Comment on "Nonlocality claims are inconsistent with Hilbert-space quantum mechanics"
aa r X i v : . [ qu a n t - ph ] F e b Locality claims based on Bell inequality inconsistency are inconsistent
Justo Pastor Lambare
Facultad de Ciencias Exactas y Naturales, Ruta Mcal. J. F. Estigarribia,Km 11 Campus de la UNA, San Lorenzo-Paraguay (Dated: February 16, 2021)The view exists that the Bell inequality is a mere inconsistent application of classical concepts toa well established quantum world. In the article, “Nonlocality claims are inconsistent with Hilbert-space quantum mechanics” [Phys. Rev. A, 101, 022117, (2020)] Robert B. Griffiths advocatesfor the locality of quantum theory. Although R. B. Griffiths presents valuable insights in favor ofquantum mechanics’ local character, he based some of them on unjustified views concerning the Bellinequality interpretation.
I. INTRODUCTION
Quantum mechanics’ locality is a contentious question.Hence, it is essential to base its defense on correct andlogically consistent arguments.Robert Griffiths [1] presents valid arguments but, un-fortunately, based some of them on incorrect views con-cerning the Bell inequality significance. Since quantumlocalists seem to share many of those views, it is of gen-eral interest to clarify some points.One of Griffiths’s provocative points is that since thepurportedly quantum nonlocal influences are useless fortransmitting information according to the no-signalingprinciple: “This means that such influences (includingwave-function collapse) cannot be directly detected in anyexperiment. The simplest explanation for their lack of in-fluence is that such influence do not exist.
The former point is immune to Bell inequalities vio-lations and constitutes a valid argument to avoid theirnonlocality implications. It suggests that quantum me-chanics demands a revision of our causality concept justas relativity demanded a revision of simultaneity. Thatcould result as counterintuitive as the latter.Accepting it prompts that Bell’s local causality defini-tion is different from relativistic causality and that theformer necessitates a proper revision. Or that we may beliving in a super-deterministic world.However, such valid arguments are different from re-ducing the Bell inequality to an erroneous applicationof classical physics to quantum mechanics. Griffithspresents various arguments depicting the Bell inequalityas being physically irrelevant.First, we review the meaning of the Bell inequality thengo over Griffiths’s arguments analyzing their consistency.
II. MEANING OF THE BELL INEQUALITY
John Bell first conceived his theorem in 1964 [2] as acontinuation of the
Einstein-Podolski-Rosen (EPR) [3]argument. Bell investigated the possibility to locallyexplain the existence of perfect correlations predictedby quantum mechanics. We shall consider the
Clauser,Horne, Shimony, Holt (CHSH) [4] generalization of the Bell inequality.The correct inequality formulation rests on two hy-pothesis, local causality and freedom . Hence, violationof the inequality implies the violation of at least one ofthose assumptions. Local causality and freedom are ei-ther true or false. They cannot be valid in a quantumsense and at the same time classically invalid.We shall use the standard expression local realism forthe conjunction of those two hypotheses. Notice that lo-cal realism is not the conjunction of locality and realism,at least in an obvious way. III. NONCOMMUTING OPERATORS
To prove his thesis that violations of the Bell inequal-ity have nothing to do with quantum nonlocality, Grif-fiths conceived an experiment with neon atoms violat-ing a Bell-type inequality, where locality is not an issue.According to his interpretation, the presence of noncom-muting operators is the cause of the inequality violations.In Griffiths’s ingenious experiment, only one particleproduces a Bell-type inequality, so obviously, locality isnot an issue. That is confusing; one could argue the pro-posed experiment is absolutely foreign to the physics ofa CHSH singlet state correlation experiment. Locality isnot an issue when measuring only one particle, but cer-tainly, it is when simultaneously measuring two differentparticles far apart. The mathematical analogy does notjustify drawing conclusions about the physical propertiesof one system from the physical properties of the other.If Griffiths’s point is to prove that the Bell inequal-ity may have other interpretations that are not relatedto locality issues, his example is, indeed, correct. How-ever, that is not proof that locality cannot be involved inother contexts where the inequality arises, for instance,a CHSH spin correlation experiment.Other authors have proposed similar ideas for inter-preting the inequality violations. For instance, Andrei The freedom or free will hypothesis is usually introduced as themeasurement independence postulate.
Khrennikov [5] proposes the same interpretation as Grif-fiths’s but applied in a CHSH context, so his example isfree from Griffiths’ previously mentioned defect. Accord-ing to A. Khrennikov “We demonstrate that the tests onviolation of the Bell type inequalities are simply statisticaltests of local incompatibility of observables” . Khrennikovshows that the Bell operator B = A B + A B + A B − A B (1)satisfies B = 4 I − [ A , A ][ B , B ] (2)(2) means that when at least one of the commutators onthe RHS vanishes, we obtain the Bell inequality. Let CO stand for the existence of at least one pair of commutingoperators, then according to (2), Khrennikov proved that CO → BI (3)If LR stands for local realism, Bell proved that LR → BI (4)In Griffiths’s example, (4) does not even make sense. In aCHSH experiment where (3) and (4) are both applicable,Krennikov suggests that by proving (3) he has disproved(4). Such arguments ignore the fact that a statementmay have different, sometimes unrelated, sufficient con-ditions. None of them disproves the validity of the othersas a sufficient condition. Of course, when analyzing (3),nonlocality is not an issue. Likewise, when analyzing (4),operators’ commutativity is not an issue.According to (3), violation of the Bell inequality meansthat [ A , A ] = 0 and [ B , B ] = 0. At the same time,either local causality or freedom is false according to (4).No matter what other interpretations of the inequalitywe may find, they cannot invalidate the locality implica-tions it has for a CHSH singlet state correlation experi-ment.For instance, some point out that George Boolefirst discovered the Bell inequality in the mid-eighteenhundreds[7]. Boole showed that a Bell-type inequality isa necessary condition for the existence of a joint proba-bility(JP). JP → BI (5)Then again, the argument goes, violations of the Bell in-equality are nothing else than proofs of the nonexistenceof a joint distribution for the experiment’s probabilities.Whereas the previous interpretation is correct, theproblem resides in the “nothing else” clause. They alsoprove the nonexistence of commuting operators and in-validates the conjunction of local causality and freedom. IV. CLASSICAL HIDDEN VARIABLES
Equation twenty-four of Griffiths’s paper reproducesthe factorization condition necessary to derive the Bell inequality
P r ( A, B | a, b ) = X λ P r ( A | a, λ ) P r ( B | b, λ ) P r ( λ ) (6)Griffiths presents a detailed explanation of what is wrongwith (6). He summarizes the argument in the final sen-tence of the corresponding section of his paper: “Thusthe usual derivations of CHSH and other Bell inequalitiesemploy classical physics to discuss quantum systems, soit is not surprising when these inequalities fail to agreewith quantum predictions, or the experiments that con-firm these predictions.” Griffiths and quantum localists[5, 8, 9] reject the con-sequences of Bell theorem on the basis that Bell usedclassical arguments incompatible with quantum mechan-ics. The realistic assumption in (6) has been contestedby many nonlocalists[10–13]. Part of the problem isthat quantum localists usually declare its realistic naturewithout giving convincing justifications.Accepting that (6) implies classical physics or realism,the following issues arise with Griffiths’s argumentation:1. Explaining why a classical prediction differs fromthe quantum one does not necessarily justifies thelocal nature of the latter. Nonlocal influences ei-ther exist or not, independent of having a classicalor quantum mechanical explanation. Griffiths’s –and quantum localists’ – bottom line seems to be:if a nonlocal influence is a quantum mechanical pre-diction, then it is not nonlocal by definition.2. (6) is direct consequence of the two hypothesis un-derlying the Bell inequality: local causality andfreedom. Therefore, at least one of them should beconsidered the culprit for introducing its classicalcharacter.3. Accepting that freedom is responsible for the pres-ence of realism, and rejecting it, allows us to keeplocality. Indeed, it is well known that by reject-ing freedom, i.e. measurement independence, it ispossible to explain locally the CHSH quantum cor-relations [15]. So, this is a valid solution to thenonlocality problem, albeit it is doubtful all quan-tum localists would be happy with this solution.We shall return to the issue of freedom when dis-cussing Griffiths’s quantum common causes.4. If we want to keep freedom, we must consider localcausality as classical. However, we could hardlyconsider quantum mechanics as locally causal bydeclaring local causality a classical concept whichdoes not apply to quantum theory. The last point is At least when it is correctly formulated. Notice the absent ofcounterfactual definiteness. See for instance Ref. [14]. an endemic inconsistency affecting quantum local-ists’ arguments who pretend to reject only the “re-alism” part of the unfortunate expression local real-ism. The issue generated longstanding and heateddebates [8–13].The correct reason for Griffiths’s rejection of (6) is notits purportedly classical character. Bell’s local causal-ity is the conjunction of two conditions: parameter in-dependence and outcome independence [16]. Since herejects no-signaling effects as being nonlocal influences,Griffiths rejects outcome independence accepting uncon-trollable nonlocality. As a consequence, he rejects Bell’slocal causality, and the Bell inequality cannot be proved.
V. QUANTUM COMMON CAUSES
According to Griffiths, basing the Bell inequality onquantum common causes(QCC) instead of classical com-mon causes(CCC) explains the inequality violation. Al-though he does not explain the difference, he describeswhat a QCC is: “Experiments that test Bell inequalitiesusing entangled photon pairs already assume a commoncause in the sense that pairs of photons produced at thesource in the same, rather than a different, down con-version event are identified using their arrival times. Allthat is needed in addition is an argument that the po-larizations measured later were also created in the same(local) event.”
Whence, the role of the QCC would be the same asthe CCC represented by λ in Bell’s formulation. A dis-tinction, thus far, is not established except perhaps thatquantum mechanics already include them.A closer examination of Griffiths’s argument revealsthat he considers the particles’ spin are determined atthe source and assumes their values as preexistent EPRelements of reality . Indeed, “...., if Alice’s apparatus isset to measure S z for particle a and the outcome corre-sponds to, say, S z = − / , she can conclude that particle a possessed this property just before the measurement took place, and, assuming it was not perturbed on its way toher apparatus, at all previous times following the initialpreparation.” As Griffiths explains, his “measurement” frameworkallows tracing both measurement outcomes back to thesource by fixing Alice’s and Bob’s settings at the momentof the singlet creation. In Bell’s framework, this has awell-known interpretation as freedom violation. Whenthe settings are predetermined at the source, it meansthat the hidden variables distribution function is not in-dependent of Alice’s and Bob’s setting parameters: ρ ( λ, a, b ) = ρ ab ( λ ) (7)Thus, Griffiths’s QCC, in the Bell inequality context, isconsidered a measurement independence violation physi-cally interpreted as freedom violation. As previously no-ticed in Sect. IV, the rejection of freedom is a logicallyvalid alternative for a local explanation of the quantumcorrelations. VI. CONCLUSIONS
Some improper considerations regarding the Bell in-equality unnecessarily obscure Robert B. Griffiths’s argu-ments of quantum mechanics’ locality. By pointing themout, we expect to have clarified the meaning of the Bellinequality showing how it consistently fits into Griffiths’sviews of quantum locality.Griffiths’s concept of locality accepts uncontrollablenonlocality and rejects outcome independence allowingfor Bell inequality violations. Also, regarding quantummeasurements, he seems to uphold superdeterminism. Ina Bell inequality context, that view is coherently inter-preted as a violation of measurement independence.Each of the above concepts infringes one of the Belltheorem’s hypotheses; Bell’s local causality, and freedomrespectively.Consequently, claims that Bell inequality is an incon-sistent application of classical physics to quantum me-chanics are unnecessary and erroneous. [1] Robert Griffiths. Nonlocality claims are inconsistent withHilbert-space quantum mechanics.
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