Pearle’s Hidden-Variable Model Revisited

Pearle (1970) gave an example of a local hidden variables model which exactly reproduced the singlet correlations of quantum theory, through the device of data-rejection: particles can fail to be detected in a way which depends on the hidden variables carried by the particles and on the measurement settings. If the experimenter computes correlations between measurement outcomes of particle pairs for which both particles are detected, he or she is actually looking at a subsample of particle pairs, determined by interaction involving both measurement settings and the hidden variables carried in the particles. We correct a mistake in Pearle’s formulas (a normalization error) and more importantly show that the model is simpler than first appears. We illustrate with visualizations of the model and with a small simulation experiment, with code in the statistical programming language R included in the paper. Open problems are discussed.


Introduction
Bell's (1964) landmark paper [1] "On the Einstein Podolsky Rosen paradox" led a few years later to a version of his inequality more suitable for experimental purposes, and consequently the focus of a very great deal of both experimental and theoretical work. That is the inequality nowadays called the Bell-CHSH inequality, presented by Clauser, Horne, Shimony and Holt (1969) [2]. Almost immediately, however, Pearle (1970) [3] pointed out that the problem of detector efficiency meant that it was easy under local realism to reproduce the famous negative cosine curve of the correlations between spin measurements on particles in the singlet state. The measurements on each particle would not have two outcomes but three: spin up, spin down, and no detection. One would be tempted to restrict attention to only those "trials" in which both particles were detected, and compute the correlation between the observed spins for that subpopulation. Whether or not a particle was detected was, in Pearle's model, determined by a hidden variable correlated with the actual "hidden" spins of the particles. Detection depended on the extra hidden variable and on the detector setting. Selection of particle pairs such that both particles got detected effectively selects a subpopulation of particle pairs, whose hidden spins actually depend on the detector settings.
This would result in experimental violation of the CHSH inequality, moreover with the maximal violation predicted by quantum mechanics, even though there is a perfect local realistic explanation of the correlations found.
Pearle's model is the subject of this paper. It was the starting shot in a huge literature on the detection loophole, which continues to grow to this day. Pearle's model did have some unphysical features, and he was well aware of them. In his model, the probability of a double detection would depend on the angle between the two detectors and hence the experimenter would immediately notice that his or her results did not make sense. The paper was for many years considered a purely theoretical exercise which established a purely theoretical lower limit to detector efficiency which would have to be exceeded before a so-called loophole-free experiment could be carried out. Soon, other detection If th e re p re s e n ta tive p o in t lie s in re g io n 1, th e p a rtic le A will be m e a s u re d a s pos s e s s ing s pin p a ra lle l to ä a n d p a rtic le B will n o t be d e te c te d . If it lie s in re g io n 2 th e th e p a rtic le A will b e m e a s u re d a s pos s e s s ing s p in p a ra lle l to ä a n d p a rtic le B a ntipa ra lle l to b, a n s o on. Th e re e xis t n in e pos s ibilitie s d e p e n d in g o n th e p o s itio n of th e p o ln t a n d th e re la tive a ngle a b e twe e n ä a n d b .Th e fu n c tio n I(a , il(r)) of th e te xt is ju s t th e s urfa ce o f re g io n 2. Risco-Delgado explains as follows: "Pearle's sphere represents the nine possible outcomes of an EPR experiment allowing undetected events. If the representative point lies in region 1, the particle A will be measured as possessing spin parallel toâ, and the particle B will not be detected. If it lies in region 2 the particle the particle A will be measured as possessing spin parallel toâ and particle B antiparallel tob, and so on. There exist nine possibilities defending on the position of the point and the relative angle α betweenâ andb. " Pearle and Risco-Delgado are modeling source emitting pairs of particles. The particles carry hidden variables X and Y which we take to be random points in the unit ball in R 3 . We assume that Y = −X and X = 0 with probability 1. The ball is drawn in the figure.
Write X = RU where U 2 = 1 and R > 0. One might think of the unit length vector U as the direction of spin of the first particle, and −U as the direction of spin of the second, equal and opposite points on the unit sphere S 2 , while the scalar R is some kind of amplitude of spin.
Assume that the direction U is uniformly distributed on S 2 and statistically independent of the amplitude R ∈ (0, 1]. Notation: bold (as opposed to italic) indicates a vector; random vectors and random variables are denoted by uppercase symbols, while lowercase is used for non-random quantities.
Each particle gets measured in directions a and b, respectively (points on S 2 , chosen freely by the experimenter); these were the directionsâ andb in the figure. The possible outcomes are +1 ("spin up"), −1 ("spin down"), and "no detection", according to the following rule: if the angle between X and a is less than Rπ/2, then the outcome of measuring the first particle is +1; if the angle between X and −a is less than Rπ/2, then the outcome of measuring the first particle is −1; otherwise, the particle is not detected at all. The rule for the second particle is exactly the same story as for the first particle, with X and a replaced throughout by Y and b.
The smaller of the angles between X and ±a is cos −1 |U · a| so the recipe becomes: the outcome of measuring the first particle is sign U · a if cos −1 |U · a| ≥ Rπ/2 while there is "no detection" if cos −1 |U · a| < Rπ/2; the outcome of measuring the second particle is −sign U · b if cos −1 |U · b| ≥ Rπ/2 while it is not detected if cos −1 |U · b| < Rπ/2. Pearle (1970) gives a formula, Equation (22) in his paper, for a particular choice of the probability density of R; however, take note of his idiosyncratic normalization (Equation (1))! There is an error in his derivation, as can be verified by integrating the density over the whole range: combining Equations (1) and (22), we get a density which does not integrate to 1. Working through Pearle's paper in detail, it turns out that the only error in Equation (22) is the normalization constant, and this probably derives from an incorrect normalization in Equation (14) where Pearle switches from R to S = cos(Rπ/2), but it is difficult to be certain about this, since his notion of probability density is ambiguous and unconventional.
Here, I present an alternative and much simpler description of the distribution of R and also of the whole model, via the distribution of S = cos(Rπ/2) ∈ [0, 1). It turns out that the distribution of S can be expressed by the formula S = (2/ (1,4); and moreover it is S that we primarily need to know in order to simulate the model.
In terms of S = (2/ √ V) − 1, the recipe for simulating the measurement of one pair of particles is as follows: generate U uniformly at random on the sphere S 2 and independently thereof, generate V uniformly at random in the real interval [1,4].
Particle 1 is detected if and only if |A| ≥ S, and, if it is detected, the outcome of measurement is sign(A). Particle 2 is detected if and only if |B| ≥ S, and, if it is detected, the outcome of measurement is sign(B).
Pearle's main result is that this model reproduces the singlet correlations: I do not reproduce Pearle's (magnificent but of necessity very involved) proof. Instead, I just derive the density of R according to my specification, so that the reader can compare with Pearle's formula. I then "prove" Pearle's result by a simulation experiment. In fact, I would dearly like to see a short-cut derivation of Pearle's result. Through some quite brilliant calculations, he characterizes all possible probability distributions of R (equivalently, of S) which reproduce the singlet correlations as (up to normalization) the positive functions within the range of a certain differential operator, and then shows that the operator when applied to the constant function-the most simple choice one could make-is indeed positive. Further details are given in Appendix A at the end of this paper.
According to my definitions, R = cos −1 (S)/(π/2) and it follows that for r ∈ (0, 1), and hence the probability density of R is 2 )) 3 on the interval (0, 1). Compare this to Pearle's Equations (1) and (22) combined: The code below generates a graph, Figure 2, of the probability density of R, as well as making a rough numerical check that it integrates to 1. Since the probability density is monotone increasing, we get guaranteed lower and upper bounds to the integral by summing the value of the density on a regular grid of points between 0 and 1, omitting the right hand and left hand endpoints, respectively, and dividing by the number of intervals generated by the grid. r <-seq(from = 0, to = 1, length = 1001) f <-(4 * pi / 3) * sin(r * pi / 2) / (1 + cos(r *pi / 2))^3 plot(r, f, bty = "n", type = "l", main = bquote("Probability density of"~italic(R)), xlab = bquote("Radius"~italic(r)), ylab = bquote("Density"~italic(f)), sub = "(Check: area under curve = 1)") sum If the points X had been chosen uniformly distributed within the unit ball, the probability density of their distance R to the origin would have had probability density 3r 2 , 0 ≤ r ≤ 1. In Figure 3, I compare the two densities (the one corresponding to a uniform distribution over the ball in green). g <-3 * r^2 plot(r, f, bty = "n", type = "l", main = bquote("Comparison of two models for density of"~italic(R)), xlab = bquote("Radius"~italic(r)), ylab = bquote("Density"~italic(f)), sub = "Pearle (black) vs. uniform in ball (green)") lines(r, g, col = "green") We see that Pearle's points have a tendency to be closer to the surface of the ball than if they had been uniformly distributed throughout it.
According to Pearle's model, Particle 1 is represented by a point in the ball. It is observed when its spin is measured in a certain direction, if and only if its point lies in either of two "mushroom shaped" regions around the measurement direction and its opposite. It ±1 depending on the region. Particle 2 is represented by the exactly opposite point in the ball but for the rest, its observation and measurement follow exactly the same rule, with respect to its own two mushrooms. Thus, if both particles are measured in the same direction, either neither is observed or both are observed, and if observed, the two outcomes are certain to be opposite. Measured in opposite directions, either neither is observed or both are observed, and if observed the two outcomes are certain to be the same.
In the following plot, Figure 4, I draw the intersections of the two mushrooms for Particle 1 with a plane through the origin containing its measurement direction, which is taken to be the direction of the positive x-axis. I superimpose on this plot a sample of 1000 particles distributed in a circularly symmetric way about the centre of the unit disk, with distance to the origin distributed as in Pearle's model. The result is a 2D caricature of Pearle's 3D model: some statistical features are the same, some are different.
The picture is neither a 2D section nor a 2D projection of the 3D model. However, it should help the reader to visualize the model. The points are colored blue, red, or black according to whether the corresponding particle measurement result is an outcome spin up, spin down, or the particle is not detected. Two particles are simultaneously measured in this way: same point in the ball, different directions of measurement, altogether four mushrooms.
The actual detection regions (the red and the blue mushroom) are formed by rotating the 2D boundaries about the x-axis. The actual distribution of the 3D hidden variable of the particles being measured has the same radial component as in the plot, but its direction is now uniform over the sphere, instead of the circle. Thus, the 3D density of points is less than what the picture suggests as we move further from the origin; however, it still increases as we move outward relative to a uniform density.
Altogether, Pearle's derivation of his model was a tour-de-force in imagination, analysis, and geometry. Whether or not there is a short-cut to getting his results and whether or not they can be improved are interesting challenges. As we see in the next section, the model has one major defect, namely the rate at which a pair of particles are both detected depends quite strongly on the pair of settings with which they are measured. This phenomenon would be experimentally observable; conversely, the usual quantum mechanical modeling of this experiment, and assuming that particles are detected independently of the direction in which their spin is measured, predicts that the rate of pair detection is independent of measurement settings. Thus, we are left with the open problem: Is there a distribution of R reproducing the singlet correlations which does not have this defect? Pearle does not answer this question explicitly but his text suggests that he believes the answer is negative. A numerical analysis (see Appendix A) confirms.

A Simulation Experiment
We now present a simulation of the model in the statistical programming language "R". First, we (re)set the random seed, for reproducibility. To see results based on a fresh sample, replace the (integer) seed by your own, or delete this line and let your computer dream up one for you (it uses system time + process ID to do this job).
# set.seed() # Initialise random seed from system time + process ID set.seed(9875) # Initialise random seed deterministically We generate uniform random points on sphere generated using the "trig method" (Method 3) of Dave Seaman: see http://rpubs.com/gill1109/13340 for an R illustration. This very effective but little known method uses the coincidence that in 3D, a uniform point on the sphere has a z coordinate which is uniformly distributed between −1 and +1. Thus, we proceed as follows.
In the following simulation, the measurement directions are all in the equatorial plane, so only Z and X have been generated and are treated as X and Y.
First, we set up the measurement angles for setting "a": directions in the equatorial plane.
beta <-0 * 2 * pi/360 # Measurement direction 'b' fixed, in equatorial plane Then, we set the sample size (number of pairs of particles).

M <-10^6
I use the same, single sample of M = 10 6 realizations of hidden states for all measurement directions.
The M columns of e represent the x and y coordinates of M uniform random points on the sphere S 2 .
plot(angles * 180/pi, Ns / M, type = "l", col = "blue", main = "Rate of detected particle pairs", ylim = c(0, 1)) abline(h = (2/3)) abline(h = (4/3) * (1 -2/pi)) The two horizontal lines are the maximum and minimum detection rates computed by Pearle: 2/3 and 4/3(1 − 2/π) = 0.4845 . . . of M, respectively. Now, if an experimenter is not using pulsed emission of particle pairs but they are being emitted in a continuous fashion according to a Poisson process, then the experimenter will have no way of knowing that, when neither particle is detected, there was still an emission of a particle pair. Thus, the loss of 1/3 of all emitted particle pairs will go unnoticed. However, the experimenter will be able to see that the rate of double detections depends strongly on the difference between the two measurement directions-the maximum rate is more than 4/3 times the smallest. Put another way: The rate at which particles are detected at one measurement station with no accompanying detection at the other depends on the difference between the two measurement directions. Thus, Pearle's model has some very unsatisfactory features: assuming a constant emission rate, the experimenter can see that particles are suspiciously being rejected in a way which depends on both the settings. It was only in 2008 that Gisin and Gisin [6] came up with a new local hidden variable model for the singlet correlations based on data rejection which possesses all the symmetries one would require. Moreover, it is amazingly simple. However, it seems further from physical interpretation than Pearle's model. However, Pearle did more than exhibit one concrete local hidden variable model, which reproduces the singlet correlations: he also characterizes the class of all distributions of R which does the job. This allows us in principle to find out if there is a distribution within the class which leads to a model with all required symmetries. I believe the answer is negative (and I believe that Pearle believed this too) but I do not have a mathematical proof. Numerical evidence (see Appendix A) is very strong and inspection of the numerical result might help in constructing a proof.
Funding: This research received no external funding.
Acknowledgments: I was stimulated to figure out exactly what Pearle (1970) [3] had done during discussion on Internet fora with Michel Fodje, Chantal Roth, Joy Christian, and others. Michel Fodje had come up with his own detection loophole simulation model and I started by comparing this with the similar "chaotic ball" model of Caroline Thompson (see the arXiv preprint Thompson and Holstein (2002)) [11]. Thompson wrote a whole series of papers on this topic but only ever got one paper published (Thompson (1996) [12]). She discusses Pearle's model at length. At least her work is preserved on arXiv. In an interesting survey, Risco-Delgado (1993) [8] also gives it a lot of attention, and includes a very nice picture explaining the idea of the model. His text simply copies Pearle's incorrect formulas. My own versions of all these models, programmed in R, can be found at my RPubs website http://rpubs.com/gill1109. Florin Moldoveanu helped check my decoding of Pearle's derivation of the density of R. There is some ambiguity of notation (Pearle's notion of "probability density" is unconventional by modern standards and moreover seems not entirely consistent throughout the paper). This is probably how the normalization error in the final result crept in, midway through the computations. Since the error does not seem to have been reported elsewhere, and since it becomes manifest as soon as one attempts to implement a simulation of the model, I believe that this was the first time anyone did actually attempt to simulate the Pearle model. The simulation reported here was posted to RPubs in early March 2014.