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

1. Introduction

Bell, of course, like Einstein, believed that quantum mechanics implies “Spooky action at a distance.” However, contrary to widespread opinions, we sustain that the proof of quantum nonlocality should not involve the Bell inequality.

To avoid typical misunderstandings, we must be explicit about what we mean by quantum mechanics in this paper. Generally and particularly when we use the expressions “quantum nonlocality” and “quantum locality”, by quantum mechanics, we mean orthodox operational quantum mechanics, i.e., the formalism that allows us to make empirical predictions without any commitments to ontological interpretations about what that formalism might or might not imply.

The Bell theorem is a straightforward mathematical result proving an uncontroversial fact: that quantum mechanics is not a non-conspiratorial local hidden variable theory.1

Straining that clear mathematical theorem for including quantum nonlocality as its thesis leads to irreconcilable antagonistic interpretations. A characteristic exposition of these extreme positions is given through an exchange between Tim Maudlin, with his article “What Bell did” which suggested the title of this paper, and Reinhard Werner’s comments, exemplifying the innumerable articles claiming quantum nonlocality based on the Bell inequality, on the one hand, and its rejection for the same reason, on the other Maudlin (2014a,b); Werner (2014a,b).

Despite the long-standing controversy between these two radical positions, a rigorous analysis reveals that Bell’s and Einstein’s arguments regarding quantum nonlocality cannot be so trivially dismissed based on philosophical prejudices about realism.

The nonlocality problem concerns the existence of superluminal observable influences and should be clarified in that context. However, an incorrect approach has diverted the problem from its true nature forcing it to reduce to the “classicality” vs. “quantumness” paradigm, without explicating how quantum weirdness is supposed to justify locality. Since anyway “nobody understands quantum mechanics”, the perfunctory locality counterargument seems acceptable before an already flawed nonlocality argument.

The nebula is clarified if we strictly follow Bell’s approach. The arguments sustaining either quantum nonlocality or locality should be based exclusively on quantum mechanics’ objective predictions, irrespective of eventual interpretations regarding the nature of the quantum state, the collapse of the wave function, or metaphysical speculations about physical reality.

Section 2 has a historical character presenting a chronological review of Bell’s main papers on nonlocality, showing that, in his most lucid writings, Bell distinctly formulated his inequality only after explicitly establishing the nonlocal character of quantum mechanics, either by previously assuming Bell (1964) or directly proving it Bell et al. (1985); Bell (2001). So, John Bell is not responsible for the controversial approach of ascribing to quantum mechanics properties derived from a “classical inequality.”

Section 3 analyzes Bell’s actual quantum nonlocality argument which only appeared in 1975, more than ten years after the publication of his celebrated 1964 paper. We remark on a more formal approach to proving quantum mechanics is not locally causal and highlight the endemic inconsistency present in the usual approach but absent in Bell’s lucid reasoning.

Since we deal with quantum mechanics interpretation, we also have logical and rational counterarguments justifying quantum locality. If only for completeness, we briefly review some well-known counterarguments in Section 4. Of course, none of such “correct” counterarguments rely on precepts such as “...the Bell inequality is based on classical physics...” or “...the Bell inequality assumes realism...”; not because they are necessarily incorrect but because they are irrelevant to the right quantum nonlocality argument. Finally, we present our conclusions in Section 5 and Section 6.

2. Chronological Review of Bell’s Works on Nonlocality

We present a brief review of Bell’s main papers on nonlocality. The key point is that nowhere in those papers can we find an explicit statement declaring that quantum mechanics’ nonlocal character is a consequence of the violations of his inequalities.

On the other hand, he explicitly proved quantum mechanics violates local causality without introducing hidden variables and before formulating his inequality at least on two occasions Bell et al. (1985); Bell (2001). Therefore, contrary to the usual folklore, the objective evidence indicates that Bell did not consider his inequalities as a direct proof of quantum nonlocality but only of the impossibility of a non-conspiratorial local completion.

2.1. 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 prove otherwise Bell (1964):

a)

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.

b)

In correspondence with the above-quoted sentence, 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.

So, Bell’s stunning conclusion is not that quantum mechanics is nonlocal, which he took for granted, but that we cannot fix its nonlocality by completing it.

Bell resumed the argument where EPR has left it: 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 a deterministic hidden variable model and proved, through his inequality, the untenability of a local completion. Again, the impossibility of an acceptable 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. Probably, that was what Richard Feynman meant when he said that Bell’s theorem Whitaker (2016),

“It is not an important theorem. It is simply a statement of something we know is true – a mathematical proof of it.”

However, we disagree with Feynman on the unimportance of the Bell theorem since it was a significant advance in the Bohr-Einstein debate that, by 1964, remained stagnant for almost thirty years.

2.2. Bell’s Theorem After 1964

Bell’s arguments evolved over the years. In later works, he abandoned the EPR polemic reasoning, which, to Einstein’s dislike, was based on metaphysical speculations about physical reality.2

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 explicitly claimed his inequality proved quantum nonlocality nor said otherwise. To prove our previous assertion, next, we chronologically review Bell’s main papers discussing nonlocality after 1964.

2.2.1. Introduction to the Hidden Variable Problem

In 1971 he wrote the paper “Introduction to the hidden-variable question” Bell (2004). Here Bell did not explicitly 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.”

Note the adjective “local” refers to the hidden-variable theory so irrespective of whether he believed that quantum mechanics is nonlocal, he did not say his inequality proved it and any assertion in that direction would be purely speculative.

2.2.2. The Theory of Local Beables

This article appeared in 1975.3 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 1927.4 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” unless we force the interpretation. Otherwise, why would he bother to prove quantum mechanics violates local causality two sections before without using any inequality or hidden variables? We guess because he was well aware of the logical loophole of concluding quantum nonlocality directly 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 explain in the section 3.1.3, the Bell inequality cannot be formulated without additional variables foreign to quantum mechanics, notwithstanding that one of those variables may include the quantum state as indicated by Bell himself Bell (1981).

2.2.3. 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. 2.2.4), he returned to his previous formulations, i.e., either assuming (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 Bell and Aspect (2004):

“...are nontechnical introductions to the subject. They are meant to be intelligible to nonphysicists.”

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.

2.2.4. 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”

Although we could force the former statement to interpret that it means that quantum mechanics itself is nonlocal, the order in which he presents his argument does not favor that interpretation. First, he unambiguously established quantum nonlocality without any inequality and then, in a separate section, proved the impossibility of a local completion through his inequality, clearly separating the arguments.

2.2.5. Further Writings

In 1975, an editor asked Bell to write a response letter Bell (1975). He started the paper by expressing:

The editor has asked me to reply to a paper, by G. Lochak, refuting a theorem of mine on hidden variables.

Despite the paper’s title being Locality in quantum mechanics: reply to critics, it is remarkable that he referred to his theorem as “a theorem of mine on hidden variables” instead of “a theorem of mine on quantum nonlocality.” Again, although both subjects are intertwined, he carefully maintained a distinction.

In 1977 Bell wrote Free variables and local causality Bell (1977). He started the paper by writing:

It has been argued that quantum mechanics is not locally causal and cannot be embedded in a locally causal theory

Once more, his expression again clearly shows that he considered the statement “quantum mechanics is not locally causal” to be different from the assertion “cannot be embedded in a locally causal theory”, which he usually concluded from his inequality violations Bell et al. (1985); Bell (2001).

3. Bell’s Proof of Quantum Nonlocality

In this section, we analyze Bell’s explicit argument on 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 formal definition of locality that is directly applicable to quantum mechanics. He called it local causality (LC).
• An argument showing that quantum mechanics violates LC and hence is nonlocal. Bell presented his nonlocality argument before formulating his inequality, therefore, despite the usual belief, the former cannot be considered a consequence of the latter.
• A justification for assuming statistical independence (SI) in his hidden variable model. 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.

3.1. Local Causality and Quantum Nonlocality

Firstly, we briefly review the meaning of LC. Since, as observed by Norsen Norsen (2011), LC is a little-known concept, appendix A contains a detailed explanation for those unfamiliar with it. Secondly, we prove like Bell, that quantum mechanics violates LC without the Bell inequality. Thirdly, we analyze the logical loophole in the usual quantum nonlocality argument.

3.1.1. Local Causality

The concept of LC was conceived to be directly applicable to not deterministic theories like quantum mechanics. Here it suffices to say that when distant experiments are independently performed, to exclude nonlocal effects, the existence of correlations must have a local common cause explanation.

More concretely, in the case of a Bell-type experiment, if $P(A,B\mid a,b)$ is the joint probability for Alice and Bob finding results A and B when their experimental settings are a and b respectively, we in general have that,

$$\begin{array}{ccc}\hfill P(A,B\mid a,b)& \ne & P(A\mid a)P(B\mid b)\hfill \end{array}$$

(1)

(1) means the results of the experiment are correlated. However if correlations are to be explained locally, after including all relevant factors represented by $\lambda$,5 we must have,

$$\begin{array}{ccc}\hfill P(A,B\mid a,b,\lambda )& =& P(A\mid a,\lambda )P(B\mid b,\lambda )\hfill \end{array}$$

(2)

(2) means that after all factors(known and unknown) are included, whatever Bob decides to do in his distant laboratory cannot influence Alice’s local measurements; and vice versa. Therefore, (2) must be satisfied by locally causal theories. Note that $\lambda$ represents local common causes in general, not necessarily “preexisting values” and can well include the quantum state (cf. appendix A).

3.1.2. Quantum Nonlocality

After defining local causality, Bell gave an argument explaining why, when considered complete, quantum mechanics violates it. He gave a qualitative explanation and concluded that in quantum mechanics “We simply do not have (2)”, where (2) in his paper is LC.

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 (2) avoids a deterministic formulation or any “classical” prejudice and is directly applicable to quantum mechanics. If quantum mechanics is complete and local, the locally causal explanation of its correlations must lie within the quantum state. In our case

$$\mid \psi \rangle ={\displaystyle \frac{1}{\sqrt{2}}}(\mid +\rangle \otimes \mid -\rangle -\mid -\rangle \otimes \mid +\rangle )$$

(3)

Thus, if locally causal, ordinary quantum mechanics must satisfy (2) when

$$\lambda =\mid \psi \rangle$$

(4)

However, choosing $a=b$, $A=1$, and $B=-1$, an elementary quantum mechanical calculation gives

$$\begin{array}{ccc}\hfill P(1,-1\mid a,a,\mid \psi \rangle )& =& P(1\mid a,\mid \psi \rangle )P(-1\mid a,\mid \psi \rangle )\hfill \\ \hfill {\displaystyle \frac{1}{2}}& \ne & {\displaystyle \frac{1}{2}}{\displaystyle \frac{1}{2}}={\displaystyle \frac{1}{4}}\hfill \end{array}$$

(5)

Given that (5) is not widely known as a non-classical argument for quantum nonlocality, and some believe (4) is incorrect despite Bell’s explanation that $\lambda$ is completely general and could include the quantum state Bell (1981), the appendices C and B contain detailed explanations.

Since in (5) $1/2\ne 1/4$, 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 (5) relies exclusively on quantum mechanical objective predictions. It depends only on the quantum formalism irrespective of any interpretation of the wave function. It is an argument in line with the Copenhagen approach, an operational definition that does not rely on metaphysical assumptions.

3.1.3. The Endemic Logical Loophole and the Gist of the Controversy

Arguing that quantum mechanics is nonlocal because it violates the Bell inequality is unconvincing because proving Bell-type inequalities requires writing joint probabilities as

$$P(A,B\mid a,b)=\int P(A\mid a,\lambda )P(B\mid b,\lambda )P\left(\lambda \right)d\lambda$$

(6)

which is impossible without going beyond quantum mechanics because (5) proves that

$$\begin{array}{ccc}\hfill P(A,B\mid a,b,\lambda =\mid \psi \rangle )& \ne & P(A\mid a,\lambda =\mid \psi \rangle )P(B\mid b,\lambda =\mid \psi \rangle )\hfill \end{array}$$

(7)

Thus, we are confronted with the following facts:

• The proof of any property based on (6) is not a property that can be unambiguously ascribed to quantum mechanics.
• (7) is proved independently of (6), so the Bell inequality is not necessary to prove that good old orthodox operational quantum mechanics’ predictions indeed violate the local causality condition.

For some strange reason, those who argue quantum mechanics is nonlocal dismiss (7) despite, in some cases, being well aware of its existence. They prefer to turn to the Bell inequality as their sole argument, departing from Bell’s rationale and weakening their argument Norsen (2011).

On the other hand, according to our personal experience, those dogmatically supporting quantum locality when confronted with the correct reasoning immediately reject (7), alleging that $\lambda$ ought to be a “real entity”, so putting $\lambda =\mid \psi \rangle$ in (7) is nonsensical, thus excluding the quantum state as a possible local common cause which is a sophism since it either implies that $\mid \psi \rangle$ itself is a “real entity” and/or that the quantum state cannot contain local common causes.

In section 4, we shall see that it is indeed possible to reject LC on rational grounds, but we must concede that it is not possible to “have one’s cake and eat it too” and the issue is not as trivial as many dismissively assume.

3.2. Statistical Independence

As we mentioned above, in Bell’s 1964 paper, he implicitly assumed the hidden variables distribution function6 $P\left(\lambda \right)$ was not conditional on the experimental settings a and b.

$$P(\lambda \mid a,b)=P\left(\lambda \right)$$

(8)

We can justify (8) by requiring the experimental settings to be independent of the same common factors $\lambda$ affecting the results

$$P(a,b\mid \lambda )=P(a,b)$$

(9)

According to Bayes theorem we have

$$P(a,b\mid \lambda )P\left(\lambda \right)=P(\lambda \mid a,b)P(a,b)$$

(10)

Then from (9) and (10) we get (8). The ansatz (9) seems to be a reasonable assumption justifying (8).

Thus, (9) and (8) are equivalent and are known as statistical independence, measurement independence, freedom, or no-conspiracy. We shall come back to SI in sect. 4.3.

3.3. 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 (5) 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 on his separation principle and avoided reference to the reality criterion. Thus, it is worth noticing that Einstein and Bell distanced themselves from the excessive metaphysical baggage carried with the EPR elements of physical reality. Even in 1964, when Bell referenced the EPR article, he never mentioned the elements of physical reality.

4. 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.

4.1. Rejecting Local Causality

Bell’s definition of LC can be rejected as the correct definition of locality. In that sense 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),7

$$LC\equiv PI\wedge OI$$

(11)

Shimony also proposed the more picturesque expressions controllable and uncontrollable nonlocality, respectively. We refer the non-specialist to Shimony 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 recognizing the correct arguments instead of superfluous discussions about metaphysical irrelevancies such as the preexistence prejudice of elements of physical reality.

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).

4.2. Rejecting Causation

Causation in physics was criticized by Bertrand Russell in 1912 Russell (1913) and was proposed to solve the quantum nonlocality problem by Van Fraassen Van Fraassen (1982). There is no action at a distance simply because there is no need for a causal explanation.

According to Van Fraassen, “In some cases, the methodological tactic of developing a causal theory will achieve this aim of empirical adequacy, in other cases it will not, and that is just the way the world is.” He claimed that a mythical picture of causal processes got grip on our imagination.

The need for a causal explanation is also called realism. However, this rational form of realism does not conflict with quantum superposition and has nothing to do with the pre-existence tenet usually implied with the expression “local realism”.

Van Fraassen was also puzzled by the “incredible metaphysical extravaganzas to which this subject has led”. 8

4.3. 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

$$LC\wedge SI\to Bellinequality$$

(12)

Thus, it is possible to keep 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 Degorre et al. (2005); Feldmann (1995).

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.

5. Conclusions

The incorrect mixing of two distinct issues, the proofs of quantum nonlocality on the one hand and quantum completion on the other, has diverted the debate from the correct arguments and hindered it from advancing to more rational alternatives; giving a wrong perspective of the real interpretational difficulties.

We have argued for two key questions: a) the correct quantum nonlocality proof is not based on the Bell inequality which should be sensibly and unambiguously interpreted as a no-local-hidden variables theorem, and b) John Bell’s quantum nonlocality argument was also not based on his inequality, which he used only to prove the impossibility of an acceptable local completion.

The first issue (a) is a consistency puzzle. It is formally sustained on Bell’s concept of local causality, which orthodox operational quantum mechanics certainly violates without the need to introduce any elements extraneous to quantum theory or inequalities based on hidden variable models.

The second one (b) has a historical character. Although it is more subject to interpretation, we claim it follows from an unbiased and rigorous reading of Bell’s writings.

We conclude there are valid reasons to support both views, locality, and nonlocality. However, the correct arguments supporting or rejecting either of the two opposite positions are not as definite or trivial as the usual specious explanations uphold, exacerbating the different attitudes.

6. Epilogue

Although locality advocates may find it comforting to base the claim of quantum nonlocality on the Bell inequality to dismiss it as the consequence of different worldviews, classical vs. quantum Griffiths (2021); Khrennikov (2019); Werner (2014a), accepting the correct argument is not insurmountable, and may hint at deeper insights.

Distant simultaneity and nonlocality are closely related concepts. Both lack direct and objective physical determination. Admitting a certain degree of convention is necessary to maintain a coherent level of discourse. That is why the locality problem will remain controversial. However, it is essential to recognize its contentious nature for the correct motives instead of incorrect or dubious reasonings.

Relativity taught us to reject absolute simultaneity because of its lack of objective determination, likewise, quantum mechanics may be teaching us to reject action at a distance for the same reason, concretized through its nonsignaling property. 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.9 These options are still valid open questions, and pretending they are closed or inexistent is not the best scientific attitude.