Chapter 10 PROOFS of IMPOSSIBILITY and POSSIBILITY PROOFS

Chapter 10 PROOFS OF IMPOSSIBILITY AND POSSIBILITY PROOFS

There are proofs of impossibility and possibility proofs, a situation that sometimes produced a bit of confusion in the debates on hidden-variables theories and sometimes led to something worse than confusion. So, we have to clarify it.

Proofs of Impossibility von Neumann Brief Beginning of the History In his famous book, providing an axiomatization of quantum mechanics, von Neumann [26] introduced in 1932 a proof of impossibility of hidden- variables theories, the famous “von Neumann’s proof.” This were a few years after the Solvay Congress of 1927 during which de Broglie’s pilot wave was defeated. Although de Broglie had already renegated his work, and rejoined the camp of orthodoxy, the publication of the proof certainly confirmed that he had lost time with his heretical detour. Furthermore, in the words of Pinch [154], “the proof was accepted and welcomed by the physics elite.” The continuation of the story shows that the acceptance was not based on a careful examination of the proof but rather on other elements such as a psychological pressure exerted by the very high reputation of its author or the unconscious desire to definitively eliminate an ill-timed and embarrassing issue. Afterward, for a long time, the rejection of hidden variables could be made just by invoking von Neumann (the man himself more than the proof). As stated by Belinfante [1], “the truth, however, happens to be that for decades nobody spoke up against von Neumann’s arguments, and that his conclusions were quoted by some as the gospel … the authority of von Neumann’s over-generalized claim for nearly two decades stifled any progress in the search for hidden variables theories.” Belinfante also [1] remarked that “the work of von Neumann (1932) was mainly concerned with the axiomatization of the mathematical methods

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of quantum theory. His side remarks on hidden variables were merely an unfortunate step away from the main line of reasoning.” This was more than unfortunate, however; it was erroneous, and even deeply erroneous, “because of the obviousness of inapplicability of one of von Neumann’s axioms to any realistic hidden variables theory [1].” As far as I know, the first person to challenge the validity of von Neu- mann’s proof was the philosopher Grete Hermann in 1935 [117], three years after von Neumann’s book, followed nine years after, in 1944, by another philosopher, Hans Reichenbach [134], according to Pinch [154]. As noted by Bitbol [133], and can be checked in the original reference, recently republished with enlightening complementary discussions by Léna Soler, Hermann correctly identified the weakest point of von Neu- mann’s proof, namely, the misleading use of an additivity condition of average values (which are soon to be discussed), pointed out by Bell again in 1966 [188]. Furthermore, she charged von Neumann’s proof with the accusation of circularity. Likely because they were philosophers, Hermann and Reichenbach seemingly did not have much influence on the community of physicists. Very often, they are even forgotten or dismissed. For instance, Pipkin [156] could erroneously write that it was Bell, in 1966, “who first pointed out the axiom by which von Neumann’s formulation violated the elementary principles of any realistic hidden variables theory,” a statement strongly and strangely in agreement with Belinfante [1] when he claimed, five years before that “The first to publicly pinpoint the axiom by which von Neumann’s formulation violated the elementary principles of any realistic hidden variables theory was Bell (1966).” As far as I know, the first physicist to challenge von Neumann was Bohm, twenty years after the proof was proposed, in one of his 1952 papers [30]. The best rebuttal of the early Bohm has obviously been the fact that he produced a logically consistent hidden-variables theory. This was a constructivist rebuttal, explicitly showing that what was demonstrated as being impossible was in fact possible. It definitely established that something was wrong with von Neumann’s proof; but what exactly was wrong? This is something that was compulsory to establish. The formal answer to this issue by Bohm was rather poorly conceived, but he correctly identified the most important point, namely, that von Neumann implicitly restricted himself, in his proof, to an excessively narrow class of hidden-variables theories. It just so happened that the early Bohm’s pilot wave did not pertain to this class. Louis de Broglie then also manifested

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a reluctance against von Neumann’s proof, e.g., in [62, 64]. The fact that Bohm’s rebuttal was unsatisfactory, and required more clariﬁcation, was also pointed out by Bell [188], who commented, “The analysis of Bohm seems to lack clarity, or else accuracy ….”

Bell’s Rebuttal of von Neumann Accepting the idea that von Neumann’s proof has been rebutted, PEDES- TRIANS (∼∼∼∼) are allowed to skip this subsubsection. Bell [188] was able to make clear in a fairly laconic way what was unclear in Bohm’s attack against von Neumann. In essence, his identifi- cation of the reason why von Neumann failed is in agreement with what Hermann had already understood, some thirty years before. It was, however, the influence of Bell’s paper, about thirty-five years after the proof was announced to the world, a long Middle Age for hidden variables, that was the most powerful. The dragon was struck down.At that time there was a dragon still living and it had to be destroyed; this was not obvious to all scientists, as we can infer from the following quotation [188]: “The present paper … is addressed to those who do find the question interesting, and more particularly to those among them who believe that ‘the question concerning the existence of such hidden variables received an early and decisive answer in the form of von Neumann’s proof on the mathematical impossibility of such variables in quantum theory’.” The influence of Bell’s paper is acknowledged by Bitbol [133] when he stated that the very sociological turnaround of the community of physicists regarding von Neu- mann’s theorem nevertheless only took place from 1966 on, the date of the publication of Bell’s paper … Bell’s attack pointed out a faulty additivity postulate used by von Neumann that, when applied to hidden-variables theories, was “one of the weakest points in the proof,” in the words of Jammer [24]. The essential assumption used by von Neumann, the one we called above the “additivity condition on average values,” from now on called the additivity postulate, is described in the following statement: “Any linear combination of any two Hermitian operators represents an observable, and the same linear combination of expectation values is the expectation value of the combination.” In the words of Mermin [197], this was “von Neumann’s silly assumption.” The sentence quoted above actually contains two statements. The first one says that any linear combination of any two Hermitian operators represents an observable. This seems obvious, but it actually is not guaranteed

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at all. Let us, however, forget this problem as irrelevant for our purpose and focuss on the second statement, the one we properly call the additivity postulate, saying that the same linear combination of expectation values is the expectation value of the combination. Let us consider two Hermitian operators A and B, and let us assume that the sum of the two Hermitian operators A and B is another Hermitian operator, deﬁning an observable, which we denote by C. Let us remark that we have been here a bit more careful than von Neumann. We do not claim that C is an observable for any couple (A, B). We just consider a couple such that C is indeed an observable. We then have

C = A + B (10.1)

The additivity postulate tells us that

C=A+B (10.2)

where X is the expectation value of X. In quantum mechanics, the expectation value of X for a given wave function can be evaluated according to the following rule: ∗ X= Xdr (10.3)

where ∗ is the complex conjugate of . Then, the additivity postulate is seen to be always true in quantum mechanics, since C= ∗Cdr = ∗(A + B)dr ∗ ∗ = Adr + Bdr =A+B (10.4)

In particular, it is true regardless of the value of the commutator [A, B]; that is, it does not depend on whether or not the operators A and B commute. As obvious as it seems, the additivity postulate is not trivial. This can be exempliﬁed by noting that it does not necessarily apply to eigenvalues. To see this, let us consider a counterexample taken from Jammer [24]. Let us consider the spin component of an electron in the direction along

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the bisector line between the x axis and the y axis. This observable is represented by the operator 1 S45◦ = √ (σx + σy) (10.5) 2

The outcome of the measurement of S45◦ for the electron, a spin-1/2 particle, may be either h/¯ 2or−h/¯ 2; that√ is, the eigenvalues are ±1, in units of h/¯ 2. This is different from (±1 ± 1)/ 2, which is what we would obtain if we applied the additivity postulate to the right-hand side of Eq. 10.5. Therefore, indeed, eigenvalues do not combine linearly. Nevertheless, we still have, as we should, 1 √ (σx + σy) =±1=0 (10.6) 2 1 1 √ (σx+σy) = √ (±1+±1) = 0 (10.7) 2 2 that is, 1 S45◦ =√ (σx+σy) (10.8) 2 Although the additivity postulate (which has to be true for commuting as well as for noncommuting observables) is satisﬁed by Eq. 10.8. This example is particularly interesting, because σx and σy precisely do not commute. For commuting variables, the additivity postulate may hold for eigenvalues having well-deﬁned values for common eigenstates. In such a case, it follows immediately that it holds too for expectation values. In contrast, we just have exhibited an example, with noncommuting variables, where it does not hold for eigenvalues, although it holds for expectation values. As stated by Bell [188], Ameasurement of a sum of noncommuting observables cannot be made by combining trivially the results of separate observation on the two terms—it requires a quite distinct experiment. For example, the measurement of σx for a magnetic particle might be made with a suitably oriented Stern–Gerlach magnet. The measurement of σy would require a different orientation, and of (σx + σy) a third and different orientation. But this explanation of the nonadditivity of allowed

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values also established the nontriviality of the additivity of expectation values. The latter is a quite peculiar property of quantum mechanical states, not to be expected a priori. There is no reason to demand it individually of the hypothet- ical dispersion-free states, whose function is to reproduce the measurable peculiarities of quantum mechanics when aver- aged over. Another example, similar to the one above (picked up from Jammer), but with a different ﬂavor, is available from Bohm and Hiley [68]. Let us explain it because of its complementary interest. Consider a particle whose observables (or operators) for the components of the orbital angular momentum are denoted as Lx,Ly, and Lz, in the usual notation. We also restrict ourselves to the case in which the eigenvalues√ are h¯,0,and√ −h¯. For A and B of Eq. 10.1, let us take A = Lx/ 2 and B = Ly/ 2. Let now L45◦ be the operator associated with a measurement of the orbital angular momentum component along the bisector line between the x axis and the y axis. We have 1 L45◦ = √ (Lx + Ly) (10.9) 2

so that L45◦ is the operator C of Eq. 10.1. The additivity postulate, which is valid for expectation values in quantum mechanics, tells us that we indeed may write 1 L45◦ =√ (Lx+Ly) (10.10) 2 Now, let us assume the existence of dispersion-free subensembles in which the observables Lx, Ly, and L45◦ have well-deﬁned values V(Lx), V(Ly), and V(L45◦ ), respectively. If these values satisfy the additivity postulate, we should have 1 V(L45◦ ) = √ [V(Lx) + V(Ly)] (10.11) 2

Now, let us assume for instance that V(L45◦ ) = h¯, one of the allowed values indeed. Next, we have V(Lx) = h,¯ 0, −h¯ and also V(Ly) = h,¯ 0, −h¯. We can then see that there is no way to satisfy Eq. 10.11. Hence, there are no dispersion-free states, and hidden variables do not exist. This is substantially von Neumann’s proof speciﬁed for a convenient example.

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The rebuttal, again specified for this example, goes as follows. First, in the words of Bohm, “As is well known (and as von Neumann agrees), there is really no meaning to combine the results of noncommuting operators such as Lx,Ly, and L45◦ . These measurements are incompatible and mutually exclusive.” Nevertheless, Eq. 10.10 for expectation values is still true, owing to the validity of the additivity postulate in quantum mechanics, even if it requires the use of three separate mutually exclusive series of experiments. Now, from the fact that Eq. 10.11 cannot be satisfied for dispersion-free sets, we should not conclude that dispersion-free sets do not exist. Conversely, we have to conclude that the additivity postulate does not apply to such sets. Indeed, the true value of V are not of a quantum nature. The additivity postulate is a peculiarity of quantum mechanics that need not be satisfied by the true values of hidden-variables theories. From the previous arguments, the mistake of von Neumann should now be clear: He unduly extended an additivity postulate from quantum mechanics, where it is valid but nontrivial, to the realm of hidden variables where it is trivially nonvalid. The same issue of an undue extension is also put forward by Mugur-Schächter, although in the context of an argumen- tation that is more logical than physical, and sheds complementary light to illuminate the landscape.

Mugur-Schächter’s Rebuttal of von Neumann I believe that PEDESTRIANS (∼∼∼∼) are allowed to skip this subsubsection too, just accepting the fact that Mugur-Schächter provided a rebuttal to von Neumann’s proof. They will find a very short comment, worth visiting, at the end of it. Two years before Bell’s rebuttal in 1966 [188], another rebuttal had already been provided in 1964 by Mugur-Schächter in her thesis [291], more precisely in the first part of her thesis. This work has been less influ- ential than Bell’s, but I must confess, without any disrespect concerning Bell, that I found it particularly impressive and convincing. What Mugur- Schächter did is to analyze the logical structure of von Neumann’s proof to demonstrate that the logic used is inconsistent. We may say that the deep physical content of Mugur-Schächter’s rebuttal is similar to Bell’s, but it provides a quite different, but complementary, point of view by insisting on the flawed logical organization of von Neumann’s proof. The thesis, interestingly enough, is prefaced by De Broglie, who stated that now for several years there were already doubts rising against von Neumann’s

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proof and that he already came to the conviction that von Neumann’s reasoning was misleading and circular. The circularity of von Neumann’s proof lies in the fact that he implicitly introduced in his premises the result that he intended to demonstrate. De Broglie also stated that Mugur- Schächter’s work achieved a genuine logical dissection of von Neumann’s proof and rigorously demonstrated its fallacious character. (For Jammer, however [24], the charge of circularity is not justiﬁed.) I am now going to present the argument of Mugur-Schächter. However, I shall simplify it a bit without, I hope, destroying the gist of it. Also, I shall do a bit of rewording to better match the terminology used today. Mugur- Schächter starts with a logical analysis of von Neumann’s proof, which can be summarized in a few steps. To set the stage, let us consider a wave function and N realiza- tions of , i.e., N copies of named 1,2,...,N (in other words, we consider a statistical set {1,2,...,N }). Let us also consider an observable R (any observable) that we can measure on . From the copies, we may then evaluate the expectation value R of R, which actually can also be quantum mechanically evaluated by using Eq. 10.3, or its more general formulation expressed in Dirac notation as

R= | R | (10.12)

We say that the statistical set is dispersion free if, for any R, we have

R2=R2 (10.13)

Now, the steps to be considered are as follows.

(1) In quantum mechanics, there is no dispersion-free set (as is clear if we just think of the Heisenberg uncertainty relations). This is, in particular, true for pure states when, and only when, the density operator is a projector [43]. (2) Assume that we possess hidden variables associated with the behavior of quantum objects. Then, a pure state set can be decomposed into dispersion-free subsets by sorting them according to the values of the hidden variables. (3) However, by (1), there is no dispersion-free set, and hence we have a contradiction, which establishes that hidden variables do not exist. As a corollary, quantum mechanics is intrinsically indeterministic.

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The rebuttal may proceed as follows. Surely, items (1) and (2) are correct. The point is that item (2) does not apply to quantum mechanics. Indeed, hidden variables are not necessarily of a quantum nature; i.e., they are not necessarily compatible with the postulates of quantum mechanics. In particular, they are not necessarily distributed according to the probability rules of quantum mechanics. For instance, in the pilot wave theory hidden variables are of a classical kind, insofar as they are associated with classical (or pseudoclassical) deterministic trajectories whose existence is rejected in the framework of quantum mechanics. They are therefore not distributed according to Born’s postulate, which applies to the guiding wave , or, say, to the observed values of positions, and not to the hidden trajectories, or, say, to the objective values of positions. We may also say that the hidden variables, being of a classical kind, are deﬁned outside of the logical framework of quantum mechanics in which measured values of observables are of a quantum type. In other words, the theory THV consisting of quantum mechanical theory TQM and its completion with hidden variables is larger than quantum mechanics (THV >TQM ). This is a convenient way to say that THV does not operate inside TQM or that it is not internal to TQM . Therefore, item (1) pertains to TQM and item (2) to THV. Then, item (3) is faulty because it unduly transports an element of a smaller theory into the structure of a larger theory. PEDESTRIANS (∼∼∼∼), start here. Borrowing a question from the title of a paper from Pinch [154], we then may ask, “What does a proof do if it does not prove?” Well, the answer might be that von Neumann’s proof indeed did not prove what it was aiming to prove, but yet it is proving something. It is proving that if hidden variables do exist, they cannot be of a quantum type, because, if they were then von Neumann’s proof would apply to them. This restricts the class of admissible hidden-variables theories, but it does not rule them out. The restriction of the class of admissible hidden-variables theories, as we shall see, will require other conditions: They will have to be contextualist and nonlocal, as is the case for the early Bohm’s pilot wave. Mugur-Schächter went on discussing other proofs against the proof, from Bohm, Weizel, Fenyes, Bocchieri and Loinger, and de Broglie, pointing out their inadequacies. I can, without any loss of completeness, leave them out. Complementary discussions on von Neumann’s proof are available from de Broglie [62, 63, 65], Pauli [164], Bohm [60], Bohm and Bub [297], Bohm and Hiley [68], Siegel [303], Ballentine [304], Belinfante [1],

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Jammer [24], Flato et al. [305], Pinch [154], Pipkin [156], Bell [142], Wheeler and Zurek [100], Squires [13], Mermin [197], Bitbol [133], and d’Espagnat [84]. This rather signiﬁcant list of references is certainly a testimony to the fame of von Neumann’s impossibility proof for the debate on hidden variables.

von Neumann Still Worrying about Hidden Variables Once things are published, they are published. Some authors may feel, sooner or later, that they did not present things in the best possible way. They may find that a particular sentence is not properly phrased or that a certain demonstration could have been made shorter and more elegant, or they may even discover an error—a real nightmare for theoretical physicists who crave perfection. However, we must accept the fact that everything produced in theoretical physics will ultimately be shown to be wrong, in a long process of mistakes being pinpointed and corrected. If there is only one truth, it is a matter of elementary computations in probability theory to demonstrate that the probability of having reached the truth is zero. This paragraph applies in particular to von Neumann, who did publish an erroneous proof. As far as I know, we do not possess any information concerning what von Neumann eventually thought of von Neumann’s proof. It seems well established, however, that he went on worrying about hidden-variables theories. In any case, according to Wigner [74, 306], von Neumann pos- sessed another (although unpublished) objection against hidden variables. To explain this objection, let us consider a Stern–Gerlach experiment, or, more precisely, an indefinite number of repetitions of Stern–Gerlach experiments. We begin, for instance, with a measurement of a spin component along the z direction, followed by a measurement of a spin component in the x direction, followed by still another measurement of a spin component along the z direction, followed again by another measurement of a spin component along the x direction, and so on. Let us now assume again, for instance, that we are dealing with spin-1/2 particles. The first measurement is used to prepare the system in a pure state, say, spin up in the z direction. Once this is done, the outcomes of the second measurement can be spin forward or spin backward along the x direction, both with probability 1/2. If these values are deterministically determined by hidden variables, the result obtained (say, + or − along the axis 0x) provides restrictive information on the values of the hidden

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variables. The next measurement, now with probability 1/2 for both spin up or spin down (along the z direction), will further restrict the values of the hidden variables. Eventually, after N measurements (where N is very large but finite), the hidden variables will become restricted to a very narrow range. Yet, this narrow range should still be able to completely determine all further measurements, after the N first measurements, even if the number of further measurements is assumed to be infinite. (We will not worry about the fact that it is in practice impossible to carry out an infinite number of measurements.) Is this possible? Well, for von Neumann it seemed unreasonable. According to Wigner [306], it seems that von Neumann was thinking in terms of a variety of hidden variables to determine the outcomes of the measurements, not in terms of a single variable. Then von Neumann would have pointed “to the unreasonable large variety of hidden variables which must be assumed if one wishes to account for the postulate (implicit in quantum mechanical theory) that no matter how many successive measurements we undertake of a system, the distribution of the hidden variables remains sufficiently unsharp so that the outcomes of measurements are as unpredictable as they were to begin with.” It is difficult to know why von Neumann did not publish his proof. A likely possibility is that, although the proof seems intuitively convincing, von Neumann did not succeed in giving it a satisfactory mathematical shape. Also, we may advance as a guess that when trying to make it mathematically safe, he found that the proof was basically flawed. I am daring to propose a rebuttal that maybe von Neumann eventually discovered. Let us begin by assuming that each hidden variable in the variety of hidden variables is continuous. Then, we may invoke the Cantor–Bernstein theorem, telling us that Rn is equipotent to Rm, whatever the values of n and m, which are positive integers [307]. Let n denote the number of hidden variables in the von Neumann variety of hidden variables, and let m be equal to 1. Then, the Cantor–Bernstein theorem implies that we may reduce the von Neumann variety to a single hidden variable ranging over R. This reduction is not strictly necessary for the rebuttal, but it is useful to support the intuition. Now the cardinal of R is infinity (not a countable infinity, however, but the infinity of the power of the continuum). Therefore, there is plenty of room in R to deal with an infinite number of successive measurements, and von Neumann still worrying is rebutted. The full real line is not even necessary to develop the argument. For instance, let us assume that the single hidden variable does not spread

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over R but is instead distributed on the open segment ]0, 1[. After N successive measurements, the hidden variable may have been restricted to a number of open segments located inside the original unit open segment, with children segments narrowing more and more when the number N of measurements increases. However, it is a property of the power of the continuum that, whatever N, whatever the number of children segments, and whatever the narrowness of these segments, the cardinal of the set formed by all children open segments remains strictly equal to the cardinal of the original unit open segment. As another possibility, let us assume that each hidden variable in the variety of von Neumann may take an inﬁnite countable number of values. The argument used above for continuous variables may be, mutatis mutandis, repeated for this new case by relying on the equipotence between Nn and Nm. Finally, the rebuttal may be made complete, in a similar way, if we assume that the variety of von Neumann consists of a mixture of continuous and countable hidden variables. I believe that von Neumann, although he had a new objection in mind, soon realized that it could not be mathematically defended.

Early Other Proofs of Impossibility There are also other early proofs of impossibility that were never so famous or disputed as von Neumann’s proof. One year after von Neumann’s proof appeared in the literature, Solomon [308] also examined the possibility of hidden-variables theories and concluded that any deterministic hidden-variables theory is incompatible with the intrinsic indeterminacy of quantum mechanics. Another demonstration of the fact that quantum mechanics is intrinsically indeterministic was given by Destouches-Fevrier in 1945 [309, 310], who reached the conclusion that it is no longer possible to return to determinism in the microphysics realm. One year after Bohm’s 1952 publication of his early papers on the pilot wave, Destouches [202] took as guaranteed the results of von Neumann and Solomon, gave a rebuttal to a recent objection from de Broglie [152], and concluded that quantum mechanics is essentially indeterministic. He afﬁrmed that any future theory that would replace the present quantum theories would have to be indeterministic too. I am not aware of any attempt to refute these works, a kind of indiffer- ence that might be easy to explain. First, these new proofs of impossibility, dated 1933, 1945, or 1953, when von Neumann was still largely dominating

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the stage, were likely found to be neither interesting nor worthwhile enough to be deeply examined. Second, later on and today, it may be simply sufﬁ- cient to state that we possess a convincing rebuttal of von Neumann’s proof of impossibility and, more importantly, that we possess a consistent example of deterministic hidden-variables theories, namely, the early Bohm’s pilot wave theory. Therefore, without any doubt, these other early proofs of impossibility must be ﬂawed somewhere. I shall be content in this book with this expedient point of view, although a more careful examination of these early other proofs, to identify where the shoe pinches, might deserve a bit of effort (if we had plenty of time to do it…).

Late Other Proofs of Impossibility Even after Bohm’s pilot wave of 1952, and after the criticisms already addressed against von Neumann, other proofs of impossibility have been published. In presenting these new proofs, some authors considered themselves in the filiation of von Neumann and were aiming to improve an imperfect proof. After all, maybe von Neumann’s demonstration was wrong, but this does not mean that the conclusion was wrong too. It could be right, and an improved demonstration could be able to establish this point correctly. The persistence of such an effort to get rid of hidden variables might seem to be welcomed, particularly from those who were not aware of the existence of Bohm’s pilot wave theory—or who simply ignored it. Indeed, if these damned hidden variables do not exist, we better be able to prove it and to eliminate this long-lasting and irritating issue. However, Bohm was performing on the center stage and, simply, could not be ignored. The effort to find new proofs of impossibility, which, given Bohm’s influence, was doomed to failure, seemed weird to Bell. For instance, he would ask [142]: “… extraordinarily, why did people go on producing impossibility proofs, after 1952 …?” and also speak of “the strange story of the von Neumann impossibility proof and of the even stranger story of later impossibility proofs.” The topic of late other proofs of impossibility is not an easy one. It does not make for easy reading or easy reporting, particularly when one needs to invoke logical-mathematical approaches and the proposi- tional calculus of quantum mechanics, and when one has use swear words such as lattice of propositions or orthocomplemented lattice or even swear words that are more familiar to mathematicians than to physicists. Further- more, as we shall illustrate, the literature is often confusing, polemical,

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and contradictory. The reader wanting to quickly sample a flavor of the topic may refer to a collection of papers on the logico-algebraic approach to quantum mechanics, edited by Hooker [136]. The second paper, by Strauss [311], leads to the result that “all attempts at interpreting the quantum mechanical formalism in terms of classical probabilities … are doomed to failure,” a way to dismiss hidden variables. Conversely, in a subsequent paper of the same collection, Kamber [312] concluded that “the existence of hidden parameters … cannot … be excluded … mathematically.” These two quotations illustrate the contradictions that we may have to face. The story goes on in the same style. An early work among the late works is by Gleason [313] on the basis of which Jauch and Piron [314] thought that they had dismissed the possibility of hidden variables. The works of Gleason, and of Jauch and Piron, are discussed in a simplified manner by Belinfante [1]. Also, other comments on these works are available from Shimony [93]. However, Belinfante, considering that Gleason’s proof was rather abstract, preferred to derive Gleason’s result from a later, more understandable work by Kochen and Specker, to which we shall return soon. Bohm and Bub [315] provided a rebuttal to Jauch and Piron, which was rebutted by Jauch and Piron, followed by a rebuttal of the rebuttal of the rebuttal by Bohm and Bub again [316]. The refutation of Jauch and Piron by Bohm and Bub is also discussed by Gudder [317]. In 1966, Bell in the same paper in which he pointed out the faulty axiom in von Neumann’s proof [188], also criticized the proofs from Gleason, and from Jauch and Piron, but made clear, relying on Gleason’s work, that a class of hidden-variables theories, possibly called noncontextualist theories (which we shall discuss later), are inconsistent, more specifically inconsistent for quantum systems whose Hilbert spaces are more than two dimensional (i.e., dimension three or greater). In his paper, Bell took the opportunity to provide a new proof of Gleason’s result, which is independent of the scheme used by Gleason. This is similar to the result that was soon to be obtained by Kochen and Specker [28]. These impossibility proofs are actually “no-go” theorems (i.e., do not go to the classes of forbidden hidden variables). In the words of Mermin [197], the results obtained, “by demonstrating that a hidden-variable program necessarily requires outcomes for certain experiments that disagree with the data predicted by the quantum theory, are called no-hidden-variables theorems (or, vulgarly, ‘no-go theorems’).” It should now be clear to the reader that the so-called proofs of impossibility cannot prove the impossibility of something that, with Bohm, has been constructively shown to be possible.What they can, however, do when

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properly examined is to exclude some classes of hidden-variables theories. Focusing on this point of view, Bell examined which classes of hidden- variables theories are actually ruled out by these proofs. He emphasized that these classes form a rather small subset of all possible hidden-variables theories. Jauch and Piron’s proof, as von Neumann’s proof, required the additivity of expectation values of noncommuting variables, which, as we have seen, is valid in quantum mechanics but need not be applied to hidden variables. Gleason considered theories in which the outcome of a measurement is independent of which compatible observables were simultaneously measured. Such theories are called noncontextual theories. Here again, however, this was an undue restriction. Hidden-variables theories need not be noncontextual theories. Conversely, they have to be contextual. The rather subtle issue of contextuality will be examined and discussed in the next chapter. As stated by Shimony [93], “Bell (1966) noted however that Gleason’s theorem does not preclude a more complex type of hidden- variables theory, called ‘contextual’ according to which a complete state assigns a definite truth value to a proposition only relative to a specific context.” Bohm and Bub [315] also contradicted Jauch and Piron, saying that they actually prove nothing at all and that the argument of Jauch and Piron is circular, an accusation already addressed previously to von Neumann’s proof. In a subsequent paper, Jauch and Piron [318] provided a rebuttal to this rebuttal, something like turning down the objection point blank, being content with a “restatement of our result in a nontechnical language.” See also the book by Jauch [319] in which Chapter 7 is dedicated to hidden variables. In this chapter, Jauch maintains a theorem according to which the existence of hidden variables (of a certain kind) is in contradiction with empirical facts. This is soft enough because of the cautious expres- sion of a certain kind. Indeed, what occurs most generally for proofs of impossibility is that they refer to certain definitions or premises that imply the impossibility of hidden variables when these definitions or premises are accepted. However, the use of other definitions or premises might be allowed, so that proofs of impossibility only refer to a certain kind of hidden variables. A second somewhat parallel line of exposition starts with Kochen and Specker [28], who give a proof of the nonexistence of hidden variables. Actually, once again, they did not prove what they claimed to prove, but they proved that hidden-variables theories must be contextual, a result already obtained ten years before by Gleason, although by different means. As stated by Freedman and Holt [300], they “considered only theories in

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which the outcome of a measurement was independent of which compatible observables were simultaneously measured,” that is, noncontextual theories. However, it is acknowledged that the proof of Kochen and Specker is complicated (and, indeed, it is). For an easier path, the interested reader should better turn to Belinfante [1] who provided a simple-enough discussion concerning contextuality. In plain terms, Belinfante remarked that the argument used by Kochen and Specker (and also by Jauch and Piron) simply denies that the way in which a measurement is made could have an effect on the result of the measurement; this is a very strong hypo- thesis indeed! A conclusion may be borrowed from Jammer [24]: “Kochen and Specker have proved once again that noncontextual hidden variables do not exist in quantum mechanics.” For another line of exposition, let us consider a work from Gudder [320]. Relying on previous works, in particular on Jauch and Piron, Gudder examined proofs of impossibility in the framework of formal logic, using the concept of an orthocomplemented partial set that is complete with respect to compatible elements.Although the content makes for frightening reading, the conclusion can fortunately be easily delivered as follows: A quantum system admits hidden variables if and only if it acts physically as an ideal classical system. Hence, because it is well known that quantum systems do not behave like classical systems (e.g., consider Heisenberg’s uncertainty principle), hidden variables must be excluded from quantum mechanics. We identify here again something that is like a circle, as in von Neumann’s proof. A heuristic rebuttal can be constructed by referring to the example of Bohm’s pilot wave. In this pilot wave, there are two levels: the higher level of quantum mechanics and the lower level of hidden variables. The higher level, the one of quantum mechanics, being not of a classical nature, indeed does not receive hidden variables. The lower level is the one of the hidden trajectories. What Bohm did is to succeed in connecting the two levels, which, somehow, can each be considered each in its own right. At least this is the result of my effort to dismiss Gudder in a simple and I hope essentially correct way, avoiding any technicality. A bit later on, Gudder [321] himself remarked, taking into account the fact that Bohm’s theory had never been shown to be inconsistent, that “clearly, there is something wrong here. One obviously cannot have a HV [hidden- variables] theory if it is impossible.” Going on further, Gudder eventually demonstrated that hidden variables are always possible. Bub [322], facing such a confusion, took the expedient but rather arbi- trary point of view that “the term hidden-variables theory is justiﬁably

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used to denote the kind of theories rejected by von Neumann, Jauch and Piron, Kochen and Specker, [but] it is suggested that the term should not be used as a label for the theories considered by Bohm and other workers in the field. Such theories should be regarded as fundamentally compatible with the original Copenhagen interpretation of the quantum theory, as expressed by Bohr,” adding, “The conclusion of this paper will therefore be that there are no hidden variables theories of quantum phenomena in the usual sense, that the term hidden variables theory for the kind of theory considered by Bohm and his collaborators is unfortunate and misleading, and that this latter approach might well be characterized as an extension of Bohr’s conception of wholeness as opposed to von Neumann philosophy.” This is a kind of normative attitude introducing a dichotomy between what deserves and what does not deserve to be called hidden variables, if not a kind of propaganda to shield Bohm from attacks from hidden-variables opponents. In any case, even if the concept of wholeness is somehow common to Bohr and Bohm, it is extraordinarily difficult to see Bohm as a continuator of Bohr. In a clever and amusing manner, Greechie and Gudder [137] considered two hidden variable proofs in the quantum logic framework: one a proof that they do not exist and one a proof that they do. This is indeed rem- iniscent of a Jesuitical exercise in rhetoric, in which the same man can prove that God does not exist and, just after, with the same cleverness in persuasion, that He does exist. At this point, is the reader confused by a literature that is indeed confusing? There are, however, a small number of simple key points that can provide some insight: (1) Proofs of impossibility can never prove impossibility. (2) They can only prove that some classes of hidden variables are impossible. (3) Among the set of impossible hidden-variables theories, there are those theories that are noncontextual. This last item needs clarification. It will require a chapter, after this chapter. Further details are available in Jauch [319], Belinfante [1], Jammer [24], Freedman and Holt [300], Flato et al. [323], Gréa [324], Pinch [154], Fine and Teller [325], Peres [326], and Bohm and Hiley [68].

Possibility Proofs Once again, the easiest possibility proof is simply the very existence of Bohm’s pilot wave, a constructivist proof in which an ediﬁce is constructed so that we can simply say, “Just look at it.” It is logically consistent, and

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this consistency has never been contradicted. Once we understand Bohm’s theory, and understand that it is logically consistent, there is really no need to further investigate the issue. If we just want to know whether or not hidden-variables theories are possible, the issue is indeed closed. However, we are going to dig a bit further with formal approaches, to provide a still better understanding. Actually, we have many other things to learn. PEDESTRIANS (∼∼∼∼) may skip the next subsection, being content with the conclusive last lines, but they should be able to deal with the next one after it.

A Simple Particular Formal Approach In this subsection, we are going to provide a simple example due to Bell [188], an example that is also discussed by Flato et al. [323]. The reader might find that this example is too particular to be convincing. In the words of Flato et al.,itmay be too simple. However, even if it is too simple, it is enlightening and sufficiently illustrating to support later discussions. Following Bell, we consider a system with a two-dimensional state space, to be specific, a particle of spin 1/2 without any translational motion, or, more generally, a two-dimensional system that can be reduced to a spin-1/2 system (see, e.g., several examples in [43]). In such a case, any quantum mechanical state may be represented by a spinor , which is a two-component wave function. Any observable R in the two-dimensional state space may be represented by a square Hermitian matrix of rank 2, which can always be decomposed into the form

R = αI + β · σ (10.14)

where I is the unity matrix, the supervector σ has for components the Pauli matrices σx, σy, and σz, α is taken to be a real constant, and β is taken to be a real vector (and Bell indeed takes α and β as real). Explicitly, Eq. 10.14 can be rewritten as 10 01 0 −i 10 R = α + β + β + β (10.15) 01 x 10 y i 0 z 0 −1

Let us consider the rather simple case in which βx = βy = 0. Then 10 10 R = α + β (10.16) 01 z 0 −1

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from which we see that the eigenvalues are

Ev = α ±|βz| (10.17)

Physics should not depend on the orientation of the vector β. Therefore, in the most general case of Eq. 10.15, eigenvalues should be

Ev = α ±|β| (10.18)

According to the basic rules of quantum mechanics, the associated expectation values are

αI + β · σ=, (αI + β · σ) (10.19)

Let us now complement quantum mechanics with a real hidden variable, denoted λ, pertaining to the interval [−1/2, 1/2]. A microstate is then deﬁned by the couple (, λ) in which is the spinor. By a rotation of coordinates, the spinor can always be given the simple form 1 = (10.20) 0

For such a spinor, by using Eq. 10.19, quantum mechanics computation rules allow one to ﬁnd 1 αI + β · σ=(10)(αI + β · σ) = α + β (10.21) 0 z

Indeed, 10 01 0 −i 10 1 (10) α + β + β + β 01 x 10 y i 0 z 0 −1 0 1 0 0 1 = 10 α + β + β + β 0 x 1 y i z 0

= α + βz (10.22)

Now, let us set ⎫ (1) X = βz if βz = 0 ⎬⎪ (2) X = βx if βz = 0,βx = 0 (10.23) ⎭⎪ (3) X = βy if βz = 0,βx = 0

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and also signX =+1ifX ≥ 0 (10.24) signX =−1ifX<0

Now, let us introduce microstate eigenvalues denoted as E(,λ) deﬁned by 1 E = α +|β|sign λ|β|+ |β | signX (10.25) (,λ) 2 z

where the vector β is still given by (βx,βy,βz), but in the new coordinate system in which the spinor reduces to the form of Eq. 10.20. This vector being given, the dispersion-free microstate (, λ) completely determines the microstate eigenvalues E(,λ). If we were able to measure the microstate, particularly the value of λ, the probability for the associated microstate eigenvalue would be exactly equal to 1, in contrast with the probabilities associated with quantum eigenvalues. The philosophy of hidden-variables theories is that quantum expectation values are obtained by averaging over hidden variables. Using a uniform averaging over λ, we are then led to the evaluation of an integral over the microstate eigenvalues, according to

+1/2 1 α + β · σ= α +|β|sign λ|β|+ |β | signX dλ (10.26) 2 z −1/2 After a simple evaluation (starting with case (3) of Eq. 10.23 to check), we obtain

αI + β · σ=α + βz (10.27) that is, the same result as found in quantum mechanics. PEDESTRIANS (∼∼∼∼), reconnect here. We therefore possess a deterministic hidden-variables model, based on a hidden variable λ, that reproduces quantum mechanical predictions. There is however an important difference with respect to Bohm’s pilot wave, namely, the fact that λ does not receive any physical meaning, so that this hidden-variables model does not allow one to propose any new interpretation of quantum mechanics. Such hidden variables may be called dummy hidden variables. (Some people might prefer to call them artiﬁcial hidden variables, and this is actually the terminology we shall use later.)

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A Simple General Formal Approach The reader might worry that the previous model has been developed in a very special case (spin 1/2) and that perhaps it would be impossible to gen- eralize it. As it is, it might even look a bit ad hoc and contrived. However, it is a fact that one formal example is sufﬁcient to demonstrate that hidden variables are possible, simultaneously ruining all pretensions of proofs of impossibility. We may even have a stronger result: Not only are hidden variables possible, but they are always possible. Bell [327] stated this result as follows: “If no restrictions whatever are imposed on the hidden variables, or on the dispersion-free states, it is trivially clear that such schemes can be found to account for any experimental results whatever. Ad hoc schemes of this kind are devised every day when experimental physicists, to optimize the design of their equipment, simulate the expected results by deterministic computer programs drawing on a table of random numbers.” He added, however, that “such schemes … are not very interesting. Certainly what Einstein wanted was a comprehensive account of physical processes evolving continuously and locally in ordinary space and time.” The same argument has been served again in 1976 [328]: “That the apparent indeterminism of quantum phenomena can be simulated deterministically is well known to every experimenter. It is now quite usual, in designing an experiment, to construct a Monte Carlo computer programme to simulate the expected behavior … Every such programme is effectively an ad hoc deterministic theory, for a particular set-up, giving the same statistical predictions as quantum mechanics.” According to Wigner [306], “it is rather obvious that, given any quantum mechanical measurement represented by the operator Q one can introduce a hidden-variable model.” This can be done as follows. Together with any operator Q, and associated measurements, let us introduce a hidden variable denoted q. Let us demand that the statistical distribution of q reproduces the probabilities for the various possible (eigenvalues) λ1,λ2,...of the quantum measurements for Q and examine whether this is always possible. The answer is yes. To demonstrate this, we biunivocally associate domains D(λ1), D(λ2),...of q with each possible measurement outcome λ1,λ2,.... Let us now consider a state | and a distribution P(q). We then postulate that the distribution P(q) assigns a probability to the domain D(λi) that is equal to the probability that the measurement of Q on yields the value λi. We are done! This may be generalized to the case of several operators Q1,Q2,.... It sufﬁces to introduce a hidden variable qj for each operator Qj . Assume,

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for instance, that the spectra of all operators are discrete. Hence, let us call λij the ith eigenvalue (i = 1, 2,...,Nj )ofthejth operator (j = 1, 2,...,N). To each discrete value λij , we can associate a hidden variable qij in such a way that the value of qij determines λij . If we associate a probability P(qij ) to each qij , we may adjust the P(qij )’s to recover the various probabilities for the λij ’s. Of course, for each j, the sum of all probabilities over i must be equal to 1. Let Nhv be the number of hidden variables required for the process. We may use Nj hidden variables for the jth operator having Nj eigenvalues. The summation of all the Nj ’s over j = 1, 2,...,N, that is, for all operators, will provide a value for Nhv. We then have a list of hidden variables qij , which may be arranged by using a single integer s ranging from 1 to Nhv. This means that the number of hidden variables can actually be reduced to 1, denoted by s, taking Nhv possible different discrete values. If we are dealing with continuous spectra, j still takes on discrete values, but the discrete index i is to be replaced by a continuous index. We are then dealing with a number N of continuous segments that, by invoking again the Cantor–Bernstein theorem, can be merged into one single segment. Once again, the number of hidden variables may be reduced to 1. Similar considerations may be used for mixed spectra. Then, one can reduce all discrete spectra to one discrete hidden variable and reduce all continuous spectra to one continuous hidden variable. The number of hidden variables has then been reduced to 2. Whether it is possible to achieve a further reduction from 2 to 1 is something that I am unable to answer. The previous discussion is similar to the one we used when considering von Neumann’s consecutive measurements. Let us remark that the procedure we used here is the one also used for the construction of dispersion-free statistical sets, which, according to von Neumann’s proof, are forbidden. Indeed, in his book [26], von Neumann was already aware of such a possibility for a decomposition of a statistical set into dispersion- free subsets. Ironically enough, though, he did not use this fact to conclude that hidden variables are always possible. Instead, as we know, he ended with the claim that such decompositions are actually not possible and that hidden variables must always be excluded. Wigner [306] stated that the number of hidden variables increases enor- mously when we assume an increasing number of operators and of consecutive measurements. I hope I have correctly proven that such is not the case. It remains, however, true that hidden variables introduced as above do not receive any physical interpretation. Therefore, they do not

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permit alternative interpretations of quantum mechanics and look ad hoc and artificial. Certainly, this is a kind of hidden variables that most physicists would not like to consider. I do not know whether this difficulty can be overcome. It might simply be “a lack of imagination,” in the words of Bell [142]. In any case, as stated by Flato et al. [323], “… it is quite trivial to construct theories containing hidden variables which do not have any physical meaning. The real problem will of course be to construct theories with hidden variables having physical meaning, capable of reproducing all known quantum-mechanical results and possessing also predictions in domains not yet covered by quantum mechanics. No example of such a theory is known to our day” (and, as far as I know, still today). Gudder [321] explains the contradiction between proofs of impossibility and possibility proofs in the following terms: “The proponents of hidden variable theories have an idea of what these theories should be and have given examples of such theories. The antagonists have a different idea of what a hidden variable theory should be and have proved that such theories are impossible in the present general framework of quantum mechanics. These proofs are irrelevant since they do not refer to the hidden variable theories as formulated by the advocates of these theories.” To avoid any ambiguity, he then provided a definition of what constitutes a hidden-variables theory, which is in Bohm’s spirit. Now that we have several examples of hidden-variables theories, it is appropriate to precisely define what a hidden-variables theory is, and this can be done in a concise and perfect way, following Gudder, as follows: “The state m of a quantum mechanical system is not complete in the sense that another variable ξ can be adjoined to m so that the pair (m, ξ) completely determines the system. That is, a knowledge of (m, ξ) enables one to predict precisely the outcome of any single measurement. Furthermore, an average of (m, ξ) over the values of ξ gives the usual quantum state m.” Gudder afterward demonstrated that, in a certain sense, a hidden-variables theory is always possible (although the usual, even clever, physicist might feel distressed by some mathematical statements, heavily relying on a calculus of propositions, used to establish the result). Gudder’s work is acknowledged by Holland [40], saying, “It proved possible to show that quantum mechanics can always be supplemented by hidden variables.” The paper of Gudder has been complemented by Greechie and Gudder [137]. See also Fine [329], who agreed with the statement that we may always build a deterministic hidden-variables model that reproduces quantum mechanical results.

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We have already met the concept of contextuality in several places, when we stated and loosely explained that Bohm’s pilot wave is a contextual theory, but also regarding the proofs of impossibility of Gleason, and Kochen and Specker, when we restated their results, saying that what they proved is that admissible hidden-variables theories have to be contextual. This result may be viewed as a theorem, often called the Kochen–Specker (KS) theorem, or for reasons that we shall make clear the Bell–KS theorem. There is another important result: that admissible hidden-variables theories must be nonlocal. This result also may be viewed as a theorem, called Bell’s theorem (which is much more famous than the KS theorem). Nonlocality will be discussed in the next chapter. In the present chapter, we will deal with contextuality (a concept that we have now to understand clearly) and with the proof that hidden-variables theories must be contextual; that is, the properties assigned to hidden variables must be determined by the experimental context of the measurement.

Kochen and Specker for Everyday Cyclists Cyclists go faster than PEDESTRIANS (∼∼∼∼) but climb lower than mountaineers. I believe that cyclists can handle this section, but I am not sure about the fate of PEDESTRIANS (∼∼∼∼). Anyway, they may try. Otherwise, given that the next section is somehow in the same mathematical mood as the present one, they better go directly to the next chapter, just taking a bit of time to gain some specific information from a few useful comments. As we stated, Kochen and Specker’s proof is difficult. Belinfante produced a simplified proof but it is still too involved for an easy presenta- tion, in a reasonable number of pages (at least in the framework of this book). Fortunately, we also possess a still simpler version as presented by Bitbol in Annex III of one of his books [133]. This is akin to a kind of green lane for everyday cyclists. Following the green lane, let us consider spin-1 particles. When we measure a component of the spin in any direction, the possible outcomes

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are −1, 0, or +1 (in units of h¯), producing three spots in a Stern–Gerlach experiment. This is in particular true if measurements are made along the x,y and z directions, with associated observables denoted Sx, Sy, and Sz, respectively. Hence, if we measure the squares of the components 2 2 2 in any direction, in particular Sx , Sy , and Sz , there are only two possible outcomes, namely, 0 and 1. Let us now consider the observable

= 2 + 2 + 2 Sx Sy Sz (11.1)

For a spin s = 1, the result of the measurement of is s(s + 1) = 2. If we admit hidden variables, these hidden variables must determine the values (a concise way to designate the values of outcomes of measurements) 2 2 2 of Sx,Sy, and Sz.Therefore, they must also determine Sx , Sy , and Sz . Then, because each component Si (i = x,y,z) may have the values −1, 0, or 1, 2 = and Si (i x,y,z) may have the values 0 or 1, we conclude that two of 2 the values of Si must receive a value equal to 1, and the third one must receive a value equal to 0 so that, as required, the sum of the three values () is equal to 2. Let us now rotate our coordinates about the x direction and, instead of Sx, Sy, and Sz, consider Sxnew, Synew, and Sznew, with xnew, ynew, and znew the new directions after rotation. We then may consider the new observable

= 2 + 2 + 2 new Sx Synew Sznew (11.2)

Here, the reader should not worry: On the right-hand side (r.h.s.) of Eq. 11.2, 2 2 the ﬁrst term is indeed Sx , not Sxnew. We can now follow the same kind of reasoning as above, leading to the following observations. If we measure with an apparatus M,ornew with an apparatus Mnew, then in both cases the outcome of the measurement must be equal to 2. Again, among the 2 2 2 three observables Sx , Synew, and Sznew, two of them must receive a measurement value equal to 1, and the third one a measurement value equal to 0. We are now going to introduce an assumption that we call the noncontextual assumption, which looks most reasonable and from which we shall draw dramatic consequences concerning quantum mechanics and hidden- variables theories possibly underlying quantum measurements. To begin, 2 take notice of the fact that the observable Sx pertains to the r.h.s. sum- 2 mations of both Eqs. 11.1 and 11.2. Now, let us assume that Sx receives one of its possible values, say 1, determined by the hidden variables. The most reasonable noncontextual assumption tells us the following: If we

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have 1 with apparatus M for , then we also have 1 with apparatus Mnew for new. In other words, the value 1 does not depend on the context, that is, on whether the experimental setup is M or a rotated setup Mnew. Simi- 2 larly, if Sx yields its other possible value 0, it must be 0 independently of the experimental setup being M or Mnew. Our most reasonable assumption 2 therefore provides an additional constraint concerning the observable Sx , which is shared by both and new. However, we have the following extraordinary result: This constraint is incompatible with quantum numerical predictions, meaning that the noncontextual assumption must be rejected. Therefore, hidden-variables theories have to be contextual, as is indeed Bohm’s pilot wave. We now complete the proof of the KS theorem by demonstrating that quantum mechanical predictions cannot be recovered from our aforementioned noncontextual hidden-variables theory. For this, we make a geometrical transposition of our constraint (or, more precisely, two constraints: one for the value 1and the other for the value 0). Let us consider a set of three orthogonal vectors in Newtonian space, called a trio. Next, consider a family of trios. We assign the value 1 or 0 to each vector of a trio. Our algebraic constraints may then be converted into geometrical constraints as follows: (1) For any trio, one vector is assigned the value 0 and the two other vectors are assigned the value 1. (2) Let us consider the value assigned to a given vector. This vector may pertain to several trios. Then the value assigned to the vector does not depend on the trio to which it pertains.

Next, consider two orthogonal vectors a = (xa,ya,za) and b = (xb,yb,zb) with components taken from a Cartesian coordinate system (x,y,z). Let us assign the value 1 to each of these vectors, and we denote this as

[a]=[(xa,ya,za)]=1 (11.3) [b]=[(xb,yb,zb)]=1 (11.4)

There is one and only one vector c orthogonal to both a and b. Item (1) implies that this vector must be assigned the value 0 and, according to item (2), any vector orthogonal to c must be assigned the value 1. Any vector of this kind lies in the plane deﬁned by a and b and therefore can be written as (αa + βb) with α, β ∈ R. Hence, we have the following

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rule: If a and b are two orthogonal vectors and [a]=[b]=1, then [αa + βb]=1, whatever α and β pertaining to R. Now, we will consider the trios (1, 0, 0), (0, 1, 0), and (0, 0, 1) and let us assume [(1, 0, 0)]=0. By item (1), we then must have [(0, 1, 0)]= [(0, 0, 1)]=1. We next examine the vector (2, 1, 0), which lies in the plane deﬁned by (1, 0, 0) and (0, 1, 0) and makes an acute angle with respect to (1, 0, 0). We now show that this vector cannot receive the value 1, by establishing that it would lead to a contradiction. For this, we therefore have [(1, 0, 0)]=0, [(0, 1, 0)]=[(0, 0, 1)]=1, and we assume [(2, 1, 0)]=1. Applying several times the aforementioned rule, we deduce a set of relations:

[(2, 1, 0) + (0, 0, 1)]=[(2, 1, 1)]=1 (11.5) [−(0, 1, 0) + (0, 0, 1)]=[(0, −1, 1)]=1 (11.6) [(2, 1, 0) − (0, 0, 1)]=[(2, 1, −1)]=1 (11.7) [−(0, 1, 0) − (0, 0, 1)]=[(0, −1, −1)]=1 (11.8) [(2, 1, 1) + (0, −1, 1)]=[(2, 0, 2)]=1 (11.9) [(2, 1, −1) + (0, −1, −1)]=[(2, 0, −2)]=1 (11.10)

as can be seen by noting that the two vectors present on the left-hand side (l.h.s.) of any of these relations are orthogonal. Now, the vectors (2, 0, 2), (2, 0, −2), and (0, 1, 0) are three orthogonal vectors; i.e., they form a trio. From one of our starting relations, namely, [(0, 1, 0)]=1, and from the results obtained in Eqs. 11.9 and 11.10, each of the vectors in the above trio is assigned the value 1. By item (1), which demands that one of the vectors should be assigned the value 0, this is impossible. Therefore, if we have [(1, 0, 0)]=0, implying [(0, 1, 0)]= [(0, 0, 1]=1, we cannot simultaneously have [(2, 1, 0)]=1. Hence, we must have [(2, 1, 0)]=0. So we have the following easily stated result: [(1, 0, 0)]=0 implies [(2, 1, 0)]=0. In other words, what we have obtained is the following: Given items (1) and (2), if a certain vector is assigned the value 0, then there exists at least another vector, making an acute angle with respect to the previous one, that must be assigned the value 0 too. Let us now convert this result to our original problem concerning spin-1 particles. For this, we consider a component of the spin in a certain direction. If the square of this component receives the value 0, a true value determined by hidden variables, then there is at least another component

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in a direction making an acute angle with the previous one whose square must receive the value 0 too. However, this is in deep contradiction with the predictions of quan- 2 tum mechanics. Indeed, if Su in direction u is measured to be 0, that is, Su is measured to be 0 too, then quantum mechanics predicts that Sv, in direction v making an acute angle with u, possesses a certain probability to be =± 2 measured as Sv 1, so that Sv possesses a certain probability to be mea- 2 = 2 = sured as Sv 1. In other words, among the particles having Su 0, there 2 = is a certain number of them that will be measured as Sv 1. This does not agree with our previous result telling us that there exists at least a direction v such as all particles must receive the value 0. Therefore, we must reject the noncontextual assumption. Hence we have the Kochen–Specker theorem: Any admissible hidden-variables theory must be contextual.

Mermin, Just by Himself We now examine another argument published by Mermin in 1993 [197], about twenty-five years after the work by Kochen and Specker. Such a large time gap results from this argument being a lengthy historical development, starting, more or less arbitrarily but significantly, with Bell in 1964 [189] on the concepts of locality and nonlocality and discussed in the next chapter. All through this period, many things were learned, organized, and reor- ganized, and a better understanding was gained through many efforts of several contributors that both deepened and often simplified the argument. As a result, we do not need to find a version of the Mermin argument for everyday cyclists, as required with Kochen and Specker. We may examine Mermin, just by himself (but resorting to some minor rewordings), although it may still be fairly difficult reading for PEDESTRIANS (∼∼∼∼). Before doing this, let us mention another paper by Mermin [330], providing two examples that significantly simplify the no-go theorem of Kochen and Specker. Preferably, both papers should be examined together. Following Mermin [197], we now consider a four-dimensional state space generated by two spin-1/2 particles, particle 1 being associated with σ 1 σ 2 Pauli matrices μ and particle 2 with Pauli matrices ν. Any observable in the state space may conveniently be represented in terms of these Pauli matrices (and of unity matrices; see, e.g., [43]). They enjoy a certain number of properties worth recalling.

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Property 1 The squares of each matrix are unity, and therefore each of them have eigenvalues equal to ±1: σ 1 2 = σ 2 2 = μ ν 1 (11.11)

σ 1 Property 2 Any component of μ commutes with any other component σ 2 of ν. Property 3 Let μ and ν specify orthogonal directions (μ = x,y,z and = σ i σ i = = ν x,y,z); then μ anticommutes with ν for i 1, 2, and for i 1, 2, σ i σ i = σ i x y i z (11.12) It is convenient, for either i = 1 or 2, to recapitulate the properties in a series of formulas as follows [14]: ⎫ 2 = 2 = 2 = σx σy σz 1 ⎪ ⎬⎪ σxσy =−σyσx = iσz (11.13) σ σ =−σ σ = iσ ⎪ y z z y x ⎭⎪ σzσx =−σxσz = iσy where we have omitted the superscripts and simpliﬁed the matrix notation from bold σ to normal σ. One can then build nine observables conveniently arranged in the following table: σ 1 σ 2 σ 1σ 2 x x x x σ 2 σ 1 σ 1σ 2 y y y y (11.14) σ 1σ 2 σ 2σ 1 σ 1σ 2 x y x y z z (returning to bold notation, with superscripts for particles 1 and 2). These observables exhibit the following properties:

Property A The observables in each of the three rows and each of the three columns are mutually commuting. This is immediately evident for the top two rows and for the ﬁrst two columns from the left. It is also true for the bottom row and rightmost column, because in every case we can use a pair of anticommutations. Here is an example concerning the ﬁrst row: σ 1 σ 1σ 2 = σ 1σ 1 2 − σ 1σ 2 1 = σ 1σ 1σ 2 − σ 1σ 1σ 2 = x, x x x xσx x xσx x x x x x x 0 (11.15)

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Here is an example using anticommutations: σ 1σ 2, σ 1σ 2 = σ 1σ 2σ 1σ 2 − σ 1σ 2σ 1σ 2 x x y y x x y y y y x x = σ 1σ 1 σ 2σ 2 − σ 1σ 1 σ 2σ 2 x y x y y x y x = σ 1 σ 2 − − σ 1 − σ 2 = i z i z i z i z 0 (11.16)

Property B The product of the three observables in the column on the right is (−1). The product of the three observables in the other two columns and all three rows is (+1). Here is a demonstration for the ﬁrst product: σ 1σ 2σ 1σ 2σ 1σ 2 = σ 1σ 1σ 1σ 2σ 2σ 2 = σ 1 2 σ 2 2 =− x x y y z z x y z x y z i z i z 1 (11.17)

For the second product, here is an example: σ 1σ 2σ 2σ 1σ 1σ 2 = σ 1σ 1σ 1σ 2σ 2σ 2 = σ 1 2 − σ 2 2 =+ x y x y z z x y z y x z i z ( i) z 1 (11.18)

Now, if we consider mutually commuting observables, then any iden- tity satisﬁed by the observables must also be obeyed by the values they receive. This means in particular that Property B may be converted to a corollary in which “observables” is replaced by “values of the observables.” Hence, the product of the values assigned to the three observables in each row must be (+1), and the product of the values assigned to the three observables must be (+1) for the ﬁrst two columns and must be (−1) for the rightmost column. Mermin pointed out that this “is impossible to satisfy, since the row identities require the product of all nine values to be 1, while the column identities require it to be −1.” The impossibility may also be shown from Eqs. 11.17 and 11.18. Indeed, let V(X) denote the value of the observable X. Then, by recalling that values are numbers, that is, commuting quantities, Eq. 11.17 may be converted to σ 1 σ 2 σ 1 σ 2 σ 1 σ 2 =− V( x)V ( x)V ( y)V ( y)V ( z)V ( z) 1 (11.19)

and Eq. 11.18 to σ 1 σ 2 σ 2 σ 1 σ 1 σ 2 V( x)V ( y)V ( x)V ( y)V ( z)V ( z) = σ 1 σ 2 σ 1 σ 2 σ 1 σ 2 =+ V( x)V ( x)V ( y)V ( y)V ( z)V ( z) 1 (11.20)

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Hence there is a contradiction. Afterward, Mermin similarly examined, with a similar conclusion, the case of an eight-dimensional state space (instead of a four-dimensional state space) consisting of three independent spin-1/2 particles (instead of only two particles), a case that is even simpler and more versatile, as stated by Mermin. What has been proven here is a Bell–KS theorem on the impossibility of some kinds of hidden variables, if these hidden variables have to satisfy quantum mechanical predictions. To see that hidden variables are indeed concerned, we just need to imagine that they determine the values V(X) discussed above. The theorem has something to do with locality since the particles are far apart but, to justify its discussion in this chapter, it must be noted that it also has something to do with contextuality. To understand this, let us return to Mermin’s words: “In all these cases … we have tacitly assumed that the measurement of an observable must yield the same value independently of what other measurement must be made simultaneously ….” In particular, in the four-dimensional example above, and also in the eight-dimensional example that we did not discuss,

We required each observable to have a value in an individual system that would give the result of its measurement, regardless of which of two sets of mutually commuting observables we chose to measure it with. But since the additional observables in one of those sets do not commute with the additional observables in the other, the two cases are incompatible. These different possibilities require different experimental arrange- ments: There is no a priori reason to believe that the results … should be the same. The result of an observation may reason- ably depend not only on the state of the system (including hidden variables) but also on the complete disposition of the apparatus.

Here is now the explicit introduction of the concept of contextuality by Mermin: “This tacit assumption that a hidden variable theory has to assign to an observable A the same value whether A is measured as part of the mutually commuting set A,B,C,...or as part of a second mutually commuting set A,L,M,...even when some of the L,M,...fail to commute with some of the B,C,... is called noncontextuality.” Therefore, what has been proven is that the tacit assumption must be rejected: Quan- tum mechanics is contextualist, and hidden-variables theories have to be

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contextualist. This result might appear to be strange, at least sufﬁciently strange to motivate asking: Is the Bell–KS theorem silly? Mermin further commented, “It is surely an important fact that the impossibility of embedding quantum mechanics in a noncontextual hidden- variables theory rests not only on Bohr’s doctrine of the inseparability of the objects and the measuring instruments, but also on a straightforward contradiction, independent of one’s philosophic point of view, between some quantitative consequences of noncontextuality and the quantitative predictions of quantum mechanics.” Also, according to Mermin, the previous remark may be expressed another way, in connection with the fact that the KS or Bell–KS theorem is less famous than the Bell’s theorems discussed in the next chapter, as follows: “One reason the Bell–KS theorem is … less celebrated … is that the assumptions made by the hidden-variables theories it prohibits can only be formulated within the formal structure of quantum mechanics.” This statement can only be fully appreciated when we have in mind sufﬁcient material concerning nonlocality, the EPR paradox, and Bell’s inequalities and theorems, all topics to which we now turn.

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