaces, and re-labeling its four detector clicks opens new conditional-probability Bell inequalities that standard approaches cannot produce.

desk verdict Two-space relabeling is real; the new Bell tests are not established — Eq. (3.32) needs a setting-independence assumption the paper denies. read the letter →

arxiv 2506.07496 v1 pith:6JA4RB7N submitted 2025-06-09 quant-ph

Alfredo Luis This is my paper

classification quant-ph PACS 03.65.Ta03.65.Ud

keywords quantumBelltestsgeneralizedmeasurementsprobabilityspacesconditionalprobabilitiesPOVMentanglementqubittomographyeight-porthomodynedetector

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 shows that a single four-detector photon arrangement can serve as the practical implementation of two very different probability spaces for Bell tests. In one reading of the same clicks, the outputs encode which of two observables is measured and its outcome, reproducing the statistics of randomly chosen exact measurements. In the other reading, the same clicks encode a noisy simultaneous measurement of two complementary qubit observables, with the noise invertible in a state-independent way. Because both descriptions are fed by the same raw data, the paper argues that Bell tests tied to either probability space can be run without new experiments, and that swapping the roles of outcomes and settings in the first space yields dual and mixed Bell-type inequalities not available in standard approaches.

What carries the argument

The carrying object is a generalized measurement, or POVM (a family of positive operators summing to the identity), formed by mixing the signal modes with vacuum at a beam splitter, applying SU(2) polarization rotations, and registering which of four detectors clicks. The same click pattern is read twice: as pairs $(j,\alpha)$ that make the POVM factor into a state-independent setting probability times an exact projection-valued measure, and as pairs $(j,k)$ that make it a noisy joint measurement of $X$ and $Y$ with product $Z$, whose noise is exactly invertible through $\tilde p_K(\kappa|\kappa')=\frac{1}{2}(1+\kappa\kappa'/\gamma_K)$. The constraint $\gamma_X^2+\gamma_Y^2+\gamma_{XY}^2=1$ makes the four-outcome POVM a valid measurement and, in the symmetric case, reduces it to the minimal qubit-tomography POVM.

What would settle it

Sweep the beam-splitter parameters $r,t$ and the polarization parameters $\theta,\phi$ for an entangled two-photon state, compute $C'$ and $C''$ from the paper's formulas, and check whether either ever falls outside $[-1,0]$; if no state and settings do so, the claim that these are Bell tests is falsified.

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Extended reading notes

Core claim

The central claim is that the same lossless beam-splitter plus polarization-rotation plus four-click detection scheme realizes the two probability spaces of Ref. [13,14] merely by how the clicks are labeled. In probability space 1 the POVM is $\Delta_A(j,\alpha)=p(\alpha)\frac{1}{2}(\sigma_0+j\mathbf{S}_\alpha\cdot\boldsymbol{\sigma})$, so conditioning on $\alpha$ gives the exact statistics of observable $A_\alpha$, and the setting probabilities $p(\alpha)$ are independent of the state. In probability space 2 the POVM is $\Delta_A(j,k)=\frac{1}{4}(\sigma_0+j\gamma_X\mathbf{S}_X\cdot\boldsymbol{\sigma}+k\gamma_Y\mathbf{S}_Y\cdot\boldsymbol{\sigma}+jk\gamma_{XY}\mathbf{S}_{XY}\cdot\boldsymbol{\sigma})$, a noisy joint measurement of $X$ and $Y$ plus the product variable $Z$, with $\gamma_X^2+\gamma_Y^2+\gamma_{XY}^2=1$ and noise removed by state-independent inversion. Starting from the conditional probabilities of space 1, the paper derives a Bell bound $0\ge C\ge -1$ and shows that exchanging the roles of the variables produces analogous dual and mixed bounds, which it presents as new Bell tests.

Load-bearing premise

The new dual and mixed Bell tests rest on locality factorizations that are assumed rather than derived, and on the as-yet-undemonstrated existence of quantum states that violate the resulting bounds.

Editorial extensions

If this is right

A single run of the arrangement, with one set of raw clicks, supplies data for both the random-exact-measurement Bell analysis and the noisy-simultaneous-measurement analysis, so no second apparatus is needed.
In probability space 1 the Bell test can be evaluated without imposing a setting-independence condition $p(\lambda|\alpha,\beta)=p(\lambda)$, because the whole test lives in a single context.
The dual and mixed quantities $C'$ and $C''$ are candidates for Bell tests that do not exist in the standard separate-runs formulation, which never exposes the conditional probabilities with the roles of outcomes and settings reversed.
In probability space 2, unsharpness of the joint measurement is removed by a state-independent inversion, and at the symmetric point the POVM is exactly the minimal qubit-tomography POVM.

Reading between the lines

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

If entangled states are shown to violate the dual or mixed inequalities, the same dataset can be used to compare the two probability-space conclusions directly, turning the interpretive debate about probability spaces into an experimentally controlled comparison.
The beam-splitter construction is likely to generalize: adding more vacuum ports and re-labeling output patterns would produce further probability spaces and a hierarchy of generalized Bell inequalities for more than two observables per party.
The state-independent noise inversion suggests a practical two-for-one device: the same clicks that test Bell inequalities can also reconstruct the two-photon polarization state, since the symmetric case is tomographically complete.

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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

First thing to know: the core construction holds up. The same generalized-measurement setup, with only a relabeling of the four detector outcomes, realizes the two probability spaces of Refs. [13,14]. That's a clean observation and genuinely useful. I checked the POVM algebra; it's correct. The marginal result p(α,β)=p(α)p(β) is also right. The paper deserves credit for making this explicit.

The soft spot is the Bell-test claim. The derivation of the bound 0≥C≥−1 asserts that Eq. (3.31) (locality factorization) is enough, and that no setting independence p(λ|α,β)=p(λ) is needed because there is a single context. That is false. A simple local model satisfying (3.31) — four equally likely λ's fixing (j,α,k,β) as in the stress-test — gives C=1, outside the claimed bound. The problem is that with no free settings, the hidden variable can determine which observable is measured; that is exactly the conspiracy the standard Bell derivation rules out. The paper's setup has no free choice of settings, so this is not a minor technicality. The same defect propagates to C' and C''.

Also, the paper never shows a quantum state violating C' or C''. The inequalities are asserted to be Bell tests, but without a violation example or at least a numerical demonstration, the 'novel Bell tests' claim is unsupported. Hand-waving to previous works [19,22,24] does not cover these specific inequalities.

One more thing: the paper cites Refs. [13,14] for the bound (3.32). I would want to know whether those references include the setting-independence assumption. If they do, the paper drops it without justification; if they don't, the bound is questionable there too.

Overall: the two-space construction is a legitimate contribution to the generalized-measurement Bell literature. The Bell-test claims need serious repair. A referee should send back a version that adds the missing assumption, re-derives the bounds, and shows at least one quantum violation of C' or C''. As written, the paper is incomplete on its main advertised novelty.

Recommendation: send to peer review, not desk reject. The construction is worth publishing; the Bell-test part needs work. A serious referee should engage.

Referee Report

2 major / 5 minor

Summary. The paper presents a generalized-measurement scheme in which a single experimental arrangement (a beam splitter mixing signal modes with vacuum, followed by SU(2) polarization transformations and four detectors per subsystem) yields four raw outcomes per party. By relabeling these outcomes, the same detector data are interpreted in two distinct 'probability spaces': in probability space 1, the four outputs are grouped into two dichotomic observables A_±1 with outcome j; in probability space 2, the same outputs define dichotomic variables j,k that constitute a joint noisy measurement of X and Y (and Z). The paper derives the POVMs and marginal/conditional probability relations, and claims that this permits new Bell tests C and C' (and a mixed variant C'') with bounds 0 ≥ C ≥ −1, which are said to require no setting-independence condition because all observables are measured in a single context.

Significance. The proposed unified measurement scheme is natural and potentially useful: the same raw data can be analyzed under two statistical frameworks, and the POVM constructions in Sections III.A–C and IV are derived carefully within standard quantum mechanics. The paper also contains a nice explicit example, including the relation γ_X^2 + γ_Y^2 + γ_XY^2 = 1 and the minimal-tomography limit in Eq. (4.9). However, the advertised 'novel Bell tests' are not established: the derivation of the bound for C omits a necessary assumption, and no quantum state is shown to violate C' or C''. The paper's central novelty claim therefore currently rests on an unsupported assertion, and the two-space construction, while promising, is not accompanied by a demonstrated Bell-test violation.

major comments (2)

[III.D, Eq. (3.32)] The claimed bound 0 ≥ C ≥ −1 does not follow from the locality condition (3.31) alone. Consider four equally likely hidden variables λ_{++}, λ_{+-}, λ_{-+}, λ_{--} that fix the joint event (j,α,k,β) deterministically as (+,+,+,+), (+,+,-,-), (+,-,+,+), (+,-,+,-), respectively. This model satisfies p(j,k|α,β,λ)=p(j|α,λ)p(k|β,λ) and has p(α,β)=p(α)p(β)=1/4, yet for j=k=+ Eq. (3.30) gives C=1, violating the claimed bound. Therefore the statement in Section III.D that 'there is no need to impose a setting independence condition' is incorrect. The derivation of Eq. (3.32) must include an explicit condition such as p(λ|α,β)=p(λ), or a physical justification of why the hidden-variable distribution is independent of the displayed settings in this single-context setup; otherwise the Bell-test interpretation of C is unfounded.
[III.E, Eqs. (3.34)–(3.40)] The 'novel Bell tests' C' and C'' are not established. The locality conditions (3.34) and (3.38) are asserted without derivation or physical justification, and the bounds (3.35) and (3.39) are stated as 'analogous' without proof. More importantly, the paper gives no example of a quantum state (and no numerical scan) for which C' or C'' violates its bound. Since the abstract and introduction advertise these as new Bell tests, the paper must provide a concrete quantum violation (for instance, a computation of C' and C'' for an entangled two-photon state within the POVM of Section IV) or clearly label these inequalities as conjectural. Without such evidence, the claim that these are 'Bell tests' is unsupported.

minor comments (5)

[Abstract and Introduction] The word 'withing' appears in the Introduction ('not possible withing more standard approaches'); it should be 'within'.
[Section IV] The word 'unnormlized' in the sentence before Eq. (4.3) should be 'unnormalized'.
[Section V] The conclusion contains the phrase 'de marginal and conditional statistics'; it should be 'the marginal and conditional statistics'.
[References] Reference [21] appears to have a missing volume/article number: 'Phys. Rev. A 111022204 (2025)' should likely read 'Phys. Rev. A 111, 022204 (2025)'.
[Introduction and Section III.D] The term 'probability space' is used throughout but never defined. Since the two-space claim is central to the paper, a brief definition or a summary of the two probability spaces from Refs. [13,14] would make the manuscript more self-contained and easier to verify.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the two-probability-space construction is derived explicitly from the measurement model and standard quantum traces, with no fitted parameter or load-bearing self-citation.

full rationale

The paper's derivation chain is self-contained. The generalized measurement is constructed explicitly from beam-splitter relations and SU(2) transformations (Eqs. 2.2-2.12); the POVMs for probability space 1 follow from those projectors (Eqs. 3.1-3.11); and the joint probabilities are obtained by standard trace evaluations (Eq. 3.28). The standard Bell bound in Eq. (3.32) is quoted from independent Refs. [13,14] under the explicit locality factorization (3.31), not derived from the present paper's own prior results. The dual and mixed Bell-type bounds in Section III.E are asserted by analogy and are not reductions to fitted inputs; any mathematical insufficiency in deriving those bounds, such as an implicit setting-independence assumption, would be a correctness or validity concern rather than circularity. Self-citations [19,22,24] are used only to refer to additional results for probability space 2 and are not load-bearing for the central two-space construction or for the conditional-probability bounds. No equation in the paper reduces to its own input by construction, and no parameter fitted to data is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central construction relies on standard quantum measurement theory and a specific optical model, plus ad hoc locality assumptions for the new inequalities. There are no free parameters fitted to data, and no new physical entities are postulated.

assumptions (3)

standard math Quantum mechanical Born rule and projection postulate for POVMs
Used throughout to compute click probabilities from the state in Eqs. (2.11)-(2.12) and (3.8)-(3.11).
domain assumption Two-mode single-photon polarization qubit model for Bell tests
The entire scheme assumes each subsystem is a single photon in two polarization modes, with a lossless beam splitter and vacuum auxiliary inputs. This limits the practical validity to that physical realization.
ad hoc to paper Hidden-variable locality factorization p(j,k|α,β,λ)=p(j|α,λ)p(k|β,λ) and the analogous factorizations (3.34), (3.38)
These are the hidden-variable assumptions used to derive the Bell bounds. They are natural analogues of Bell locality but are introduced ad hoc, especially the dual form conditioning on j,k in Eq. (3.34).

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Cite this review

Pith. "Pith review of Generalized measurements for Bell tests in different probability spaces." pith.science (2026). https://pith.science/paper/6JA4RB7N

@misc{pith2026250607496,
  author       = {Pith},
  title        = {Pith review of: Generalized measurements for Bell tests in different probability spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6JA4RB7N}},
  note         = {Machine review of arXiv:2506.07496}
}

read the original abstract

Bell tests are of profound statistical nature. Besides physical considerations, the proper understanding of their implications should involve detailed statistical analyses. In this regard, recent works have shown that their consequences and interpretations depend on the probability space adopted. Some other recent works have also shown that generalized measurements may allow to further exploit the statistics of Bell-like tests. Following these ideas, in this work we show that one and the same experimental arrangement can provide a practical scheme valid for two very different probability spaces. Moreover, we show that this allows the introduction of novel Bell tests that are not possible in more standard approaches.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Conditional probabilities in quantum-optical settings
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Conditioning on one outcome of a joint noisy measurement can reduce (even eliminate) uncertainty in a complementary variable, and the resulting conditional distribution generally has no Born-rule representation via th...

Reference graph

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Similarly regarding the clicks in modes a2, that is |φ2x⟩=|0⟩1x|0⟩1y|Ω2, 1⟩, |φ2y⟩ = |0⟩1x|0⟩1y|Ω2, −1⟩, (2.8) where |Ω2, 1⟩ = cos θ2 2 |1⟩2x|0⟩2y + eiϕ2 sin θ2 2 |0⟩2x|1⟩2y, (2.9) and |Ω2, −1⟩ = − sin θ2 2 |1⟩2x|0⟩2y + eiϕ2 cos θ2 2 |0⟩2x|1⟩2y, (2.10) where Ω 2 = (θ2, ϕ2) are polarization parameters associ- ated to the polarization transformation SU(2) 2...

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Similarly regarding the clicks in modes a2, that is |φ2x⟩=|0⟩1x|0⟩1y|Ω2, 1⟩, |φ2y⟩ = |0⟩1x|0⟩1y|Ω2, −1⟩, (2.8) where |Ω2, 1⟩ = cos θ2 2 |1⟩2x|0⟩2y + eiϕ2 sin θ2 2 |0⟩2x|1⟩2y, (2.9) and |Ω2, −1⟩ = − sin θ2 2 |1⟩2x|0⟩2y + eiϕ2 cos θ2 2 |0⟩2x|1⟩2y, (2.10) where Ω 2 = (θ2, ϕ2) are polarization parameters associ- ated to the polarization transformation SU(2) 2...

work page

[2] [2]

J. S. Bell, On the Einstein Podolsky Rosen paradox, Physics 1, 195–200 (1964)

work page 1964

[3] [3]

L. E. Ballentine, Quantum Mechanics (Prentice Hall, En- glewood Cliffs, 1990). Chapter 20

work page 1990

[4] [4]

R. F. Werner and M. M. Wolf, Bell inequalities and En- tanglement, arXiv:quant-ph/0107093

arXiv

[5] [5]

Fine, Hidden Variables, Joint Probability, and the Bell Inequalities, Phys

A. Fine, Hidden Variables, Joint Probability, and the Bell Inequalities, Phys. Rev. Lett 48, 291–295 (1982)

work page 1982

[6] [6]

Rivas, On the role of joint probability distributions of incompatible observables in Bell and Kochen–Specker Theorems, Ann

A. Rivas, On the role of joint probability distributions of incompatible observables in Bell and Kochen–Specker Theorems, Ann. Phys. (N.Y.) 411, 167939 (2019)

work page 2019

[7] [7]

J. A. de Barros, J. V. Kujala and G. Oas, Negative prob- abilities and contextuality, J. Math. Psychol. 74, 34–45 (2016)

work page 2016

[8] [8]

Clauser and M.A

J.F. Clauser and M.A. Horne, Experimental conse- quences of objective local theories, Phys. Rev. D10, 526– 535 (1974)

work page 1974