Is the deformation in Eq. 56 really innocuous?

[Roberto Pavani]

TL;DR

In Eq. 56 the observable is changed from sign(φ) to sign(φ+α). Since shifts in function spaces generally modify overlaps and correlations, I'm struggling to see why this deformation should be considered innocuous. Is there a theorem guaranteeing that the relevant correlation structure is preserved, or is the altered correlation structure precisely what produces the Bell-CHSH enhancement?

I'm reading these papers:

https://arxiv.org/pdf/2412.03840

https://link.springer.com/article/10.1140/epjc/s10052-026-15445-1

and I'm struggling with Eq. (56).

The Bell-CHSH enhancement appears only after replacing

$sign(φ(f))$

with

$sign(φ(f)+α)$

whereas the undeformed observable does not seem to violate the Bell-CHSH bound.

My concern is that this is not just a reparametrization. It changes the threshold that partitions the spectrum of φ(f), from 0 to -α, and therefore changes the induced probabilities and correlations.

From my perspective, changing a threshold generally changes the geometry of the probability space. Two observables may have one correlation structure before thresholding and a different one after thresholding, especially when the threshold itself is shifted.

Is there a theorem showing that this deformation preserves the correlation structure relevant to the Bell-CHSH analysis, or is the modified correlation structure itself part of the mechanism that generates the enhancement?

Another point that confuses me is that the paper does not introduce a single common shift, but four independent parameters

$α, α', β, β'$

for the Bell observables.

If these shifts modify the probability partitions, then each observable is being deformed differently. Why should the resulting CHSH value be regarded primarily as a property of the underlying vacuum state rather than, at least in part, a consequence of the observable-dependent threshold choices?

I am probably missing something, but I do not see why the transformation in Eq. (56) should be considered innocuous from the point of view of correlations.

 

[emillindberg]

I think your concern is reasonable. A shift inside sign(φ+α) is not obviously a harmless reparametrization, because the sign operation makes the threshold change physically relevant to the resulting dichotomic observable. In particular, allowing four independent α values seems capable of changing the correlation functions being tested. It would be helpful if the authors explicitly derived whether the CHSH enhancement persists for fixed/common α, or explain precisely why the observable-dependent shifts should be regarded as part of the measurement definition rather than an additional source of the violation.

 

[WWGD]

Just curious , is this ' sign(x)' , meaning the function that is -1 when x<0, 0 at 0, 1 when x>0, or the trigonometric sine function? Sorry if it's a dumb question, as my knowledge of Physics is on the rudimentary side.

 

[Roberto Pavani]

My interpretation is that it is the sign (signum) function, not the sine function. The paper later treats the observable as dichotomic, taking values ±1, so the value at zero is probably irrelevant (measure zero for a continuous field). My question concerns the effect of shifting the threshold from 0 to $-\alpha$, not the precise convention at zero.

 

[Matterwave]

I have not read through the entire paper and just glanced around the relevant sections so take this with a grain of salt.

My impression of what the authors are doing here is they are trying to find detector settings that maximally violate the Bell inequalities. This is something that is done even for the "regular" (non QFT) analysis of Bell inequality violations.

Certain detector settings do not show any discrepancy with classical bounds and only certain intermediate settings do.

For example, for the simple singlet state, there are no discrepancies with classical results if Alice and Bob's detectors are set parallel, antiparallel or indeed even at right angles to each other. This was shown in the original Bell paper from 1964.

The authors specifically point out they are writing out a different operator $\hat{\mathcal{P}}_f$ and not the original $\mathcal{P}_f$. One is related to the other via Unitary transformation which is the usual thing we do when changing a basis. They state that doing such a transformation leads to adding $\alpha$ into the sign function.

Do you have a problem with them making this unitary transformation or you have a problem with the math they used to obtain the result?

 

[Roberto Pavani]

My concern is slightly different (perhaps due to my limited understanding of the subject).

I fully accept that one may define a new observable by a unitary transformation.

However, the quantities entering CHSH are ultimately probabilities (or correlations derived from probabilities). My concern is whether the threshold shift in Eq. (56) is merely a different representation of the same physical probabilities, or whether it changes the probability partition itself.

As an example,

$A(x)=sign(sin(x))$
and
$B(x)=sign(cos(x))$

are uncorrelated, while

$A(x)=sign(sin(x))$
and
$B_\alpha(x)=sign(cos(x)+α)$

are generally correlated because the threshold changes the measure of the regions associated with the outcomes ±1.

So my question is not whether the new observable is mathematically legitimate, but whether the probabilities entering CHSH are still revealing the same underlying structure or are partly reflecting the thresholding procedure itself.

 

[Matterwave]

I guess I don't understand your concern because to me, the whole point of changing detector settings is to change the (quantum) correlations so that we can actually see the Bell violations.

Correlations behave differently under different detector settings. We can't see a violation under some settings, we can in others.

They're not the "same physical probabilities" because you actually change the detector settings.

If you agree that we can make the unitary transformation, that that leads to a legitimate change in detector settings, and the math is solid in calculating that change, I can't see how there could be another gap there in the argument.

Of course, I myself have not read the whole paper, so I am not vouching for the author's conclusions here. I am only trying to illustrate how I'm unable to get/understand your point.

It might require a more knowledgeable poster to alleviate your worries. :)

 

[Matterwave]

You might have a legitimate concern. But from my reading, I can't tell. What I have read of the paper seems fine to me.

 

[Roberto Pavani]

My concern is that Eq. (56) is not merely a linear re-expression of the same observable.

The operation

$\phi \rightarrow \operatorname{sign}(\phi+\alpha)$

is nonlinear.

Therefore it does not generally preserve inner products or correlations.

In my simple example,

$\langle \operatorname{sign}(\sin x), \operatorname{sign}(\cos x)\rangle = 0$

whereas

$\langle \operatorname{sign}(\sin x), \operatorname{sign}(\cos x+\alpha)\rangle \neq 0$

for generic values of α.

This suggests that the threshold shift creates a different partition of the probability space and therefore a different correlation structure.

My question is therefore whether the threshold shift is intended merely as a different measurement setting, or whether the resulting modification of the probability partition is itself part of the mechanism responsible for the CHSH enhancement.

 

[PeterDonis]

My concern is that Eq. (56) is not merely a linear re-expression of the same observable.

I don't think it's intended to be. I think, as @Matterwave has said, that it's intended to capture varying the relative angles of the different measurements. Measurements at different angles are different observables.

 

[Roberto Pavani]

I understand that Eq. (56) defines a different observable.

My concern is that CHSH is ultimately computed from coincidence probabilities. Since the threshold shift changes the partition of events into ±1 outcomes, it also changes the coincidence statistics themselves.

So I am trying to understand whether the observed CHSH enhancement should be interpreted primarily as a property of the state, or as a consequence of the particular way the outcomes are defined.

 

[PeterDonis]

I understand that Eq. (56) defines a different observable.

Which means this...

it also changes the coincidence statistics themselves.

...is exactly what we should expect since we have changed the observables. So I don't understand why you think this is an issue.

am trying to understand whether the observed CHSH enhancement should be interpreted primarily as a property of the state, or as a consequence of the particular way the outcomes are defined.

I don't understand what this means. You have a state and you have observables; the correlation between the measurement results is determined by those things. If you change any of them, you change the correlations. But the correlations are not a property of any one of those things in isolation; they're a property of all of them combined.

 

[Roberto Pavani]

After further reflection and numerical verification, I can now state my concern precisely.

The core argument (Fine's theorem):

The CHSH bound $|S| \leq 2$ holds if and only if the four dichotomic observables admit a joint probability distribution (Fine, 1982).

In the paper's setup, the vacuum state gives the smeared fields $(\phi(f), \phi(f'), \phi(g), \phi(g'))$ a well-defined joint Gaussian distribution.
The four observables $\text{sign}(\phi(f)+\alpha)$, $\text{sign}(\phi(f')+\alpha')$, $\text{sign}(\phi(g)+\beta)$, $\text{sign}(\phi(g')+\beta')$ are deterministic (measurable) functions of these fields. Their joint distribution is simply the pushforward of the Gaussian measure through the sign functions; it exists for any choice of offsets $(\alpha, \alpha', \beta, \beta')$.
By Fine's theorem, $|S| \leq 2$ must hold. No parameter tuning can change this (indeed the undeformed case $\alpha=\alpha'=\beta=\beta'=0$ gives $|S| < 4/\pi \approx 1.27$, well below the bound).

Why this differs from spin rotations:

In quantum mechanics, measuring spin along different axes corresponds to incompatible observables ($\sigma_x$, $\sigma_z$) that do not admit a joint distribution, this is what allows the violation.
A threshold shift, by contrast, never destroys the joint distribution: for any realization of the field, all four outcomes are simultaneously determined.

The analogy between the offset $\alpha$ in eq. (56) and the Bell angles in Section 2 is therefore not valid. Bell angles rotate incompatible observables; the offset merely translates the decision boundary of the same observable.

Numerical verification:

I computed $E[\text{sign}(\phi(f_i)+\alpha_i)\cdot\text{sign}(\phi(g_j)+\beta_j)]$ directly from the bivariate Gaussian distribution, using the exact parameters of eq. (59), via Monte Carlo with $2\times 10^8$ samples. The
result:
$|S| = 2.000002 \pm 0.00007$
This is consistent with the Fine bound and inconsistent with the reported value $|S| = 2.02$.

Possible source of the discrepancy:

The paper evaluates the correlators through the Dirichlet integral representation (eq. 58). For the chosen parameters, the integrand in the $E(f,g)$ term has a Gaussian envelope of width $\sim 586$ in the integration variable $k$, while $\cos(\alpha k)$ oscillates with period $\sim 0.5$. This produces approximately 1200 oscillations within the envelope. The QuasiMonteCarlo routine with $10^9$ sampling points cannot faithfully resolve such an integral.
The 0.02 excess seems to be numerical noise.

The Python script performing this verification is attached as txt (original name chsh_verification.py).

 

[Matterwave]

It seems to me that your concern has changed drastically. If I read your post right, now you are saying the authors performed their math incorrectly (or at least their simulation) and they actually can't violate the CHSH inequality with their set up. So even post unitary transformation of the observables, the CHSH inequality is still not violated.

This is beyond my expertise to validate. I suppose if you found an actual error in the paper you could contact the authors to let them know.

 

[Roberto Pavani]

It seems to me that your concern has changed drastically.

You're right, my concern evolved during the discussion. Initially I was worried that the threshold shift might artificially inflate the correlations. But after thinking more carefully, I realized that Fine's theorem actually forbids any violation regardless of the offsets, the joint distribution always exists. At that point the question became: how can the paper report 2.02 if the bound is 2?
The numerical verification suggests the answer is a convergence issue in the oscillatory integral.

 

[Roberto Pavani]

Following @Matterwave's suggestion, I have also contacted the corresponding author to share the numerical verification and the Fine's theorem argument. I'll report back if I receive a response.