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A Critical Analysis of the Quantum Nonlocality Problem: On the Polemic Assessment of What Bell Did

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15 February 2024

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19 February 2024

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Abstract
Despite their Nobel prize-winning empirical verification, the interpretation of the Bell inequality remains controversial. The alleged nonlocal character of quantum mechanics is inextricably related to the formulation of the Bell theorem. However, as is usually presented, the relation disagrees with Bell's approach and is inappropriately posed. The departure from the clear line of reasoning that Bell tried to convey has contributed to a polarization of part of the scientific community. We review part of Bell's work and show how the correct appreciation of Bell's rationale calls for reformulating a widespread argument on quantum nonlocality, yielding a more balanced perspective of the problem. We highlight a more formal proof of quantum mechanics' violation of local causality. For completeness, we mention a few alternatives that may justify considering quantum mechanics as a local theory.
Keywords: 
Subject: Physical Sciences  -   Quantum Science and Technology

1. Introduction

Bell, of course, like Einstein, believed that quantum mechanics implies “Spooky action at a distance.” But contrary to a widespread opinion, we argue that an unbiased reading of Bell’s work shows that he did not claim his inequality proves it.
The main goal of this work is to bring up an embarrassingly trivial point that is, however, almost universally overlooked, namely, the difference between arguing for quantum nonlocality and quantum completeness.
Although that difference should have been clear from an objective reading of Bell’s work, by dismissing his careful formulation, we obtain a controversial approach that facilitates taking the extreme opposite position adopted by the quantum localists, sometimes leading to heated debates Maudlin (2014b); Werner (2014a); Maudlin (2014a); Werner (2014b).
Despite our agnostic attitude towards quantum nonlocality, we argue that Bell’s and Einstein’s belief in quantum nonlocality cannot be so easily dismissed on naive prejudices about realism as many “localists” interpret.
Our main point is to stress the correct argument of quantum nonlocality does not involve the Bell inequality. Otherwise, the reasoning is not merely unconvincing and easy to debunk but inconsistent.
We contend there is no evidence to support that Bell interpreted his inequality as a direct proof of quantum nonlocality. However, there is proof where, on a few occasions, he explicitly argued for quantum nonlocality without using his inequality and unambiguously used the latter only to prove the impossibility of a rational local completion.
In Section 2, we analyze Bell’s 1964 formulation, and in Section 3 some of his papers on nonlocality after 1964. We shall see that, although in some papers he left the issue unclear, in a few others, he clearly separated his arguments of quantum nonlocality from his arguments of quantum completion and unambiguously interpreted his inequality as a no-local-hidden variables theorem instead of a quantum nonlocality theorem.
In Section 4, we analyze Bell’s actual quantum nonlocality argument. We highlight an argumentation change that puts on a more formal basis the reasons why quantum mechanics is not locally causal.
Since we are dealing with quantum mechanics interpretation, we also have “correct” counterarguments for quantum locality. If only for completeness, we briefly review some well-known counterarguments in Section 5. Of course, none rely on expressions such as “...because the Bell inequality is based on classical physics...” or “...the Bell inequality is based on realism...” Finally, we present our conclusions in Section 6 and Section 7.

2. The 1964 Bell Theorem

A usual and widespread interpretation asserts that Bell formulated his inequality to prove that quantum mechanics is not a local theory, presenting the Bell theorem as a quantum nonlocality theorem. But an attentive reading of Bell’s 1964 argument reveals two crucial facts that allow interpreting it otherwise:
  • Bell already considered quantum mechanics as nonlocal from the beginning, i.e., before formulating his inequality. Indeed, in the third line of the introduction, he wrote: “These additional variables were to restore to the theory causality and locality.” That is, the inclusion of hidden variables into the theory was supposed to modify it and recover locality instead of proving its nonlocality.
  • Bell starts the conclusion section by saying: “In a theory in which parameters are added to quantum mechanics....”; so, clearly, he was not inferring properties of quantum mechanics, but only of a modified theory in which parameters are added.
Bell took the argument where EPR Einstein et al. (1935) has left it, namely, if local, quantum mechanics must be incomplete. Since the orthodox interpretation asserts quantum mechanics is complete, then, according to EPR, it must be nonlocal. The previous inference was implicit in his expression in a) asserting that additional variables were necessary to restore locality. Then, following an EPR-like reasoning, Bell derived determinism and proved through his inequality that a local completion is untenable. Therefore, the impossibility of a local completion only proves that we cannot modify orthodox quantum mechanics to make it local.
Thus, accepting the EPR reasoning, as Bell did, the inequality is unnecessary to prove quantum nonlocality, so claiming that Bell’s inequality proves it (which Bell did not) would only be circular reasoning.
On the other hand, Bell’s theorem is a mere mathematical theorem that should be free of any polemic if we strictly follow Bell’s rationale, namely, that a local completion is untenable. Richard Feynman put that clearly when he said that Bell’s theorem
It is not an important theorem. It is simply a statement of something we know is true – a mathematical proof of it. Whitaker (2016)
However, we disagree with Feynman about the unimportance of the Bell theorem since it was a significant advancement in the Bohr-Einstein debate that remained stagnant for almost thirty years.

3. Bell’s Theorem after 1964

Bell’s arguments evolved over the years. In later works, he abandoned the polemic EPR reasoning, dissipating the fog around it. However, a persistent view exists that dispenses with Bell’s later arguments. That view advocates a controversial reading of Bell’s 1964 reasoning advertising the Bell theorem as a quantum nonlocality theorem. We call this view “radical non-localist”.
According to the radical non-localist stance, the EPR argument is unassailable. They even consider it an “analytic concept”, i.e., we cannot coherently deny it Maudlin (2014b). However, as we observed in the previous section, be it analytic or not, when we accept the EPR reasoning, we do not need the inequality to prove quantum nonlocality.
Except on the occasions where he left the issue ambiguous, Bell explicitly separated his arguments of quantum nonlocality from his inequality, which he used to prove the impossibility of a local completion. Above, ambiguous means he neither claimed his inequality proved quantum nonlocality nor said otherwise. To prove our previous assertion, next, we chronologically review some of Bell’s papers discussing nonlocality after 1964.

3.1. Introduction to the Hidden Variable Problem

In 1971 he wrote the paper “Introduction to the hidden-variable question” Bell (2004). Here Bell did not mention quantum nonlocality but investigated the de Broglie-Bohm hidden variables theory and highlights its explicit nonlocal character as “the difficulty”.
Bell concluded after formulating his inequality and proving that quantum mechanics violates it:
Thus the quantum-mechanical result cannot be reproduced by a hidden-variable theory which is local in the way described.
Literally, “the quantum-mechanical result cannot be reproduced by a hidden-variable theory” is different from concluding quantum mechanics is not local. Besides, the adjective “local” refers to the hidden-variable theory.

3.2. The Theory of Local Beables

This article appeared in 1975. 1 Here Bell abandoned the EPR reasoning and introduced the concept of local causality. He argued that quantum mechanics violates this form of locality in section 3 without mentioning any inequality. He starts that section by asserting:
Ordinary quantum mechanics, even the relativistic quantum field theory, is not locally causal in the sense of (2).
(2) above refers to local causality. Then he develops his quantum nonlocality argument. It is similar to the one given by Einstein in 1927.2 In the same section, immediately after establishing the nonlocal character of quantum mechanics, Bell explored the problem of adding hidden variables. Then in section 4, “Locality inequality”, he derived a stochastic Bell-CHSH inequality. Finally, in section 5, he established the impossibility of a local completion by proving that quantum mechanics violates his inequality, concluding:
So quantum mechanics is not embeddable in a locally causal theory as formulated above.
That is different from concluding, “So quantum mechanics is not a local theory”. Otherwise, why would he bother to prove quantum mechanics violates local causality two sections before without using any inequality or hidden variables? Because he was well aware of the logical loophole of concluding quantum nonlocality from his inequality. As Stapp once clearly explained Stapp (2012):
Thus whatever is proved is not a feature of quantum mechanics, but is a property of a theory that tries to combine quantum theory with quasi-classical features that go beyond what is entailed by quantum theory itself. One cannot logically prove properties of a system by establishing, instead, properties of a system modified by adding properties alien to the original system.
Above, “properties alien to the original system” rigorously mean variables that do not legitimately pertain to quantum mechanics. Although some have observed that the hidden variables can include the quantum state Norsen (2011); Gisin (2012); Laudisa (2018), the problem persists with the other “additional variables”. As we observe in the appendix, the Bell inequality cannot be formulated without additional variables foreign to quantum mechanics, notwithstanding that one of those variables may include the quantum state.

3.3. Bertlmann’s Socks

In 1981 Bell wrote his celebrated paper “Bertlmann’s socks and the nature of reality” Bell (1981). On this occasion, Bell did not explicitly prove quantum nonlocality before formulating the inequality. He based his arguments on EPR. But he also uses common causes to explain local correlations.
This is one of the papers where he left ambiguous whether his inequality violation should be interpreted as proof of quantum nonlocality. Can we assume that Bell changed his mind about the meaning of his inequality? We do not think so because, in his last paper (cf. Section 3.4), he returned to his previous formulations, i.e., either accepting (1964) or proving (1975) quantum nonlocality without introducing hidden variables or mentioning any inequality.
In Bell (1981), Bell chose intuition and ease of interpretation over logical rigor. In Bell’s own words, this paper was one of those that:
...are nontechnical introductions to the subject. They are meant to be intelligible to nonphysicists. Bell and Aspect (2004)
That is why he spent great effort explaining the difference between quantum and classical entanglement through naive analogies, such as those of Mr. Bertlmann’s socks.

3.4. La Nouvelle Cuisine

This is Bell’s last paper which appeared in 1990 Bell (2001). Here again, Bell’s view of his inequality and quantum nonlocality is crystal clear. This time Bell mentions EPR in the two sections that concern us here. In section 8, when proving that “Ordinary quantum mechanics is not locally causal” without mentioning any inequality and without actually using an EPR argument. Then, in section 10, when explicitly introducing hidden variables as local common causes, for proving, through his inequality, that :
Quantum mechanics cannot be embedded in a locally causal theory
Again, the order in which he presents his argument, first establishing quantum nonlocality without using any inequality and then proving the impossibility of a local completion through his inequality, is unambiguous and uncontroversial.

4. Bell’s Proof of Quantum Nonlocality

In this section, we closely analyze Bell’s arguments about the nonlocal character of quantum mechanics. In his 1975 paper, “The theory of local beables” Bell et al. (1985), Bell gave an explicit argument for quantum nonlocality for the first time. This paper has four outstanding characteristics that were missing in 1964:
  • A rigorous definition of locality he called local causality (LC).
  • A proof that quantum mechanics violates LC, therefore, is nonlocal.
  • A physical justification for assuming what later became known as the statistical independence (SI) hypothesis. In 1964, SI was an ad hoc implicit assumption.
  • An absence of any reference to the EPR paper.
Next, we briefly address each of these characteristics.

4.1. Local Causality

Bell’s definition of LC is a formalization of the idea that, according to relativity theory, interactions can happen only at a finite speed. It means that causes cannot have an instantaneous effect on distant events. He formulated LC so that it can be applied to not deterministic theories like quantum mechanics. It is a locality argument that avoids a purportedly classical EPR-like reasoning. A concept directly applicable to orthodox quantum mechanics without distorting its nature.
For the particular case that concerns us, i.e., the singlet state correlations in a Bell-type experiment, LC takes the following form. Let P ( A , B ∣ a , b ) be the probability of a joint measurement giving the results A , B ∈ { − 1 , + 1 } conditional on the respective measurements directions a , b . The laws of probabilities require
P ( A , B ∣ a , b ) = P ( A ∣ B , a , b ) P ( B ∣ a , b )
So far, it is just about probabilities. Let us now add some physics and assume that both observers, Alice and Bob, choose their measurement directions at the last moment so that both measurements are spacelike separated events. Then LC requires that neither the results A , B nor the measurement settings a , b made on one side can affect the state of affairs on the other side. However, we cannot exclude the existence of correlations. In the r.h.s of (1), we can have that
P ( A ∣ B , a , b ) ≠ P ( A ∣ a )
P ( B ∣ a , b ) ≠ P ( B ∣ b )
notwithstanding that events A and a are spacelike separated from B and b. However, relativistic causality requires the correlations implied by (2) and (3) to be explained by local common causes λ . They are local because they are supposed to lie at the intersection of the backward light cones of the measurement events. Once the common causes λ are specified, the inclusion of spacelike separated parameters in the l.h.s of (2) and (3) become redundant
P ( A ∣ B , a , b , λ ) = P ( A ∣ a , λ )
P ( B ∣ a , b , λ ) = P ( B ∣ b , λ )
Including λ in (1)
P ( A , B ∣ a , b , λ ) = P ( A ∣ B , a , b , λ ) P ( B ∣ a , b , λ )
Replacing (4) and (5) in (6)
P ( A , B ∣ a , b , λ ) = P ( A ∣ a , λ ) P ( B ∣ b , λ )
The last equation is also known as the screening-off condition. It is the formal expression of the intuitive idea behind relativistic locality and is Bell’s definition of LC for the case at hand.
The common cause λ is usually called “hidden variables”; however, it is somewhat misleading to believe the λ variables are necessarily unknown parameters. The only condition they need to comply with is lying at the intersection of the backward light cones of the measuring events to constitute a local explanation of the correlations. It is also utterly misleading to think they are EPR elements of physical reality; on the contrary, their role is to eliminate any EPR-like argument. Furthermore, local causality is independent of the stochastic properties of the common causes. More concretely, they are independent of the statistical independence hypothesis.
Although Bell did not mention Reichenbach, his λ variables are according to Reichenbach’s common cause principle Hitchcock and Rédei (2020). The last point is relevant because, as we shall see later, one possibility to block the argument in favor of quantum nonlocality is to reject Reichenbach’s principle of common causes Cavalcanti and Lal (2014).

4.2. Quantum Nonlocality

After defining local causality, Bell gave an argument explaining why, when considered complete, quantum mechanics violates it. Bell’s argument is similar to the one given by Einstein in 1927.3
We can recast Bell’s and Einstein’s arguments in more formal terms through the mathematical formulation of local causality. The crucial point is that (7) avoids the polemic around an EPR-like classical argument. If quantum mechanics is complete and local, the locally causal explanation of its correlations must lie within the quantum state. In our case
∣ ψ 〉 = 1 2 ( ∣ + 〉 ⊗ ∣ − 〉 − ∣ − 〉 ⊗ ∣ + 〉 )
Thus, if locally causal, ordinary quantum mechanics must satisfy (7) when
λ = ∣ ψ 〉
However, choosing a = b , A = 1 , and B = − 1 , an elementary quantum mechanical calculation gives
P ( 1 , − 1 ∣ a , a , ∣ ψ 〉 ) = P ( 1 ∣ a , ∣ ψ 〉 ) P ( − 1 ∣ a , ∣ ψ 〉 ) 1 2 ≠ 1 2 1 2 = 1 4
Given that (10) is not widely known as a non-classical argument for quantum nonlocality, and some find (9) puzzling or inappropriate, the appendix contains a detailed explanation.
Since in (10) 1 / 2 ≠ 1 / 4 , ordinary quantum mechanics lacks a locally causal explanation of its correlations, i.e., the quantum state alone cannot screen-off events on one side from spacelike separated events on the other far away side. Hence, it conspicuously fails the LC locality criterion.
Note that (10) is not an EPR-like argument. It relies exclusively on quantum mechanical objective predictions. It is independent of the wave function interpretation and wave function collapse. In particular, it is independent of the ontic or epistemic nature of the quantum state, depending only on the quantum formalism irrespective of any interpretation. It is an argument in line with the Copenhagen approach, an operational definition that does not rely on metaphysical assumptions.
Formally, that is the counterargument against claims asserting the singlet correlations find a local common cause explanation in their preparation with the same generating event Griffiths (2020). There is no doubt they find an explanation in their preparation. Unfortunately, that explanation is not locally causal because all we know from its preparation is its quantum state, and as (10) proves, it does not contain a common cause explanation. Nor does the magic of superposition justify those correlations, at least in a locally causal way Boughn (2017).
As we explained in Section 3.2, Bell introduced his inequality only after proving that quantum mechanics is not a locally causal theory. The controversial approach of using a “classical inequality” to derive properties of quantum theory is not ascribable to John Bell. Unfortunately, often the same does not apply either to localists or non-localists Maudlin (2014b); Werner (2014a); Zukowski and Brukner (2014); Boughn (2017); Griffiths (2021); Norsen (2011); Goldstein et al. (2011); Gisin (2012); Laudisa (2018).

4.3. Statistical Independence

As we mentioned above, in Bell’s 1964 paper, he implicitly assumed the hidden variables distribution function4 P ( λ ) was not conditional on the experimental settings a and b.
P ( λ ∣ a , b ) = P ( λ )
We can justify (11) by requiring the experimental settings to be independent of the same common factors λ affecting the results
P ( a , b ∣ λ ) = P ( a , b )
According to Bayes theorem we have
P ( a , b ∣ λ ) P ( λ ) = P ( λ ∣ a , b ) P ( a , b )
Then from (12) and (13) we get (11). The ansatz (12) seems to be a reasonable assumption justifying (11).
Thus, (12) and (11) are equivalent and are known as statistical independence, measurement independence, freedom, or no-conspiracy. We shall come back to SI in sect. Section 5.3.

4.4. The EPR Paper

Although Bell conceived his 1964 paper as a continuation of the EPR argument, one of the virtues of his 1975 formulation is not referencing the EPR paper. Besides presumably being a classical-like argument, the EPR reasoning contains an unnecessary construction that has been the source of much superfluous metaphysical speculation, namely, the elements of physical reality.
The reality criterion has a highly metaphysical burden because it assumes the existence of physical magnitudes from the mere possibility of predicting their values, notwithstanding that we do not indeed measure them. They are unnecessary because they are employed neither to prove quantum nonlocality (10) nor to derive the Bell inequality Lambare and Franco (2021).
Bohr attacked the reality criterion Bohr (1935). Einstein did not write the EPR paper, and he did not like how it came out. In a letter to Schrödinger he wrote Howard (1985):
But still it has not come out as well as I really wanted; on the contrary, the main point was, so to speak, buried by erudition.
Einstein based his argument for incompleteness in his separation principle and avoided any reference to the reality criterion. Thus, it is worth noticing that Einstein and Bell distanced themselves from the original EPR elements of physical reality criterion. Even in 1964, when Bell referenced the EPR article, he never mentioned the elements of physical reality.

5. Quantum Locality

We briefly mention three counterarguments that may justify considering quantum mechanics as a local theory. Only the third one contemplates the use of the Bell theorem and is related to the Bell inequality. We mention them only for completeness as possible logically admissible counterarguments.
These arguments are, of course, well-known and not new. There are also others we do not mention, such as the many-worlds interpretation and QBism, that also claim to preserve quantum locality.

5.1. Rejecting Local Causality

Jarrett Jarrett (1984) helped clarify the nature of local causality by decomposing it into the conjunction of two different conditions, which Shimony respectively called parameter independence (PI) and outcome independence (OI) 5
L C ≡ P I ∧ O I
Shimony also proposed the more picturesque expressions controllable and uncontrollable nonlocality, respectively. We refer the non-specialist to Ref. Shimony (1993) for a detailed explanation of these concepts.
Jarret proved that a theory complying with PI is no-signaling. He also showed quantum mechanics respects PI, hence is no-signaling. However, quantum mechanics violates OI, thus violating LC.
We can effectively block the argument in favor of quantum nonlocality by adopting parameter independence as the appropriate concept for locality and rejecting outcome independence as a necessary condition.
In summary, by accepting no-signaling (parameter independence) as a sufficient criterion for locality, we reject the more stringent condition of local causality, recovering quantum mechanics locality. Of course, those who claim quantum mechanics is not local will not accept the definition Norsen (2009).
However, more rational discussions are possible by explicitly acknowledging the different criteria. It is fair to note that even some who can be considered radical nonlocalists accept that quantum nonlocality is not right out “action at a distance” Gisin (2023).

5.2. Rejecting Realism

Realism has been justly criticized as an obscure concept Maudlin (2014a); Norsen (2007); Laudisa (2012); Gisin (2012). However, there is a concrete meaning we can ascribe to realism for rejecting quantum nonlocality, namely, causal explanation.
Please note that this rational meaning of realism has nothing to do with the usual metaphysical significance related to the term realism in the common expression “local realism”, which is purportedly ruled out by the Bell inequality violation Lambare (2022).
Causation in physics has been criticized by Bertrand Russell in 1912 Russell (1913) and was proposed to solve the quantum nonlocality problem by Van Fraassen in 1982 Van Fraassen (1982). There is no action at a distance simply because there is no need for a causal explanation.
Whether we like it or not, when we interpret realism as a causal explanation, its rejection constitutes a logically correct option to solve the quantum nonlocality problem.

5.3. Completing Quantum Mechanics

This approach is different from the former two because it implies going beyond orthodox quantum mechanics. If we are willing to accept local causality as the correct locality concept and recover a causal explanation, we must consider quantum mechanics as an emergent theory.
We can complete quantum mechanics with local hidden variables if we reject statistical independence. The 1975 version of the Bell theorem is
L C ∧ S I → B e l l i n e q u a q l i t y
Thus, we can retain local causality in a hidden variables theory by rejecting statistical independence. Indeed, well-known local hidden variables models exist reproducing the singlet correlations violating statistical independence Feldmann (1995); Degorre et al. (2005).
Whether statistical independence is a necessary physical condition is a contentious issue. According to some physicists, its rejection is a rational position Hall (2016); Hossenfelder and Palmer (2020); ’t Hooft (2021). Others, including John Bell Bell (1981), sustain its rejection as inadmissible since it purportedly compromises the experimental freedom implying unreasonable conspiracies.

6. Conclusions

A more rational approach to the current controversy around the nonlocal character of quantum mechanics requires modifying a widespread view and accommodating it closer to Bell’s original reasoning.
The current impasse arises from the incorrect mixing of two different issues, the arguments for quantum nonlocality on the one hand and quantum completion on the other.
Quantum mechanics’ violation of local causality requires an independent proof dissociated from the Bell inequality, such as in (10) or done by Bell Bell (1975, 2001). A consistent formulation requires the Bell theorem to be a no-local-hidden-variables theorem, not a quantum nonlocality theorem.

7. Final Remarks

Although localists may find it comforting to base the claim of quantum nonlocality on the Bell inequality to dismiss it as almost “silly”, accepting the correct argument is not insurmountable and may point to deeper insights.
Distant simultaneity and nonlocality are closely related concepts. Both lack direct and clear-cut physical determination. Admitting a certain degree of convention is necessary if we want to maintain a coherent level of discourse. Although the locality problem will remain controversial, it is essential to recognize its contentious nature for the correct motives instead of incorrect or obscure reasonings.
The quantum nonlocality problem cannot be summarily dismissed by looking for trivial conceptual or logical issues within the Bell-type inequalities and Bell’s arguments Griffiths (2020); Khrennikov (2008); Nieuwenhuizen (2011); Kupczynski (2020) or through superfluous metaphysical ideas Lambare (2022). Quantum mechanics may require a revision of our notion of causality, just as relativity prompted us to revise our concept of simultaneity. The other possibility is that quantum mechanics is emergent and, because of Bell’s theorem, that would require the acceptance of superdeterminism.6 These options are still valid open questions, and pretending they are closed or inexistent is not the best scientific attitude.

Appendix A. Common Causes and the Quantum State

Some researchers find it perplexing that the quantum state ∣ ψ 〉 can be considered a common cause in the definition of local causality. That is owed to the incorrect metaphysical meaning usually attached to the λ variables as preexisting EPR elements of physical reality Nistico (2014) or a necessarily classical concept.
Sustaining that λ is by necessity an element foreign to quantum mechanics amounts to forbidding the application of the local causality concept to quantum mechanics. It is particularly convenient for summarily dismissing its annoying nonlocal character decreeing it local by construction Werner (2014a). But the physical meaning of λ is not limited to classical or metaphysical concepts other than representing local common causes. Bell also explained that the hidden variables may include the quantum state Bell (1981):
It is notable that in this argument nothing is said about the locality, or even localizability, of the variable λ . These variables could well include, for example, quantum mechanical state vectors, which have no particular localization in ordinary space-time.
Unfortunately, researchers often grossly overlook Bell’s explanation of the meaning of the λ variables. They generally identify λ with metaphysical entities such as preexisting values or believe they must necessarily be unknown parameters.
The particular case λ = ∣ ψ 〉 is necessary to formalize Bell’s (and Einstein’s) qualitative arguments of quantum nonlocality according to the rigorous definition of local causality. This step is necessary to test whether quantum mechanics gives a locally causal explanation of its correlations. Thus, there is no valid argument against submitting quantum mechanics, within its own rules, to the local causality test.
To see whether ordinary quantum mechanics complies with the local causality criterion, all we have to do is set λ = ∣ ψ 〉 in (7) with ∣ ψ 〉 given by (8), where ∣ + 〉 and ∣ − 〉 denote the spin eigenstates in the z-direction. We assume that motion takes place in y direction with setting angles a and b lying the x − z plane measured with respect to the z axis. If ∣ a , + 〉 and ∣ a , − 〉 are the spin eigenstates in the a direction
∣ a , + 〉 = + cos a 2 ∣ + 〉 + sin a 2 ∣ − 〉
∣ a , − 〉 = − sin a 2 ∣ + 〉 + cos a 2 ∣ − 〉
Analogously for the particle measured at the other laboratory, we have
∣ b , + 〉 = + cos b 2 ∣ + 〉 + sin b 2 ∣ − 〉
∣ b , − 〉 = − sin b 2 ∣ + 〉 + cos b 2 ∣ − 〉
The joint probability according the quantum formalism is
P ( A , B ∣ a , b , ∣ ψ 〉 ) = 〈 ψ ∣ ( ∣ a , A 〉 ⊗ ∣ b , B 〉 〈 a , A ∣ ⊗ 〈 b , B ∣ ) ∣ ψ 〉
Letting A = + 1 , B = − 1 according to (8), (A1) and (A3)
P ( + 1 , − 1 ∣ a , b , ∣ ψ 〉 ) = 〈 ψ ∣ ( ∣ a , + 〉 ⊗ ∣ b , − 〉 〈 a , + ∣ ⊗ 〈 b , − ∣ ) ∣ ψ 〉 = 1 2 ( 〈 + ∣ a , + 〉 〈 − ∣ b , − 〉 − 〈 − ∣ a , + 〉 〈 + ∣ b , − 〉 ) 1 2 ( ) *
= 1 2 ( cos a 2 cos b 2 + sin a 2 sin b 2 ) 2
= 1 2 cos 2 ( a − b 2 )
Where ( ) * represents the complex conjugate of the first factor in parenthesis. If we further assume a = b , (A6) gives
P ( + 1 , − 1 ∣ a , a , ∣ ψ 〉 ) = 1 2
When we perform a measurement only in Alice’s laboratory, the quantum formalism prescribes
P ( + 1 , a , ∣ ψ 〉 ) = 〈 ψ ∣ ( ∣ a , + 〉 〈 a , + ∣ ⊗ I ) ∣ ψ 〉
= 〈 ψ ∣ [ ( ∣ a , + 〉 〈 a , + ∣ ⊗ I ) ∣ ψ 〉 ] = 〈 ψ ∣ 1 2 ( ∣ a , + 〉 〈 a , + ∣ + 〉 ⊗ ∣ − 〉 − ∣ a , + 〉 〈 a , + ∣ − 〉 ⊗ ∣ + 〉 )
= 〈 ψ ∣ 1 2 ( cos a 2 ∣ a , + 〉 ⊗ ∣ − 〉 − sin a 2 ∣ a , + 〉 ⊗ ∣ + 〉 )
= 1 2 cos a 2 〈 + ∣ a , + 〉 + sin a 2 〈 − ∣ a , + 〉
= 1 2 cos 2 a 2 + sin 2 a 2
= 1 2
Where I = ∣ + 〉 〈 + ∣ + ∣ − 〉 〈 − ∣ is the identity operator in the one particle two-dimensional Hilbert-space. In a similar way, performing a measurement only on Bob’s laboratory we find
P ( − 1 , b , ∣ ψ 〉 ) = 〈 ψ ∣ ( I ⊗ ∣ b , − 〉 〈 b , − ∣ ) ∣ ψ 〉 = 1 2
From (A9), (A10), and (A16), we obtain (10) formally proving that ordinary quantum mechanics lacks a local common cause explanation for its correlations.
As we explain in the main text and contrary to widespread beliefs, Bell inequality violation does not tell us that quantum mechanics is not local. Nonlocality follows from quantum formalism itself. John Bell (and Einstein) did not claim otherwise. What is puzzling about the Bell theorem is that we cannot complete quantum mechanics with additional parameters under reasonable assumptions, proving that the experimentally tested quantum predictions seem to be hopelessly nonlocal. Probably that would have disappointed Einstein.
The violation of local causality by quantum mechanics through the singlet state is well known and was observed by other authors Norsen (2011); Laudisa (2018). However, most non-localists give it only a subsidiary importance. They prefer to turn to the Bell inequality as the main argument. On the other hand, localists conveniently overlook it.
For instance, in Ref. Norsen (2011), Norsen ultimately presents the CHSH inequality as a quantum nonlocality proof when, after taking for granted statistical independence, he declares Norsen (2011):
...the empirically violated Clauser-Horne-Shimony-Holt inequality can be derived from Bell’s concept of local causality alone, without the need for further assumptions involving determinism, hidden variables, “realism,” or anything of that sort.
In our opinion, that move is unwarranted and justifies the opposite stance held by localists. Indeed, the CHSH inequality cannot be formulated without hidden variables or common causes not present in quantum mechanics precisely because quantum mechanics violates (7), as proved by (10). Certainly, proving Bell-type inequalities require writing joint probabilities as
P ( A , B ∣ a , b ) = ∫ P ( A ∣ a , λ ) P ( B ∣ b , λ ) P ( λ ) d λ
which is impossible without going beyond quantum mechanics. Admittedly, it is a trivial logical loophole. However, an endemic loophole that is frequently exploited by localists to debunk even the most lucid quantum nonlocality presentations like the one by Brunner et al. Brunner et al. (2014) where again (7) is correctly explained but finally (A17) is highlighted as the “locality constraint”, declaring
This is the content of Bell’s theorem, establishing the nonlocal character of quantum theory and of any model reproducing its predictions.
Quantum localists respond by saying that since the inequality based on (A17) is not about quantum mechanics, it signals the nonlocality of something else. References

References

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1
Bell’s work is reproduced in Bell et al. (1985).
2
Einstein’s argument is reproduced by Laudisa 2019 and also by Harrigan and Spekkens 2010.
3
Einstein’s argument is reproduced by Laudisa in Laudisa (2019) and also by Harrigan and Spekkens in Harrigan and Spekkens (2010).
4
Note that the distribution function of the λ common causes is irrelevant for the definition of local causality. P ( λ ) is necessary only to derive the Bell inequality.
5
Jarrett used the terms “locality” and “completeness”, implying that PI alone is locality. Shimony terminology is better because it is more neutral.
6
By superdeterminism we mean the violation of the mathematical condition (11) without implying any particular interpretation.
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