On the spin projection operator and the probabilistic meaning of the bipartite correlation function

Ana Mar\'ia Cetto; Andrea Vald\'es-Hern\'andez; Luis de la Pe\~na

REVIEW 3 major objections 3 minor 16 references

On the spin projection operator and the probabilistic meaning of the bipartite correlation function

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The spin-singlet correlation is not constrained by the CHSH inequality, because each pair of detector directions partitions the probability space in its own way.

desk verdict A correct and clean derivation that ultimately restates the measurement-dependence loophole; the claim that Bell inequalities don't apply to the singlet is an overreach. read the letter →

arxiv 1908.04225 v1 pith:DMLMVYTH submitted 2019-08-12 quant-ph

Ana María Cetto , Andrea Valdés-Hernández , Luis de la Peña This is my paper · ORCID

classification quant-ph MSC 81P1581P40 PACS 03.65.Ta03.65.Ud

keywords BellinequalitiesspinsingletstateCHSHinequalityquantumcorrelationprobabilityspacepartitioningcontextualityhiddenvariablesprojectionoperator

topics The Measurement Problem

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 tries to establish that Bell-type inequalities, such as CHSH, are not applicable to the bipartite spin-1/2 singlet state. The reason, it argues, is that the quantum correlation $C_Q(\boldsymbol a,\boldsymbol b)$ carries a definite probabilistic meaning: it averages the product eigenvalues $\pm1$ over four subensembles defined by the eigenbasis of $(\hat\sigma\cdot\boldsymbol a)\otimes(\hat\sigma\cdot\boldsymbol b)$. Since that eigenbasis, and hence the partition of the probability space, changes when either direction is changed, correlations for different direction pairs live on incommensurable partitions. The standard derivation of CHSH combines such correlations under one integral over a single probability space, which the paper says is unjustified. If correct, this would remove the singlet spin correlation from the set of Bell-test witnesses against local hidden variables.

What carries the argument

The load-bearing object is the eigenbasis $\{|\phi_k\rangle\}$ of the operator $(\hat\sigma\cdot\boldsymbol a)\otimes(\hat\sigma\cdot\boldsymbol b)$, built from the individual eigenstates $|\pm\boldsymbol a\rangle$, $|\pm\boldsymbol b\rangle$. Inserting the resolution of the identity in this basis turns the correlation into $C_Q(\boldsymbol a,\boldsymbol b)=\sum_k A_k C_k$, with $A_k$ the product eigenvalues and $C_k$ the Born-rule weights; the paper identifies $C_k$ as joint probabilities. The direction-dependence of this basis is the mechanism: it makes the partition of the ensemble depend on $(\boldsymbol a,\boldsymbol b)$, which the paper contrasts with the single fixed probability space $\Lambda$ used in Bell-type derivations.

What would settle it

If a shared fixed probability space with a single partition were shown to reproduce all four correlations used in CHSH for a singlet state, the incommensurability claim would fail; one way to check is to attempt to construct a joint distribution over the 16 assignments for settings $(\boldsymbol a,\boldsymbol a',\boldsymbol b,\boldsymbol b')$ whose marginals match the four quantum correlations.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the familiar result $C_Q(\boldsymbol a,\boldsymbol b)=-\boldsymbol a\cdot\boldsymbol b$ for the singlet state can be written as $\sum_k A_k C_k$, where $A_k=\alpha_k\beta_k\in\{\pm1\}$ are the eigenvalues of the product of the two spin-projection operators and $C_k=|\langle\phi_k|\Psi_0\rangle|^2$ are nonnegative weights summing to one. The weights are therefore joint probabilities for the four outcome pairs $(+,-)$, $(-,+)$, $(+,+)$, $(-,-)$ in the basis built from eigenstates of $\hat\sigma\cdot\boldsymbol a$ and $\hat\sigma\cdot\boldsymbol b$. The paper then claims that a different pair $(\boldsymbol a,\boldsymbol b')$ requires a different basis and therefore a different partition of the same ensemble, so the four subensembles for different settings are mutually incommensurable. From this it concludes that the CHSH combination of four such correlations cannot be rearranged under a single integral, and hence that there is no reason for $C_Q$ to satisfy the Bell-type bound.

Load-bearing premise

The paper assumes that the hidden-variable probability space must be partitioned according to the quantum eigenbasis of the product spin operator for each direction pair, an assumption Bell's own argument does not make.

Editorial extensions

If this is right

CHSH experiments on spin-1/2 singlet states would no longer count as tests of local hidden variables, because the inequality being tested does not follow for these correlations.
The experimentally observed violation $2\sqrt{2}$ would be reinterpreted as the natural value of a correlation that lives on direction-dependent partitions, not as a refutation of locality.
New Bell-type inequalities would be needed that explicitly allow each setting pair to define its own partition of the probability space.
The contextuality or measurement-dependence loophole becomes the central question: any single-partition hidden-variable model is, by construction, missing the direction-dependent subensembles.
The paper's decomposition also applies to the individual terms $F_1,\dots,F_4$, so the correlation's probabilistic reading extends beyond the final value $-\boldsymbol a\cdot\boldsymbol b$.

Reading between the lines

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

A testable extension of the paper's logic would be to derive explicit direction-dependent partition models for the singlet state and check whether such models reproduce the full set of quantum correlations beyond CHSH expressions.
The same eigenbasis-partition argument could be carried to other maximally entangled states or to higher spin, where the product operator has more than four eigenvalues, potentially producing new inequalities.
If the argument is right, it suggests that experimental Bell tests should report correlations as functions of each setting pair with their own data subsamples, and that combining across settings is not just a statistical choice but a physically loaded one.
The paper leaves open whether the direction-dependent partitions could be embedded in a larger single probability space with extra variables encoding the settings; that is a direct route to test the incommensurability claim.

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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

Before you spend time on this: the math is clean, but the paper's punchline is overblown. It walks through the standard eigenbasis expansion of the singlet correlation and then concludes that Bell-type inequalities do not apply to the singlet. What it has really shown is the measurement-dependence/contextuality loophole, which Bell's theorem already accommodates as an explicit assumption.

The authors do a clear job of writing C_Q(a,b) = -a·b as a sum over the eigenstates of σ·a and σ·b, making explicit that the coefficients are joint probabilities for the four outcome pairs and that those probabilities change with the settings. For someone unfamiliar with the contextuality loophole, this is a useful pedagogical derivation. They also cite the relevant prior literature, including their own 1972 paper with Brody, and acknowledge the loophole in the final paragraphs. So the honest parts are in place.

The soft spot is the inference. From 'the partition of the probability space depends on (a,b)' they jump to 'there is no reason why the quantum correlation should obey CHSH.' That jump only works if you have already imposed that any hidden-variable model must track the quantum eigenbasis, with the probability space itself changing as the settings change. Bell's theorem does not assume that. It assumes a single probability space and response functions α(a,λ), β(b,λ) that are independent of the other setting. The model in Eqs. (44)-(48) has partitions Λ_k(a,b); the response functions and even the domains depend on the settings. That is measurement dependence by definition. It is a known loophole, not a refutation. The abstract's claim that the analysis 'puts into question the applicability' of Bell inequalities is an overstatement. The correct conclusion is: if you allow measurement dependence, CHSH can be violated; under Bell's stated assumptions, the inequality applies, and the singlet violates it.

The paper also does not engage with the loophole-free experiments. If you are going to claim Bell tests on spin-1/2 singlets don't establish what they are used for, you owe the reader a treatment of how those experiments avoid measurement dependence. That is missing.

For a foundations reading group, this is a decent pedagogical piece to pair with a Bell paper, because it makes the contextuality loophole concrete. It does not advance the Bell debate. If I were the editor, I'd let a referee look at it, mostly to get a report on record that the conclusion is a restatement of a known caveat, not a challenge. I would not desk-reject it outright, because the algebra is solid and the authors are honest about their lineage. But I would expect the referee to request a reframing—or recommend rejection.

Referee Report

3 major / 3 minor

Summary. The manuscript revisits the single-particle spin projection operator and then computes the bipartite singlet correlation C_Q(a,b) = ⟨(σ̂·a)⊗(σ̂·b)⟩, obtaining the standard result -a·b. Section IV expands this correlation in the basis of simultaneous eigenstates of σ̂·a and σ̂·b, interpreting the squared coefficients C_k as joint probabilities P_ab(α,β) for the four outcome pairs. Section V argues that, because the basis and hence the partition of the probability space changes with (a,b), the four correlations entering the CHSH expression cannot be combined under a single integral, and concludes that there is no reason for the quantum correlation to obey Bell-type inequalities. The paper cites the contextuality/measurement-dependence loophole in its closing paragraphs.

Significance. If the central claim were correct, it would invalidate the standard interpretation of Bell tests for spin-1/2 singlet states. The algebraic calculation of C_Q and the basis expansion are correct and clearly presented; the paper also makes explicit the setting dependence of the quantum probability decomposition. However, the conclusion rests on a misreading of Bell's assumptions: the hidden-variable model constructed in Eqs. (44)-(48) has setting-dependent partitions, i.e., it is measurement-dependent, and measurement independence is an explicit assumption of Bell's theorem. The paper therefore demonstrates only the well-known fact that abandoning measurement independence evades the inequality; it does not show that Bell inequalities are inapplicable to the singlet state under the standard local hidden-variable hypotheses. Its main value is as a pedagogical exposition of basis dependence and contextuality, not as a refutation of Bell's theorem.

major comments (3)

[Section V, Eqs. (44)-(48) and (51)] The hidden-variable translation introduced here is measurement-dependent: the subensembles Λ_k(a,b,α_k,β_k), and hence the response functions in Eq. (48), depend on both settings a and b. In the Bell-CHSH derivation, by contrast, Eq. (49) assumes a single probability space Λ and setting-independent functions α(a,λ), β(b,λ) defined for all settings simultaneously; this is the measurement-independence assumption. Consequently, the inability to group the four terms in Eq. (51) under one integral shows only that a measurement-dependent model can violate the CHSH inequality, a point the authors themselves acknowledge in the final paragraph. It does not establish that the quantum correlation need not obey Eq. (50) under the assumptions of the theorem.
[Section IV, Eqs. (36)-(41), and Section V] The 'unequivocal probabilistic reading' of C_Q is anchored to the eigenbasis selected by the pair (a,b), so the probabilities C_k are not the probabilities over predetermined outcomes for all settings that Bell's local hidden-variable model postulates. The statement that combinations of eigenvalues from different pairs are 'physically meaningless' is the load-bearing premise, but it is an interpretive assumption about how hidden variables must relate to quantum bases, not a consequence of the operator algebra. Under Bell's assumptions, a single joint distribution over all four settings exists by hypothesis, and the question is whether it can reproduce the quantum correlations; the basis dependence of the quantum expansion does not by itself rule this out.
[Abstract and Section V, Eq. (50)] The paper's strongest claim—that 'there is no reason' for C_Q(a,b) to obey inequality (50)—does not follow from the preceding analysis. Under the standard assumptions encoded in Eq. (49), the inequality is a theorem, and the authors have not shown that those assumptions are inconsistent with the singlet state; they have only shown that their basis-dependent expansion does not directly map onto the hidden-variable integral. This is a failure of the paper's central inference, not a limitation of Bell's theorem.

minor comments (3)

[Eq. (11)] The symbols θ and ϕ are used for both angles and unit vectors; the right-hand side of Eq. (11) should be typeset with explicit unit vectors (e.g., e^{iϕ}(θ_hat + i ϕ_hat)·a) to avoid ambiguity.
[Throughout] There are typographical errors: 'anaysis' in Section V, 'appropiate' in the Introduction, and 'colinear' in Section II should be corrected, and the abbreviation 'a.s.o.' should be replaced by 'and so on' or similar.
[Footnote [3]] The definition s_i · s_j = −s_j · s_i for i ≠ j conflicts with the ordinary scalar product; the intended geometric product should be distinguished from the dot product.

Circularity Check

1 steps flagged · score 5.0 of 10

The paper's claim that Bell-type inequalities are inapplicable is built into its own setting-dependent definition of the hidden-variable probability-space partition.

self definitional [Section V, Eqs. (44)-(51)]
"The partitioning of the probability space Λ corresponding to Eq. (36) can be expressed as C_Q(a,b)=∑_k ∫_{Λ_k} A_k(a,b,λ)ρ(λ)dλ, where Λ_k=Λ_k(a,b,α_k,β_k) ... Consequently, one cannot group the four terms under the same integral sign ... Ergo, there is no reason why the quantum correlation C_Q(a,b) should obey the inequality (50)."

The paper's hidden-variable 'translation' (Eqs. 44-48) posits that each correlation C_Q(a,b) lives on a partition Λ_k(a,b,...) that depends on the settings. This is not a consequence of quantum mechanics or of Bell's theorem; it is a modeling choice imported into the hidden-variable language. Bell's derivation starts from a single probability space Λ with setting-independent response functions, not from eigenbasis subensembles. Once Λ_k is defined as setting-dependent, the conclusion that the four CHSH terms are on incommensurable partitions is already contained in the definition, so the inference 'there is no reason to obey (50)' restates the assumption rather than proving that Bell inequalities fail for the singlet state.

full rationale

The paper's algebraic decomposition of C_Q(a,b) in the eigenbasis of the product spin operator (Section IV) is correct and self-contained: Eq. (36) is a standard spectral expansion with C_k nonnegative and summing to one. No parameter is fitted, no numerical prediction is disguised as a derivation, and the authors do not rely on a load-bearing self-citation; Refs. [8-15] merely supply prior statements of the contextuality/measurement-dependence objection. The circularity lies in Section V, where the basis-dependent decomposition is recast as a physical partition of a hidden-variable space with Λ_k = Λ_k(a,b,...). That recasting is the paper's own interpretive input, not a theorem; Bell's CHSH derivation does not partition Λ according to each detector-setting pair. The later claim that the quantum correlation need not obey inequality (50) is therefore a restatement of this input rather than an independent result. Because the paper's central conclusion is thus partly built into its definition of the probability space, while the underlying algebra is independent, a moderate self-definitional circularity score is appropriate.

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

The paper introduces no free parameters and no new entities. It relies on standard quantum mechanics and one additional interpretive premise that is the crux of the argument.

assumptions (3)

domain assumption The Born rule: joint probabilities are squared amplitudes |<phi_k|Psi_0>|^2
Used in Eq. (37) to identify C_k as joint probabilities.
standard math The set {|phi_k>} of eigenstates of sigma dot a and sigma dot b forms a complete orthonormal basis
Standard linear algebra; used in Eq. (33) to insert the identity.
ad hoc to paper Combining probability weights from different basis decompositions is physically meaningless
This is the load-bearing premise in Section V. It is not a theorem of quantum mechanics and is exactly the assumption that makes the conclusion depend on the authors' interpretation.

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Pith. "Pith review of On the spin projection operator and the probabilistic meaning of the bipartite correlation function." pith.science (2026). https://pith.science/paper/DMLMVYTH

@misc{pith2026190804225,
  author       = {Pith},
  title        = {Pith review of: On the spin projection operator and the probabilistic meaning of the bipartite correlation function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DMLMVYTH}},
  note         = {Machine review of arXiv:1908.04225}
}

abstract

Spin is a fundamental and distinctive property of the electron, having far-reaching consequences in wide areas of physics. Yet, further to its association with an angular momentum, the physics underpinning its formal treatment remains obscure. In this work we propose to advance in disclosing the meaning behind the formalism, by first recalling some basic facts about the one-particle spin operator. Consistently informed by and in line with the quantum formalism, we then proceed to analyse in detail the spin projection operator correlation function $C_Q(\boldsymbol{a}, \boldsymbol{b}) = \langle(\hat{\sigma}\cdot\boldsymbol{a})(\hat{\sigma}\cdot\boldsymbol{b})\rangle$ for the bipartite singlet state, and show it to be amenable to an unequivocal probabilistic reading. In particular, the calculation of $C_Q(\boldsymbol{a}, \boldsymbol{b})$ entails a partitioning of the probability space, which is dependent on the directions $(\boldsymbol{a}, \boldsymbol{b})$. The derivation of the CHSH- or other Bell-type inequalities, on the other hand, does not consider such partitioning. This observation puts into question the applicability of Bell-type inequalities to the bipartite singlet spin state.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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Let us brieﬂy look at the spin product for the bipartite singlet state from the perspective of algebraic geometry. If s and s′ are two antiparallel vectors, sa = exp(iϑ a), s′b = −sb = − exp(iϑ b), their product takes the form [4] (sa)∗ ( s′b ) = − exp (iθab) . Products containing s and s′ depend of course in general on both vectors. However, in this case...

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Tomonaga, The Story of Spin

S.-I. Tomonaga, The Story of Spin . The University of Chicago Press, 1997

work page 1997

[2] [2]

It is convenient to bear in mind that the angles ( θ, ϕ ) that deﬁne the direction of r are not set; the singlet state is spherically symmetric, and therefore r may be chosen arbitrarily. Now, in the composite system, the projection of the ﬁrst spin operator along an arbitrary direction a is de- scribed by ( ˆσ · a)⊗I, and the projection of the second spi...

work page

[3] [3]

Audretsch, Entangled Systems

J. Audretsch, Entangled Systems . Wiley-VCH, Bonn, 2006, Ch. 4

work page 2006

[4] [4]

In trying to get a more geometrical picture of the eﬀect of the operator ˆσ · a in 3D space, it may help to look at the problem from the perspective of algebraic geometry. Take s as the spin vector, and a the unitary vector as before; the product sa is then deﬁned as a combination of the scalar (or internal) product and the vector (or external) product [4...

work page

[5] [5]

Hestenes, New Foundations for Classical Mechanics

D. Hestenes, New Foundations for Classical Mechanics . Kluwer, Dordrecht, 1986, Ch. 2

work page 1986

[6] [6]

If s and s′ are two antiparallel vectors, sa = exp(iϑ a), s′b = −sb = − exp(iϑ b), their product takes the form [4] (sa)∗ ( s′b ) = − exp (iθab)

Let us brieﬂy look at the spin product for the bipartite singlet state from the perspective of algebraic geometry. If s and s′ are two antiparallel vectors, sa = exp(iϑ a), s′b = −sb = − exp(iϑ b), their product takes the form [4] (sa)∗ ( s′b ) = − exp (iθab) . Products containing s and s′ depend of course in general on both vectors. However, in this case...

work page

[7] [7]

J. F. Clauser, M. A. Horne, A. Shimony and R. A. Holt (1969), Proposed experiment to test local hidden-variable theories, Phys. Rev. Lett. 23 (15), 880

work page 1969

[8] [8]

Weak Value

B. E. Y. Svensson (2013), What is a Quantum- Mechanical “Weak Value” the Value of? Found. Phys. 43, 1193

work page 2013