Breakdown of Bell's theorem for certain objective local parameter spaces

INAUGURAL ARTICLE

Breakdown of Bell’s theorem for certain objective local parameter spaces Karl Hess†‡§ and Walter Philipp†¶ †Beckman

Institute, ‡Departments of Electrical Engineering and Physics, and ¶Departments of Statistics and Mathematics, University of Illinois, Urbana, IL 61801 This contribution is part of the special series of Inaugural Articles by members of the National Academy of Sciences elected on April 29, 2003.

We show that the known proofs of Bell’s inequalities contain algebraic manipulations that are not appropriate within the syntax of Kolmogorov’s axioms for probability theory without detailed justification. Such justification can be achieved by a variant of the techniques used in Bell-type proofs but only for a subclass of objective local parameter spaces. It cannot be achieved for an extended parameter space that is still objective local and that includes instrument parameters correlated by both time and setting dependencies.

I

n their classical paper, Einstein, Podolsky, and Rosen (EPR) (1) advanced arguments that quantum mechanics was incomplete and did suggest the existence of hidden parameters corresponding to certain elements of reality. That assertion led to the well known controversy with Bohr (2) and extensive subsequent debate. Of particular significance in the discussions was theoretical work of von Neumann and later of Bell (3). von Neumann had published a no-go proof for hidden parameters that was criticized by Bell (see pp. 1–13, and particularly p. 5, of ref. 3). The main focus of Bell’s critique was directed against von Neumann’s ‘‘arbitrary assumption of a particular (and impossible) relation between the results of incompatible measurements.’’ Later Bell (4) provided a different no-go proof for hidden parameters that had several advantages over von Neumann’s. Bell’s proof placed greatest emphasis on Einstein locality, which is in its essence the assumption of an upper limit for the velocity of causal affection: the speed of light in vacuo. Using only elementary probability, Bell derived his celebrated inequalities for correlations between spatially separated events. Apparently Bell thought that in this way the problem of incompatible measurements could be avoided. These developments precipitated experimental work in quantum optics involving correlated pairs of photons. An optical variation of a proposal by Bohm and Hiley (5) was spectacularly realized in the experiment by Aspect et al. (6). We assume throughout that this experiment and all of its results are valid and that no practical deviations from the ideal embodiment of all experimental procedures are of any significance. We exclude from all of our considerations detector inefficiencies and any other considerations of nonideal experimental conditions (e.g., not completely random choices of the settings, etc.). The impossibility of reconciling the inequalities that follow from Bell’s mathematical model with the results of quantum mechanics that are confirmed by the experimental results of Aspect and coworkers as well as others is often called Bell’s theorem and we will follow this practice here. We shall focus in our discussion on the Clauser–Horne– Shimony–Holt (CHSH) inequality (7), which contains the Bell inequality (3) as a special case. Correlated pairs of particles are sent out from a source S0 and the information they carry is characterized by the random variable ⌳. Following Bell, we introduce random variables that describe spin measurements, A ⫽ ⫾1 in station S1 and B ⫽ ⫾1 in station S2. A and B are assumed to be functions of ⌳ and are indexed by instrument settings that are denoted by three-dimensional unit vectors, usually a and d in S1 and b and c in S2. In principle, of course, all settings and also equal settings are admitted in both stations. www.pnas.org兾cgi兾doi兾10.1073兾pnas.0307479100

All published proofs of the CHSH inequality involving random variables are based on measure theory and probability and proceed in the following way. Four products A䡠B with subscripts denoting four actual pairs of instrument settings are concatenated by addition and subtraction, resulting in an entity that we denote by ⌫ (see Eq. 7 below). By a simple algebraic manipulation that assumes equality of all A with equal setting (and the same for B) it is then shown that ⌫ ⫽ ⫾2. The next step consists of integrating the constant 兩⌫兩 ⫽ 2 with respect to a probability measure. Finally, a simple inequality for integrals yields the CHSH inequality. In the same vein, Bell’s original proof uses only three products A䡠B, and Gill et al. (ref. 8 and references therein) use the indicator function 1{A ⫽ B} ⫽ 1⁄2(A䡠B ⫹ 1). One of the purposes of this article is to show that all such proofs of Bell’s theorem involving the above algebraic manipulations are technically incorrect as they stand, in essence because they imply, as von Neumann’s proof did, relations between incompatible measurements that necessitate simultaneous measurability. A second purpose of this article is to show rigorously that some of the published proofs can be modified and the difficulty with incompatible measurements can be avoided by use of statistical arguments that involve the reordering of data. This resolution may be exactly what Bell and others intuitively thought they had achieved. However, this argument, presented formally in Source parameters only, works only at the expense of a serious restriction of the parameter space. The only type of hidden parameters that Bell and his followers (3) were able to take into consideration were systems of correlated parameters emanating from a common source plus additional instrument parameters (9) that were assumed to follow severe restrictions with respect to their stochastic properties. We show that these restrictions are sufficient but not necessary to fulfill the demands of Einstein locality or any of the other assumptions that are commonly and reasonably made in proofs of Bell-type inequalities. We also show that subsequent works, in particular, proofs of Bell-type inequalities as presented in standard texts in quantum mechanics (10), proofs in broad overviews (11, 12), and proofs in recent papers (8, 13) either assume simultaneous measurability of incompatible experiments, as criticized by Bell himself, or contain the above-mentioned restrictions of the parameter space. We use throughout tools of elementary mathematical statistics based on Kolmogorov’s axioms for probability theory. We shall not make any use of the methods of quantum mechanics. Instead, we adopt the following postulates, which make the theory ‘‘objective local’’: (i) Einstein locality is strictly obeyed, which means that the upper limit for the velocity of causal affection is the velocity of light in vacuo. Abbreviations: EPR, Einstein, Podolsky, and Rosen; CHSH, Clauser–Horne–Shimony–Holt. See accompanying Biography on page 1797. §To

whom correspondence should be addressed. E-mail: [email protected].

© 2004 by The National Academy of Sciences of the USA

PNAS 兩 February 17, 2004 兩 vol. 101 兩 no. 7 兩 1799 –1805

MATHEMATICS

Contributed by Karl Hess, November 13, 2003

(ii) the particles emitted from the source carry some element(s) of reality. We also assume that the act of measurement involves additional elements of reality, the instrument parameters. Because the purpose of this article is an analysis of proofs of Bell-type inequalities, we assume that the elements of reality are represented at the source by a random variable ⌳ and the values that ⌳ may assume. The instrument parameters are also represented by random variables and the values that they may assume. Our postulate ii together with the use of random variables is a mathematical formulation of the standard postulate that each particle ‘‘possesses’’ (11) a definite value for all possible outcomes of measurements whether or not a particular measurement is actually performed. This standard terminology involving the word ‘‘possesses’’ needs explanation when instrument parameters are involved. The spin information is evaluated by random variables A and B involving instrument parameters that may depend on time. The particles therefore do not possess values that A and B may assume in the strictest sense of the word. They just possess information that is evaluated in the instruments for a given time resulting in the values that A and B assume. One more assumption needs to be added to obtain an ‘‘objective local theory’’ (11). This assumption is related to general considerations of statistical inference and is named in publications related to Bell’s inequalities ‘‘inductive inference’’ (11, 12): (iii) the average of the actual data measured for certain correlated particles (e.g., photons) and a given pair of settings equals the average that would be obtained for all particles if they had been measured with the same pair of settings. The all-important question is then whether any model based on these ‘‘innocuous’’ assumptions (11) can reproduce the results of quantum mechanics that are in agreement with the known experimental results (6). As mentioned above, the Bell theorem states that this outcome is impossible. However, the parameter space used in proofs of the Bell theorem is restricted much more than required by i–iii. We introduce an extended parameter space that is much larger than the one used by Bell and his followers. We shall show that the standard Bell-type no-go proofs come to a halt when applied to our extended parameter space. We believe that our findings may have some deeper meaning for the interpretation of quantum mechanics. At the very least our findings can be used to remove the syntactical errors from the standard textbook literature such as the algebraic manipulations in Bell-type proofs that involve assumptions of simultaneous measurability of incompatible experiments, similar to von Neumann’s assumptions that were severely criticized by Bell himself. Parameter Spaces Bell’s Parameter Space. We use essentially the notation of Bell (4)

and that defined in our previous publications (14–17). However, we denote the random variables with capital letters and use the lowercase for the values that these variables may assume. In the Introduction we have introduced several random variables. First, the source parameter ⌳ with probability distribution ␳(⌳) that does not depend on the settings in S1 and S2 because of Einstein locality. Second, A ⫽ ⫾1 and B ⫽ ⫾1 corresponding to spin measurements in S1 and S2, respectively. Finally, we introduce the setting unit vectors a, d and b, c indexing A and B, respectively. In the language of information theory, the stations S1 and S2 correspond to ‘‘decoding instruments’’ (18). Bell introduces important additional assumptions that greatly restrict the functions of the decoding instruments. Bell assumes that the functions A and B are functions of the settings and ⌳ only, i.e., A ⫽ A i(⌳) and B ⫽ Bk(⌳)

[1]

with i ⫽ a, d and k ⫽ b, c. This assumption is not dictated by Einstein locality. Here we have written the settings as a subscript although A and B are simply functions of both the setting and ⌳. We have 1800 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0307479100

done this only to indicate the special status of the settings: they are controlled by the experimenters in S1 and S2, who choose the settings stochastically independent from all other parameters and who, in addition, apply appropriate delayed choice strategies to guarantee the independence of ⌳ from the settings. Bell and his followers considered Eq. 1 to be very general. In fact, we were often told that ⌳ can be ‘‘anything,’’ including, for instance, the times of measurement. In Failure of Bell-Type Proofs we shall show that this is not the case. Instrument parameters were considered by Ballentine and Jarrett (9) and later by Bell (3) and others. However, the additional restriction that, conditional on ⌳, the instrument random variables in station S1 be stochastically independent of those in S2 was soon used by Bell (see p. 152 of ref. 3). We note that this restriction is sufficient for conditions i-iii to hold but not necessary. We shall discuss Bell’s inequalities for such parameters in Reordering including instrument parameters. The Extended Parameter Space. Time-dependent phenomena based on many-body effects involving particle spins are well known. However, for reasons of generality we do not want to consider specific physical possibilities. Instead we maintain that any Einstein local random variables that may be simulated on two appropriately independent computers must be admissible random variables in a model for EPR experiments. Naturally, both stochastic and functional independence of all variables of the setting on the other side is a strict necessity arising from Einstein locality. We will see, however, that this fact does not mean that the random variables simulated by the two computers are, conditionally on ⌳, stochastically independent from each other. We assume that both computers run on correlated clock times that symbolize possible periodic processes in the instruments. The clock times of the computers can be chosen identical if the two computers, corresponding to the instruments of measurements, are ‘‘aligned’’ at some point in time. Assume now that the computers C1 and C2 contain evaluation (decoding) routines A and B, respectively, that map the source parameter ⌳ and Einstein local time- and setting-dependent instrument parameters ⌳*a(t) (that are generated by computer C1) and ⌳*b*(t) (that are generated by computer C2) into ⫾1 (and similarly with settings d and c). These instrument parameters can be thought of as arbitrarily complicated numerical routines that supply output from the input of setting and measurement times. Note that we assume throughout that the times of measurement themselves are randomly chosen by the experimenter. Only the correlation of the measurement times in S1 and S2, respectively, matter for our reasoning. The settings for any given time of measurement are also randomly chosen. There is one more precaution that we take to avoid misinterpretations of our approach. In all critical steps below we let the distribution ␳(⌳) be time independent. This independence cannot be assumed in general, because the experimenters do not control ⌳. Time independence of ␳(⌳) does not need to be assumed for our argument. Our reason for this assumption is to remove all suspicion that we admit in any way selected ␭s for selected settings or correlate the choice of settings to measurement times, which would constitute well known exceptions to the Bell theorem. Our reasoning requires only two premises: (a) the instrument parameters are setting and time dependent and (b) different setting pairs cannot occur at the same time; they must necessarily occur at different times. The case of perfect correlation of the particle pairs in actual EPR experiments can be formulated mathematically by postulating that we have

A i ⫽ ⫺Bi for i ⫽ a, b, c, d

[2]

with probability 1. Because of this condition Bell wrote (p. 38 of ref. 3): ‘‘. . . for this case the possibility of the results depending Hess and Philipp

A i共␭, ␭*i共tm兲兲 ⫽ ⫺Bi共␭, ␭*i *共tm)) for i ⫽ a, b, c, d.

[3]

Here we have assumed, without loss of generality, that the measurement times of the correlated pair are about the same in both stations and are denoted by tm. We now give a specific example of how such parameters may be created on a computer. Note that this construction is not the most general one because Eq. 4 below is only sufficient for Eqs. 2 and 3 to hold, but not necessary. Assume that the times of measurement in the two stations are equal. The more general case of different measurement times can be obtained by an appropriate translation of all measurement times. Assume also that both stations contain four stacks of files denoted by ␭*a(t), ␭*b(t), ␭*c(t), and ␭*d(t) in station S1 and entirely identically arranged stacks of files denoted by ␭*a*(t), ␭*b*(t), ␭*c*(t), and ␭*d*(t) in station S2. Given a particular setting pair and measurement time tm, two actual files are picked, one in each station. Because the stacks are identically arranged we have for all times t

␭ *i共t兲 ⫽ ␭*i *共t兲 for i ⫽ a, b, c, d.

[4]

Then pairs of settings (i, k) are chosen sequentially and at random according to an arbitrary distribution and the measurements are performed during certain small time periods labeled as measurement time. The possible setting and time dependencies, i.e., the order of the files in the stacks, can be determined by arbitrary algorithms (including, e.g., appropriate elements of the history of past experiments). These algorithms determine then the setting and time dependence of the joint probability density of ⌳*i(t) and ⌳*k*(t) for different settings i and k. Note that for the special case of a time-independent source parameter, the instrument parameters may be stochastically dependent of each other and stochastically independent of the source parameter. In addition virtually arbitrary stochastic dependencies of instrument and source parameters can be constructed. Next we chose the functions A, B ⫽ ⫾1 with B the negative copy of A and we see that Eqs. 2 and 3 hold. We maintain that any proof of Bell-type theorems needs to be able to accommodate such parameters if it should be taken seriously. Notice that we have now at least five (for the special case of the proof of the CHSH inequality) random variables ⌳, ⌳*i(t) and ⌳*k*(t) with i ⫽ a, d and k ⫽ b, c. We also note in passing that it is easy to create in this way outcomes on two computers that are randomly equal to ⫾1 on each side but completely correlated (e.g., equal to ⫹1 on one side always entails equality to ⫺1 on the other side) if i ⫽ k. Proofs of the Theorem of Bell and Our Critique We start by analyzing the expositions in widely known books (10, 11) because they demonstrate clearly some of the major misconceptions. Then we discuss the original proof of Bell (4) and a number of subsequent journal publications. We show that all these proofs are technically incorrect and contain syntactical procedures that are at odds with historic mathematical texts such as that by Feller (19). When only source parameters are considered and conditions i–iii from above are postulated, then the proofs can be saved by a reordering procedure and the syntactical errors become a mere technicality. However, for the extended parameter space the proofs cannot be saved, neither by reordering nor by any other technique known to us. In this extended space, the technically incorrect procedures of Bell-type proofs remain false or at least Hess and Philipp

INAUGURAL ARTICLE

unproven. Mathematical and physical deficiencies in proofs of the Bell theorem have also been noted by Khrennikov, e.g., in refs. 20 and 21. Textbook Proofs and Simultaneous Measurability for Different Settings. Peres and Leggett consider in their proofs correlated pairs of

photons. Each photon of a given pair carries the same information package ␭. We transpose their almost identical proofs into our notation. Discussing his equation 6.29, Peres (10) states: ‘‘If several photon pairs are tested, we have, for the j-th pair’’:

␥ : ⫽ a a共 j兲bb共 j兲 ⫹ aa共 j兲bc共 j兲 ⫹ ad共 j兲bb共 j兲 ⫺ ad共 j兲bc共 j兲 ⫽ ⫾2.

[5]

Here we have replaced the random variables A and B by the values that these variables assume denoted by the lowercase a and b (not to be mixed up with the boldface settings). This precision in notation is appropriate and, we believe, the lack of it in standard textbooks, is partly to blame for some of the misconceptions described below. Eq. 5 appears to be plausible. Leggett (11) invokes postulates i and ii to justify Eq. 5 as follows: by the second assumption, each photon propagating to S1 possesses (see our discussion of postulate ii) a value ⫾1 of aa( j) and also a value ⫾1 of ad( j), which by the first assumption is independent of whether bb( j) or bc( j) is measured. Similar arguments are used for the measurements on the other side. Leggett concludes that therefore ‘‘each photon pair has a value . . . of each of the quantities . . . and hence of the combination’’ ␥. However, we ask the following question: If we are dealing with outcomes of experiments, i.e., with the values that the random variables assume, why should we have for the different settings the same photon pair j? Furthermore why is the information of that photon decoded (see The Extended Parameter Space) the same way for the four different measurements? For the extended parameter space postulates i (Einstein locality) and ii (counterfactual definiteness) do not guarantee that factors such as aa(j) are identical for experiments with different settings on the other side: even if the same index j can be used, the decoding may be different because time- and setting-dependent instrument parameters are involved and different settings occur at different times. The inclusion of time adds the cardinality of the continuum to the 16 different possible outcomes for the four different setting pairs that are considered in the proofs of Leggett and Peres. It is therefore not guaranteed that quartets such as ␥ are sampled and accumulated in a statistical model that corresponds to the actual experiment. Unless ␥ originates from an experiment corresponding to an indecomposable element of the sample space, implying simultaneous measurability of all its terms as outlined below, some other method must be found that justifies the use of identical aa( j), ad( j), etc. throughout Eq. 5. Otherwise, ␥ ⫽ ⫾2 is not an outcome of a statistical sample and thus relevance to actual experiments cannot be claimed. The fact that hypothetically one could have chosen all four different settings and then one would have obtained Eq. 5 with identical aa( j), ad( j), etc. at one given time is not relevant to the question of what is sampled or what data are accumulated by actual experiments that are performed at different times and for different correlated pairs. We will see from the following that neither established procedure of probability theory nor postulates ii and iii can validate the use of ␥ ⫽ ⫾2. Postulate i can indeed provide a justification by a data reordering procedure (see Data Reordering) but only for the restricted parameter space. For the extended parameter space, the value assignment to ␥ implies simultaneous measurability (see Failure of Bell-Type Proofs, particularly the last three paragraphs). In the next step Peres and Leggett proceed to average over j (indicated by 具 . . . 典) and obtain 兩具a abb典 ⫹ 具aabc典 ⫹ 具adbb典 ⫺ 具adbc典兩 ⱕ 2.

[6]

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MATHEMATICS

on hidden variables in the instruments can be excluded from the beginning.’’ Clearly Bell did not acknowledge the possibility of instrument random variables that may be independent from the source yet not stochastically independent from each other and still obey Eq. 2. However, this is possible because of time dependencies and the correlation of the two measurement times involving a correlated pair. We certainly can have

This result is known as the CHSH (3, 7) inequality, which is then compared to the experimental results. For this comparison to the actual experiment, Leggett invokes now postulate iii (11). However, postulate iii can be used only to link the average accumulated for a given setting pair to the average over all possible experiments thought to be performed with that same setting pair. It therefore cannot justify, a posteriori, the use of and the value assignment to ␥ that involves different setting pairs. Here, we believe, lies another key problem. We highlight this problem first by discussing remarks of Peres. Then we show that the accepted approach of probability theory outlined by Feller unequivocally permits the procedure of Leggett and Peres only if all of the terms in ␥ can be measured simultaneously. Peres (p. 163 of ref. 10) and Leggett (p. 164 of ref. 11) do observe that only one of the four products in Eq. 5 can be determined in one experiment; the others are counterfactual. Peres writes, ‘‘Obviously, the hidden variables, which we do not control, are different for each j. The serial number j can thus be understood as a shorthand notation for the unknown values of these hidden variables.’’ The error here is that each term in ␥ is an outcome of a statistical experiment. j denotes ‘‘the j-th pair of photons,’’ and thus j cannot be a shorthand for ‘‘the unknown values of these hidden variables.’’ Peres states further, ‘‘ . . . taking the average over the hidden variables is the same as taking an average over j.’’ However, this is just what needs to be proven, as we will see momentarily. The standard procedures of probability theory permit the calculation of averages from quartets such as ␥ if and only if these quartets originate from the same indecomposable element of the sample space, which means simultaneous measurability of all of the terms of ␥. This can be seen from the syntax of probability theory as explained in the work of Feller (19), who writes, ‘‘If we want to speak about experiments or observations in a theoretical way and without ambiguity, we first must agree on the simple events representing the thinkable outcomes; they define the idealized experiment . . . . By definition every indecomposable result of the (idealized) experiment is represented by one, and only one, sample point. The aggregate of all sample points will be called the sample space.’’ To follow Feller, we replace the particle index j by the information package ⌳j and we assume that each ⌳j is a random variable, defined on some sample space ⍀, whose elements are denoted by ␻ that represent simple, indecomposable experiments (19). For instance, ␻ can be thought as being the experiment of sending out a correlated pair from the source. The procedure of Peres and Leggett corresponds in analogy to Eq. 5 to the definition of an entity ⌫ such that ⌫: ⫽ A a共⌳共␻兲兲Bb共⌳共␻兲兲 ⫹ Aa共⌳共␻兲兲Bc共⌳共␻兲兲 ⫹ Ad共⌳共␻兲兲Bb共⌳共␻兲兲 ⫺ Ad共⌳共␻兲兲Bc共⌳共␻兲兲.

[7]

Here we omitted the index j in order not to overload the notation. Clearly one obtains ⌫ ⫽ ⫾2.

[8]

Eq. 6 corresponds to applying Eq. 7 to a sequence of independent identically distributed random variables that we denote by ⌫j, or, more generally, applying the pointwise ergodic theorem to a stationary ergodic sequence ⌫j, j ⫽ 1, 2, . . . (22). This would be correct if the experiments with the four different setting pairs could be and would be performed all at the same time and would therefore correspond to the same ␻. However, in a model of the Aspect experiment only one term of Eq. 7 can be determined, as acknowledged by both Peres and Leggett. Peres expressed his views of what was incorrect in the paper titled ‘‘Unperformed experiments have no results’’ (23), blaming essentially the counterfactual reasoning ‘‘just as in the EPR article’’ (see p. 162 of ref. 10) for the failure of the CHSH inequality to agree with experiment. However, the problem lies much deeper. It is definitely admissible to argue, 1802 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0307479100

as Einstein did, that one could have measured with another setting. In fact, all of probability theory is based on procedures that use random variables that can assume a variety of values depending on circumstance (e.g., chosen instrument setting). However, it is not admissible, either by the syntax of probability theory or by postulates i-iii, to use ⌫ as a function of a single ␻. ⌫ corresponds to a nonperformable experiment and ⌫ is a noncomputable entity. The definition of ⌫ in Eq. 7 is meaningless because ⌫ is not a well defined function on the sample space ⍀. For this reason, and because the factors Aa, Ad, Bb, and Bc cannot be guaranteed to be the same at each occurrence, Eq. 8 is incorrect. It is therefore not guaranteed that the quartets of Eq. 5 are sampled and that the averages of Eq. 6 are accumulated by the actual experimental procedure. It is obvious that under the assumption of simultaneous measurability of the parameters that appear in ⌫ or, equivalently, under the assumption of the existence of a well defined joint distribution of these parameters (24), the CHSH inequality remains valid. Indeed, in this case ⌫ is a well defined random variable for which Eq. 8 holds. As 兩⌫(␻)兩 ⫽ 2 and the integral over the probability space ⍀ equals 1 the claim follows. In the extended parameter space, a well defined joint distribution ␳(⌳, ⌳*a, ⌳*d, ⌳*b*, ⌳*c*) implies simultaneous measurability of all four setting pairs, as can be seen directly from the notation: the four random variables ⌳*a, ⌳*d, ⌳*b*, and ⌳*c*must coexist. However, if simultaneous measurability were assumed then Bell’s criticism of von Neumann’s work would apply directly to the proofs of Peres and Leggett: ‘‘It was the arbitrary assumption of a particular (and impossible) relation between the results of incompatible measurements either of which might be made on a given occasion but only one of which can in fact be made’’ (p. 5 of ref. 3). Technical Difficulties in the Original Proof of Bell. Problems virtually identical to those discussed above are contained in Bell’s original proof, which includes integrals of the form (see ref. 4 in between equations 14 and 15; significantly the critical step has no equation number):

冕

关A a共⌳共␻兲兲Bb共⌳共␻兲兲 ⫺ Aa共⌳共␻兲兲Bc共⌳共␻兲兲兴P共d␻兲

⫽

冕

关Aa共⌳共␻兲兲Ab共⌳共␻兲兲关Ab共⌳共␻兲兲Ac共⌳共␻兲兲兴 ⫺ 1兴P共d␻兲, [9]

where Bell has used Eq. 2 to eliminate B. We have highlighted the problem of Eq. 9 by taking the liberty to include ␻. In all other ways our notation is equivalent to Bell’s original one. The indecomposable element ␻ of the sample space was included to indicate that the algebraic manipulation that leads from the left side of Eq. 9 to its right side implies, in the absence of other justification, simultaneous measurability of Aa(⌳(␻))Bb(⌳(␻)) and Aa(⌳(␻))Bc(⌳(␻)). Such manipulations implying simultaneous measurability can be found, as far as we know, in all proofs published by Bell (see, e.g., ref. 3, p. 37 between equations 8 and 9; p. 56, equation 10; and p. 156 after equation 19) and by Bell’s followers [see, e.g., Stapp’s (13) ordering of possible outcomes into quartets; equations 3 and 4 on p. 1428 of ref. 25; and figure 2 of ref. 8]. Fine (24) uses joint distributions for random variables with more than two settings. The proper use of joint distributions is again based on the assumption that there must be one and the same ␻ of the probability space ⍀ underlying the allocation of a value to the random variables representing source and instrument parameters. Therefore, these joint distributions are essentially unrelated, or at least not directly related, to the actually performed EPR experiments such as that of Aspect et al. (6). Hess and Philipp

Data Reordering: A Statistical Argument to Validate Bell and Others. Source parameters only. We describe here what we believe to be the

main statistical idea that validates the algebraic manipulations related to ⌫ within the restricted parameter space. Basic to this idea is postulate i, which implies that ⌳ is independent of the settings. Assume that the number M of values that the hidden parameter ⌳ can assume is finite (extension to a countably infinite number is easy). Denote this set of values by {␭s} and the probability P(⌳ ⫽ ␭s) by ps, all with s ⫽ 1, . . . , M. Let N be the number of experiments performed. If N is large, then we have by the strong law of large numbers that with probability 1 the number of occurrences of ␭s is approximately equal to N䡠ps, s ⫽ 1, . . . , M. (We remark that the strong law of large numbers holds under rather general assumptions relating to the stochastic independence of the random variables. For instance, it suffices to assume that the random variables are quasi-orthogonal. On the other hand, if the dependence of the random variables is too strong and兾or the random variables have distributions that vary over time, then the law of large numbers need not even hold.) Assume now, and this is decisive because otherwise this proof would not work, that each of the four pairs of settings (a, b), (a, c), (d, b), and (d, c) occurs with probability 1⁄4. The fact that the measurements can be made only in sequence now plays no role, because one can reorder the measurements into rows for a given ␭s and one could recognize such rows in the experiments if ␭s could somehow be made visible. To be more specific, for each s ⫽ 1, . . . , M we would have observed about 1⁄4Nps times the value ␭s with each of the four pairs of settings because of the stochastic independence of ⌳ and the settings. Thus, after reordering the data we would have accumulated about 1⁄4Nps rows of the form as given in Eq. 5, each of them corresponding to the value ␭s. For each of these rows Eq. 5 holds. Of course, there may be some terms left over. However, by the strong law of large numbers, the number of such incomplete rows is negligible for large N. This possibility of reordering proves then the validity of Eq. 6. With this additional argument of reordering (or something equivalent), the proof of Bell’s theorem is complete for this particular subclass of parameters. The concept of data reordering is closely linked to the concept of commutative observables in quantum mechanics. Fine (24) introduced operators 具A, PA典 by pairing the functions A together with their probability measure PA. He proved that violations of the Bell inequalities correspond to violations of commutativity of his operators. We have just shown that the validity of the Bell inequalities corresponds to the possibility of data reordering. It is therefore clear that there is a connection between Bell’s restricted parameter space and the commuting operators in Fine’s notation. Reordering including instrument parameters. Bell also introduced instrument parameters (pp. 36 and 37 of ref. 3), say ⌳*i in S1 and ⌳*k* in S2 and used them in equations such as Eq. 9. Then A ⫽ Ai(⌳, ⌳*i ) and B ⫽ Bk(⌳, ⌳*k*). To proceed with his proof, Bell ‘‘first averages over these instrument variables.’’ However, this step requires some kind of conditional independence, which is not explicitly stated in Bell’s proof. For instance, the following factorization of the joint probability distribution of ⌳, ⌳*i, and ⌳*k*: P共⌳ ⫽ ␭ , ⌳ *i ⫽ ␭*i, ⌳*k* ⫽ ␭*k*兲 ⫽ P共⌳*i ⫽ ␭*i 兩⌳ ⫽ ␭兲䡠P共⌳*k* ⫽ ␭*k* 兩⌳ ⫽ ␭兲䡠P共⌳ ⫽ ␭兲 Hess and Philipp

[10]

៮ i 共␭兲 ⫽ A and B៮ k共␭兲 ⫽

冕

冕

Ai共␭, ␭*i 兲P共⌳*i ⫽ ␭*i 兩⌳ ⫽ ␭兲d␭*i

[11]

Bk共␭, ␭*k*兲P共⌳*k* ⫽ ␭*k*兩⌳ ⫽ ␭兲d␭*k*

[12]

៮ 兩, 兩B៮ 兩 ⱕ 1. Bell then completes his proof by using the same with 兩A ៮ and B៮ (see p. 37 steps as shown in Eq. 9 with A, B replaced by A of ref. 3). The same criticism as leveled above applies therefore here too. As before, data reordering can be used to complete the proof. However, assumptions have been introduced restricting the space of instrument parameters. These assumptions are not required by i–iii, in particular not by i, and are not required in our extended parameter space for which, as we will see, the reordering arguments given above do fail. Failure of Bell-Type Proofs for the Extended Parameter Space We first discuss the general question whether all possible dependencies on the time of measurement tm can somehow be absorbed into the source parameter ⌳. We shall answer the question in the negative. This conclusion shows that a parameter space that includes setting- and time-dependent instrument parameters is significantly larger than that of Bell. The role that ⌳ plays in the mathematical model of the actual experiment implies that ⌳ should be related to information regarding the spin. Indeed, all proofs of Bell-type inequalities deal with ⌳ in that way. We have given a rigorous proof of the inequalities by reordering the products A䡠B, and with them implicitly the values ␭ that ⌳ may assume. Therefore, the set of elements ␭ that ⌳ may assume cannot, in general, be replaced by the measurement times tm or certain special functions g(tm) of tm (e.g., monotone functions). The times of measurement are all different, do not repeat themselves, and cannot be reordered without additional assumptions. Yet the time of measurement may play an important role in how the information about the spin is decoded in the instruments. From the viewpoint of information theory (18) we can summarize the statements above as follows. There are two types of information that need to be associated with the correlated pair. For one we have the information from the source that is related to the spin. Second, the correlated pair carries information related to time. This information is analog in nature and adds complexity to the procedure of digitally evaluating the spin in the decoding instruments. These two types of information cannot be lumped into one random variable ⌳ without running into contradictions when attempting to prove Bell-type inequalities. We shall demonstrate now that the addition of time- and setting-dependent instrument parameters presents further problems for Bell-type proofs because in this general case the statistical argument of Data Reordering, Source parameters only cannot be completed. Assume just for simplicity and clarity the special case that the distribution of the source parameter ⌳ does not depend on time. Then there will be still about 1⁄4Nps rows of data aibk (i ⫽ a, d; k ⫽ b, c) each corresponding to the values ␭s that the source parameter ⌳ can assume. Similarly, as in Eq. 7, consider the following terms: PNAS 兩 February 17, 2004 兩 vol. 101 兩 no. 7 兩 1803

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would do. Here i ⫽ a, d and k ⫽ b, c. Of course a less stringent condition, such as a sort of conditional orthogonality of A and B, would also permit one to integrate first over the parameters related to the two instruments. However, no physical reason for such a condition is known. Averaging first over the instrument variables, ៮ and B៮ , which we write symbolically Bell obtains entities A

MATHEMATICS

Why is it then that Bell considered his proof to be valid when he clearly felt that von Neumann’s was physically untenable? The reason probably is that Bell’s proof can be modified by using a certain data reordering procedure that is based mainly on Einstein locality and the assumptions of the restricted parameter space. The same is true for some other proofs of Bell-type inequalities. This procedure was probably intuitively clear to Bell and others, although we did not find explicit mention of it. The single most important point of our article is that this procedure, which is described next, is limited to the restricted parameter space.

a a共␭s, ␭*a共tm1兲兲bb共␭s, ␭*b*共tm1兲兲, aa共␭s, ␭*a共tm2兲兲bc共␭s, ␭*c*共tm2兲兲, ad共␭s,␭*d共tm3兲兲bb共␭s, ␭*b*共tm3兲兲, ad共␭s, ␭*d共tm4兲兲bc共␭s, ␭*c*共tm4兲兲. [13] Here m1, m2, m3, m4 ⫽ 1, 2, . . . , 1⁄4Nps are the indices after reordering to obtain rows with the same ␭s. These indices are all different because they contain different measurement times for different subscripts. Because ⌳*i and ⌳**k both depend on the random measurement times tmr, r ⫽ 1, 2, 3, 4, it is easy to see that the factorization of Eq. 10 is not possible and the argument of Reordering including instrument parameters no longer works. This demonstration refutes directly Bell’s famous example of heart attacks in Lille and Lyons that he uses for the purpose of discarding time-related parameters such as heart attacks on Sundays (p. 152 of ref. 3). Bell assumes in this example conditional independence and corresponding factorization conditional to ⌳, i.e., Eq. 10. Here, the difference of the role of time and the role of the source parameters becomes very clear. While the measurement times can under no circumstance be equal for different settings, the information ␭s that is sent out from the source and relates to the spin must be independent of the settings and is assumed to appear about 1 ⁄4Nps times for each pair of settings. Furthermore, virtually arbitrary time dependencies may be permitted for the instrument parameters. These time dependencies cause virtually arbitrary complex joint parameter distributions for any given pair i, k of settings that frustrate attempts of reordering. It also becomes once more particularly clear why ␭ cannot be just replaced by time. If one replaces ␭ by the measurement times and then performs Bell’s integrations and algebraic manipulations, one automatically assumes simultaneous measurability of the four incompatible experiments. Some claim that ‘‘time is irrelevant’’ (8). Of course, once the countersyntaxial step of Eq. 5 is taken, time indeed becomes irrelevant because then the reordering into quartets as in Eq. 5 is automatic. However, Einstein locality does not justify this step because Einstein locality has no consequence for the instrument parameters as we have introduced them. Nor is this step justified by assumptions ii and iii. An interesting question arises with respect to commutativity of quantum mechanical observables and the possibility of reordering as discussed in Source parameters only. We have seen that with Fine’s definitions the use of the restricted parameter space is related to the use of commuting operators. Can we also conclude that the use of the extended parameter space is related to the use of both commuting and noncommuting operators in quantum mechanics? On the basis of our previous work (15) we believe that this is the case. Here, however, this is only a conjecture because we showed above only that the currently used methods do not permit one to prove the possibility of data reordering for the extended space. We did not show in the present work that reordering is impossible in the extended space.

the computer construction of the instrument parameters from The Extended Parameter Space and implement a parameter set of your choice. Then the game is played the following way. Two players, one for each computer, choose the setting a, d each with probability 1⁄2 for computer C1 and b, c, also each with probability 1⁄2 for C2. Value assignments for A, B need now be made without any knowledge of the setting on the other side. Then many such experiments are performed. The challenge is to show a significant violation of Bell’s inequalities. This challenge goes, of course, far beyond what we have accomplished in this article. We have shown that the Bell-type proofs cannot be completed with the parameters that have the general features of our construction in The Extended Parameter Space. We have not derived all of the time and setting dependencies that would give a complete model for EPR experiments. One can easily construct instrument and source parameters as well as functions A, B that do show violation of the Bell inequalities for any prechosen sequence of setting pairs, but we currently can not give a table of these parameters and functions that will show the violation required by quantum mechanics for all possible choices of the four pairs of settings. Finding such a table would be extremely complicated, if at all possible, in the absence of further knowledge about the nature of the spin, because it requires one to construct the correct correlations of the five families of random variables ⌳, ⌳*i, ⌳*k* as well as of A, B. The challengers are, of course, certain that no matter with what table one comes up, the Bell inequalities cannot be violated. Their opinion arises from the fact that they mistakenly believe that a well defined joint distribution exists for all four setting pairs and that the full table is actually sampled by the experiment. We have shown, however, that this would be true only if the four incompatible experiments of each row of the table (corresponding to the quartets of Eq. 7) could be and would be simultaneously measured. Otherwise, the sampling of the full table is not guaranteed because ⌫ does not originate from the same indecomposable element of the sample space and the possibility of data reordering has not been proven in the extended parameter space. We conclude that it is not currently known how to play the Bell-game. One could say that a proof for existence of the extended parameter space that always gives the quantum result has not yet been accomplished. Therefore, it can not currently be assessed whether our extended space has a deeper physical significance. However, we hope to have clearly shown the following: the no-go proofs of the Bell theorem that can be found in the literature either assume simultaneous measurability of incompatible experiments, an assumption that has been criticized by Bell in his discussion of von Neumann’s work, or they contain restrictions of the parameter space that are not justified by the postulates i–iii of an objective local theory. We believe therefore, that at least for reasons of proper education in probability theory and physics, Bell-type proofs must be changed and significant qualifications about the generality of the proofs must be added in textbooks and other publications.

Challenges and Conclusion We have been repeatedly challenged to implement the following ‘‘Bell-game’’ on two computers with equal clock time: take exactly

We thank David Ferry, Andrei Khrennikov, Louis Marchildon, and Roland Omnes for carefully reading the manuscript and many valuable suggestions and Tamer Basar for valuable discussions. Support of the Office of Naval Research (N00014-98-1-0604) is gratefully acknowledged.

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