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Guy Vandegrift, Wright State University Lake Campus, Celina, Ohio
Abstract: Three detailed figures can be used to facilitate the teaching of Bell's theorem. The first figure is a flowchart that uses a stylized rendition of a Bell's theorem experiment on two photons measured by sets of symmetrically oriented polarizing filters. After stating without proof that measurements are 100% correlated whenever the filters are aligned, one path uses a semi-classical argument to explain the observed outcome probabilities. Another path uses a Venn diagram to establish Bell's inequality, which contradicts the first path. The other two figures are space-time diagrams intended not to resolve the paradox, but instead to help students better appreciate it by giving them the opportunity to ponder superluminal communication, causality, and super-determinism.
The theorem versus the "paradox"
A popular and efficient introduction to Bell's theorem is acheived by first proving the inequality, and then stating that it is violated by experiment evidence. While this has the advantage of being technically correct and simple, it fails to highlight the fact that the physics community would have been far more astonished if Bell's inequality had been confirmed. Figure 1 attempts to remedy this by explaining both sides of this "paradox", using a Venn diagram to establish the inequality, and then producing a semiclassical argument that applies the concept of the photon to the classical behavior of light passing through polarizing filters. Ironically it is not the rigorously provable inequality, but the loosely constructed "explanation" that is consistent with experiment.
The word "paradox"  can be used to distinguish this informal introduction from the Bell's actual theorem; the latter excludes a class of alternative theories to quantum mechanics, and for that reason is extremely difficult to grasp.  The quote at the bottom of the figure was inspired by a footnote in Mermin's article that effectively invites readers to decide for themselves whether or not Bell's theorem is a "conundrum".  In the same spirit of "letting the reader decide", the comment placed at the bottom of the Fig. 1 is intended to balance the statement at the top declaring it to be a "paradox".
To avoid a long discussion, it is stated without proof that measurements are 100% correlated whenever the local and remote filters are aligned (in this context we need consider only the even parity case.) Two rather weak plausibility arguments are available to suggest such a correlation might hold: (1) if the decay producing the pair of photons conserves angular momentum, then a correlation in circular polarization is easy to justify because circularly polarized light carries angular momentum, and (2) the mathematical concept of even and odd functions should be familiar to most students, and it is not unreasonable to suggest that parity might lead to an additional rule regarding the correlation for identical alignments. Given the cultural significance of Bell's theorem, an effort should be made to better justify this correlation (or anti-correlation) in linear polarization at the lowest possible level. One attempt is available on Wikiversity as both an archived and uneditable permalink, as well as an open document available for anyone to edit. 
Figure 1 can be printed be in landscape mode on standard letter or A4 paper, although many will prefer a larger stock paper. All three figures are editable svg vector files and designed to be understood if printed in black and white.
Figure 1 begins with a symbolic representation of a Bell's theorem experiment, using fictitious mechanically rotatable linear polarizers instead of the fast electro-optic modulators and birefringent polarizers typically used. The "remote" photon strikes the detector first, which allows us to focus most of our attention on the "local" photon. We assume that a photon is detected regardless of whether it was blocked or passed, and that the filters are aligned symmetrically along one of the three orientations (α,β,γ) depicted in the figure. The symbols (α,β,γ) also refer to outcome variables, defined to be +1 if the photon is passed, and −1 if blocked. The figure directly to the right of the Venn diagram documents an observed correlation whenever both filters are set to the same orientation. In a manner of speaking, this permits the remote measurement to serve as a de facto measurement on the local photon before an actual measurement on the local photon is made. Four additional concepts are important for students to understand:
- At most, only two of the three variables (α,β,γ) can be measured for each pair of photons (one by the remote filter, the other by the local).
- For reasons involving the structure of quantum mechanics, the variables (α,β,γ) are often referred to as "hidden variables" and one may legitimately question whether they exist. Bell's inequality is derived assuming that they do exist.
- If one accepts the existence of these hidden variables, it is reasonable to assume that they acquired their values when the photons were created. This is because both photons respond in the same way to identical filter orientations, and these orientations can be changed while the photons are en route. Otherwise, it would seem that the photons would need to "communicate" with each other during measurement in order to sustain the 100% correlation for parallel measurements. This use of the word "communicate" is an example of the anthropomorphisms and references to telepathic or psychic abilities that naturally occur in discussions of Bell's theorem.
- Bell's theorem experiments have also been performed on objects that do not move at the speed of light, including protons, neutrons, and even ions and atoms.
The Venn diagram in the center organizes the three hidden variables by whether they are equal to each other. There are only four possible outcomes because all three variables cannot take different values (i.e., α≠β & β≠γ & γ≠α is impossible.) Red arrows just below the Venn diagram show the two contradictory paths of the paradox. The path leading to Bell's inequality is easily grasped using a variation of the Venn diagram in Maccone's proof of Bell's inequality. Since formal proofs are available elsewhere, the figure uses a much quicker plausibility argument involving a random selection from the first five positive integers. Since "2" is both even and prime, it is obvious that P(even)+P(prime)>1. Application to the the variables (α,β,γ) yields, P(α=β) + P(β=γ) + P(γ=α) ≥ 1 .
The semiclassical plausibility argument is based on concepts involving flux and flux density that are important in the education of a scientist or engineer. The flux density of light energy, F, is proportional to E2c, where E is the electric field and c is the speed of light. If A is an area vector, then F·A =ℏωn′, where n′ is the rate at which photons of energy ℏω pass. This permits a calculation of the probability that a photon with linear polarization in the α-direction will pass a polarizer oriented in the β direction. Since the "remote" photon was already measured along α, the electric field at the "local" filter is presumed to be either perpendicular or parallel to the α-axis. Regardless of whether the remote α-filter passed or blocked the photon, the probability of both filters passing or both blocking equals 1/4. By symmetry, we get the same result for all other choices of filter orientation so that, P(α=β) + P(β=γ) + P(γ=α) = 3/4.
Instantaneous communication and super-determinism
For some, a satisfactory resolution of the "paradox" lies in the fact that quantum theory recognizes (α,β,γ) not as variables but as observables that take on value only after a measurement changes the system. According to this line of reasoning, the rigors of mathematics do not seem to apply to nonexistent variables. Figures 2 and 3 are intended not to resolve the "paradox", but to facilitate its appreciation, and also to give students practice with position versus time graphs and exposure to the special theory of relativity.
One could argue that two people possessed psychic abilities if they routinely duplicated this violation of Bell's inequality by going into separate and isolated rooms and answering one of three "yes-or-no" questions in a way that their answers were identical only 25% of the time. Demonstration of such abilities would certainly trigger an investigation as to whether the subjects were being deceptive. Such deception, or "cheating", might include (1) secret communication during the questioning, or (2) learning the questions in advance. We shall now consider the analogy to both scenarios in a Bell's theorem experiment.
We first consider the possibility that the entangled particles might "communicate" with each other during the measurements. It is well known that faster-than-light (superluminal) communication violates special relativity, but this violation is much easier to demonstrate for a Bell's theorem experiment because the two measurements can be brought arbitrarily close to simultaneity by careful placement of the detectors. Figure 2 shows how instantaneous communication would allow Alice to send a message to her own past. This is accomplished using two hypothetical "magic phones" (unicorn icons) capable of instantaneous communication. Alice also needs help from Bob (at rest in her own reference frame), as well as a very long high-speed train occupied by two individuals inside both ends of the train who maintain and operate "magic phone #1". The message originates at the point marked "send", and since the Alice icon is in immediate proximity to the front of the train, she can transmit her message to the passenger riding at the front, who immediately sends it to the rear.
The star at the "center" of the horizontal train at the bottom represents the instant when two light pulses (blue photon icons) are directed in opposite directions. These pulses serve to identify two crucial events that are simultaneous in the train's reference frame. One event is the arrival of the front of the train to the Alice icon at the moment she sends the message, and the other occurs in Bob's past. It is best to imagine a moving train not as a line segment, but as the area between the timelines marked "front" and "rear" (each representing uniform motion at a third the speed of light). Time-travelers who suddenly "drop in" to any point in this area would find themselves at a specific place on the train at a specific time. Note however that Bob's icon is outside that area, meaning that the train already passed him when Alice sends the message. The lotus of points associated with slanted train image represents the train at the instant in the train's frame when "magic phone #1" is used. Bob had strategically chosen to remained motionless at the location where he would be adjacent to the train's rear when Alice's message arrives, allowing him to "relay" the message back to Alice via "magic phone #2" (which is stationary with respect to Alice and Bob.)
Figure 2 introduces four essential concepts about special relativity. Einstein's first postulate is invoked by assuming that if a "magic phone" can exist in one reference frame, it can exist in any reference frame. His second postulate implies that despite appearances, the two light pulses are also viewed by the train's passengers as moving at the speed of light, and therefore establish simultaneity in the train's frame. A third concept for students to appreciate is that our intuitive ideas about space-time break down. At the introductory level, it is sufficient to focus on only one such artifact of the theory, and here it is the relative nature of simultaneity that is taught. The fourth concept is subtle but important: The purpose of physics is to make testable predictions, and this argument predicts that a "magic phone" will never be constructed. Instructors can also mention to advanced students that if the two light pulses are reflected, they again meet at the center, thereby forming a moving clock. The sloped image of the train acts as a moving ruler, and this clock and ruler permit a simple and intuitive introduction to the non-orthogonal coordinate system of the Minkowski diagram.
The two individuals claiming to possess psychic abilities could also "cheat" by somehow learning the questions in advance. Is it possible that the parent atom already "knows" the future orientation of the filters in a Bell's theorem experiment? In a radio interview, John Bell spoke of "super-determinism". Figure 3 gives us insight into why Bell and others chose to add a prefix two what many might consider the well-defined "determinism". This figure depicts a timeline that meticulously coordinates every atom, photon, and living creature in the universe. This universe might be governed by a vast set of deterministic equations involving hidden variables, or it might be a vast multi-particle version of the Schrodinger equation that deterministically predicts probability amplitudes. Either way, the two distant galaxies (Sc and Sb) carry instructions from the big bang that causes each galaxy to emit a photon from one if its atoms. After millions of years, these photons will eventually reach a Bell's theorem experiment on Earth with nanosecond precision so as to trigger the decision as to how to orient each polarizer while the two entangled photons are en route. Moreover, the photons Alice and Bob were also "instructed" by the parent atom on how to behave.
While Figures 2 and 3 have limited utility in sorting out the question of causality in quantum mechanics, Fig. 3 does hint at a similar situation in Gibb's paradox, where insight was gained by making the system in question a subsystem of a larger system. It shall be left to the reader to decide if it would be worthwhile to model this on a less spectacular scale (such as using photons from excited atoms to select the measurement orientations). And the reader should decide if Figure 3 removes the "paradox" of causality or simply buries it in greater complexity. But it is worth pointing out that an experiment like this has been proposed in the literature. 
The author is also grateful to Richard Gill for discussions involving the proof of Bell's theorem, and to Hamed Attariani for conversations as to how best present the ideas conveyed by these figures.
- Wódkiewicz, Krzysztof. "Classical and quantum Malus laws." Physical Review A 51.4 (1995): 2785-2788.
- Like Gibb's "paradox", Bell's "paradox" is perhaps viewed not not a logical paradox but pair of calculations that yield different results because they are based on different a priori assumptions. Gibb's paradox involves the dependence of the entropy of a classical gas on the number of particles, and can be "resolved" by allowing the gas sample to be larger than and exchange particles with a larger gas. See, Swendsen, Robert (March 2006). "Statistical mechanics of colloids and Boltzmann's definition of entropy". American Journal of Physics 74 (3): 187–190. (link) and Swendsen, Robert H. (June 2002). "Statistical Mechanics of Classical Systems with Distinguishable Particles". Journal of Statistical Physics 107 (5/6): 1143–1166. (link)
- Hess, Karl, and Walter Philipp. "Bell's theorem and the problem of decidability between the views of Einstein and Bohr." Proceedings of the National Academy of Sciences 98.25 (2001): 14228-14233. http://www.pnas.org/content/98/25/14228.full
- Gill, Richard D. "Statistics, causality and Bell’s theorem." Statistical Science 29.4 (2014): 512-528.
- Mermin, N. David. "Bringing home the atomic world: Quantum mysteries for anybody." Am. J. Phys 49.10 (1981): 940-943. Found at http://web.pdx.edu/~pmoeck/pdf/Mermin%20short.pdf
- Tovey: A Companion to Beethoven's Pianoforte Sontatas", Associated Board of Royal Schools of Music. London 1931 p.150
- "Feynman, R. P., Leighton, R. B., & Sands, M. L. (1963). "The Feynman lectures on physics". Reading, Mass: Addison-Wesley Pub. Co." Sec. 18.3 Vol. III. http://www.feynmanlectures.caltech.edu/III_18.html.
- Aspect, Alain. Bell’s theorem: the naive view of an experimentalist. Springer Berlin Heidelberg, 2002. https://arxiv.org/ftp/quant-ph/papers/0402/0402001.pdf
- "A Physicist's View of Matter and Mind" by Chandre Dharma-wardana http://www.worldscientific.com/worldscibooks/10.1142/8594
- It may be useful to create a slightly more crowded version of Fig. 1 using a larger font for printing on the standard or A4 paper.
- For example, the input electric field in Fig. 1 is shown as a double arrow instead of a colored one. Double arrows are also easy to draw on the board on on paper.
- All three figures are available on Wikimedia Commons at https://commons.wikimedia.org/wiki/Main_Page. The filenames are: File:Bell's theorem explained on a single A4 sheet.svg, File:Minkowski_diagram_p-template.svg, and File:Bell's theorem and superdeterminism.svg
- A. Aspect, Dalibard, G. Roger: "Experimental test of Bell's inequalities using time-varying analyzers" Physical Review Letters 49 #25, 1804 (20 Dec 1982). http://www.drchinese.com/David/Aspect.pdf
- For example, a commonly found word in titles is "pseudo-telepathy" , as in : Brassard, Gilles, Anne Broadbent, and Alain Tapp. "Multi-party pseudo-telepathy." Algorithms and Data Structures. Springer Berlin Heidelberg, 2003. 1-11.
- Matsukevich, D. N., et al. "Bell inequality violation with two remote atomic qubits." Physical Review Letters 100.15 (2008): 150404.
- Maccone, Lorenzo. "A simple proof of Bell's inequality." American Journal of Physics 81.11 (2013): 854-859. http://arxiv.org/abs/1212.5214
- Relations like this between flux density and flux are essential and difficult for students to grasp, justifying the introduction of Bell's theorem to introductory students.
- Vandegrift, G. 1995. "Bell's Theorem and Psychic Phenomena". Philosophical Quarterly 45 (181): 471-476. Reprinted in Theodore Schick, Jr., (editor) 1999: Readings in the Philosophy of Science: From Positivism to Postmodernism. Mountainview, CA: Mayfield Co http://www.wright.edu/~guy.vandegrift/shortCV/Papers/bell.pdf
- Minkowski, Hermann. "Time and space." The Monist (1918): 288-302.
- Mermin, N. David. "An introduction to space–time diagrams." American Journal of Physics 65.6 (1997): 476-486.
- Kleppe, A. "9 Fundamental Nonlocality." What Comes Beyond the Standard Models 12.2: 107.
- Larsson, Jan-Åke. "Loopholes in Bell inequality tests of local realism." Journal of Physics A: Mathematical and Theoretical 47.42 (2014): 424003.
- Vaidman, Lev. "Tests of Bell inequalities." Physics Letters A 286.4 (2001): 241-244.
Supplement for reviewers of this article:
For reviewers: If you print the page on standard paper you will see how Figure 1 would appear if it filled the page in landscape mode.