Quantum Randi Challenge

Sascha Vongehr

National Laboratory of Solid-State Microstructures, Nanjing University, Hankou Lu 22, Nanjing 210093, P. R. China

Violations of Bell type inequalities in quantum physical experiments disprove all relativistic micro causal, classically real models. Desperate attempts at saving classicality have retreated to claiming what A. Shimony calls a conspiracy at the intersection of the measurements’ past light cones. It is time to embrace the quantum paradigm instead of being stuck in defending it against positions on par with the creationist belief in divinely planted fossil records. The Quantum Randi Challenge is designed to help scientists and educators discredit local realistic models and related attacks against quantum physics. Its ‘Randi-type’ properties are ensured via a simple computer game that can be made attractive and understandable to lay people. We introduce the general concept of a ‘James Randi type’ challenge as a tool for science outreach aimed against the spread of pseudoscience. This is a challenge which, according to the laws of nature as known to science, is impossible to meet. Randi challenges work simply by being known to exist while never having been overcome, despite the large rewards which would follow from meeting the challenge. This effectively refutes pseudoscientific claims according to which the challenge could easily be met. Pseudoscience exploits well meaning engagement in argument in order undermine science by artificially creating the appearance of a dispute between experts where there is none. Randi challenges allow scientists to publicly refuse to give a platform to pseudoscience without strengthening the perception of censorship and establishment conspiracy. Scientists may decline to enter rhetorical discussions “until the challenge has been met” and no-one can complain that their point of view is actively suppressed.

Keywords: Bell Inequality; Pseudoscience; James Randi Challenge

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1 Introduction: What and Why is a Randi-Type Challenge?

The James Randi Educational Foundation (JREF) famously offers one million US dollars to anyone who can demonstrate paranormal abilities under laboratory conditions.

Its existence has helped stem the spread of pseudoscience. We define a ‘Randi-type’ challenge as one having the following necessary characteristics:

C1) It cannot be met (according to the established laws of nature).

C2) If certain pseudoscientific claims were correct, it could be easily met.

C3) Meeting the challenge would quickly result in enormous rewards.

C4) Judging whether the challenge has been met does not depend on anything that could be discredited as ‘establishment conspiracy’, for instance scientific peer-review.

The original James Randi challenge evidences that such challenges can be an effective tool for furthering the public understanding of science, because these characteristics lead to the following two uses:

U1 ) Educators can point to the bare existence of the challenge to counter the spread of pseudoscience. The challenge having not been overcome in spite of items C2 and C3 gives a convincing argument that the claims of pseudoscience are wrong, convincing also to the many who cannot grasp the intricate details of the scientific issues at hand. For instance, understanding that there can be a trivial error hidden behind the smoke screen of some highly complex calculation “disproving Bell” is not easy for outsiders. However, the fact that the “anti-Bellist” does not go ahead and meet the corresponding challenge, which would immediately bring her undying fame, with or without the approval of the scientific establishment, is a powerful argument that the anti-Bellist’s theory cannot deliver what the she claims.

2 U2 ) The existence of the challenge allows scientists to refuse to enter into rhetoric arguments that actually mainly serve to provide pseudoscience a platform to promote itself. All communication is postponed until after the challenge is met. This aspect is important because one aim of pseudoscience, “intelligent design” for example, is to spread doubt and construct the appearance of a controversy among experts, giving lay persons the impression that well-established science is in dispute. Well-meaning engaging in ‘debates’ backfires by supporting the deception.

The best way to promote the use of Randi challenges is probably to go ahead and construct one. The general concept as outlined above can be applied to counter pseudoscientific claims against quantum physics. This article will further describe the issues that the hereby announced Quantum Randi Challenge (QRC) addresses, and how it is ensured that the QRC is indeed of the Randi-type, i.e., has characteristics C1 to C4.

2 Pseudoscience against Quantum Physics

Much popular pseudoscience seemingly accepts but misrepresents quantum mechanics in order to sell magic medical cures or argue for precognition. Such is not our concern here. We are concerned with the increasingly vocal pseudoscience that rejects fundamental quantum physics. In order to show relevance, we will first introduce the core issue and clarify why there is an increasing rejection against it especially from many who otherwise defend science.

Quantum mechanics has been experimentally confirmed to astounding levels of accuracy. The core of the theory is entanglement. Uncertainty and quantization could

emerge from classical substrates, but entanglement and quantum superposition, which is

3 the entanglement of states rather than that of multiple particles (which are themselves states of a field), is fundamentally non-classical. All important modern applications like quantum cryptography (Ekert 1991) 1 and quantum teleportation for example are based on entanglement. Quantum entanglement is proven to be non-classical by the experiments and theory around the Einstein-Podolsky-Rosen (EPR) (Einstein 1935) 2 paradox and John

Bell’s famous inequality (Bell 1964) 3.

The violation of Bell inequalities in quantum physical experiments (Aspect 1981,

1982)4,5 has disproved all local realistic (LR) models, for example non-contextual, possibly stochastic, hidden variables. Such hidden variables cannot violate Bell’s inequality (Bell 1966) 6, variations of which (Clauser 1969) 7 have been strongly violated by diverse experiments, most impressively with the closing of the so called communication loophole by (Weihs 1998) 8, and quite recently again by confirmation of the Kochen-Specker theorem (Kirchmair 2009) 9.

Discussing an eavesdropper’s exploitation of the still open detection loophole is important for secure key distribution protocols (Barrett 2005; Acin 2006) 10,11 . However, this high level of sophistication is ill advised when publicly defending quantum physics against those who aim to ‘save classical physics’ by exploiting the detection loophole in ever more conspiring, but LR (local realist) ways. Nature cunningly exploiting the detection loophole in just such a way as to deceive us about being classical would imply it wanting to do so rather than being the blind classical mechanism that is seemingly defended. The sophistication in the arguments is exploited to validate nonsense as profound genius which the establishment cannot grasp and therefore suppresses. This is where the QRC comes in.

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2.1 Non-Locality versus Modified Realism

Why does quantum physics encounter increasingly strong rejection? Bell’s theorem and the EPR paradox prove (QM-apparent) non-locality, but that the “local” of “local realistic” is questionable has met relatively wide acceptance. Non-locality is an instantaneous correlation. Although one cannot use it to transport matter or information with superluminal velocities, it is a form of faster than light physics, but such still does not trigger widespread rejection, especially not among lay persons. However, this

“spooky interaction at a distance” (A. Einstein) is actually quite ‘unreal’ and it is the

“realistic” in “local realistic” which has been crumbling ever since EPR.

There are different interpretations of quantum mechanics. Some accept the Everett relative state description (Everett 1957) 12 and many-worlds interpretations (DeWitt 1973;

Deutsch 1997) 13,14 ; some despise talk about other worlds, but all serious contenders agree on that the traditional form of direct, classical realism is in dire need of modification, while Einstein-locality , which is the micro-causality of modern relativistic quantum field

theory, may perhaps stay with us. Whether this necessitates modal realism (Lewis 1986) 15 or many-minds interpretations (Albert 1988; Lockwood 1996) 16,17 or yet entirely different

descriptions is under debate, but there is no longer any question about that the type of

realism is not an innocent assumption.

We will not discuss all the ways in which this triggers widespread opposition, but one

aspect should be pointed out in order to understand why this retreat of naïve realism finds

resistance inside the scientific community, for example among researchers in fields like

engineering and chemistry and among science literate lay persons. Many scientists do not

5 distinguish between realisms and often even identify scientific realism with classical direct realism, and scientists of course defend scientific realism precisely in order to reject pseudoscience. Adding in that discussions about modifying realism triggers concerns about undesired reality of unethical decisions and many other, scary thoughts questioning personal identity and agency and so on, it should be expected that the reaction against modern physics will grow right along with the acceptance that quantum mechanics demands to modify realism. This topic is of growing relevance and it should not be a surprise that people who otherwise defend science are caught up in it.

3 The Quantum Randi Challenge

The following will first clearly point out why the QRC is a Randi challenge. Afterward, the involved physics is explained as simply as possible in order to facilitate its further conversion into a high-school level exposition, which is indeed possible, but would obviously need more space as well as attractive visualizations.

3.1 The Quantum Randi Challenge is a Randi-Type Challenge

The QRC challenges all claims of quantum physics being untrue (entanglement a myth, etc.); i.e. claims of that the predictions of quantum mechanics can be reproduced by a classical (LR) model. The QRC fulfills all the criteria for a Randi-type challenge:

C1) Quantum physics predicts the violation of Bell inequalities; this has been

experimentally observed. Bell has proven that suchlike cannot possibly arise within a LR

model. The challenge cannot be overcome.

6 C2) a) The challenge is to reproduce only and nothing else but the behavior of the simplest setup known to violate Bell’s inequality maximally, which has only three different settings of detector angles.

b) Any LR model can in principle be simulated by classical computers. That a computer can model a system is the very essence of classical local realism: everything depends on locally present data, variables that have specific values. (Objections will be discussed and rejected later in this paper.)

c) The computer program that needs to be written is already essentially finished. A working version with several (of course, unsuccessful) example LR models is already available on the internet. All that the anti-Bellist needs to do is merely to modify the hidden variables and measurement prescriptions in order to reflect a specific LR model.

Because it is an LR model, it is possible, and it is moreover a trivial task: Given any LR model, turning the particular hidden variables and measurement prescriptions into instructions for a programmer is trivial (doing it in such a way that the challenge is met is impossible).

C3) Instant fame is assured. In fact, a Nobel Prize would be deserved for whoever

modifies the hidden variables and the measurement prescriptions in the program so that

the Bell inequalities are violated. The computer setup itself would constitute a classical

physical system that violates Bell.

C4) A Bell inequality violating program, published on the World Wide Web as a simple

multi-player game, would become instantly famous, without any chance for established

physicists (“the conspiring establishment”) to prevent it.

7 That the QRC is similarly effective in terms of the uses U1 and U2 is indicated by the successes that the initial trials have had on internet portals.

U1 ) The mere existence of the QRC has been successfully employed (Vongehr 2011) 18 to discredit a particular LR model in the eyes of a lay audience that was up to then unconvinced and confused by all the more “professional” argumentations.

U2 ) Strictly scientific refutations (Gill 2003; Grangier 2007; Moldoveanu 2011)19,20,21 of that LR model have shown to fuel a vicious cycle, triggering more claims that invite to be refuted again. This has indeed further popularized the involved pseudoscientific claims, some of which subsequently even found funding sources and book deals. The QRC has already succeeded in terminating the artificially created debates on several popular web portals and thus denied further promotion of these antiscientific claims.

3.2 Basics of the EPR setup and the Aspect-type Experiment

The simplest version involves a source of pairs of photons. The photons are separated by sending them along the x-axis to Alice and Bob, who reside far away to the left and right, respectively. Alice has a calcite crystal polarizing beam splitter with two output channels. Her photon either exits channel “1”, which leaves it horizontally polarized, or channel “0”, which leads to vertical polarization (relative to the crystal’s internal z-axis).

The measurement is recorded as A = 1 or 0, respectively. Bob uses a similar setup, so that there are four possible measurement outcomes (A,B) for every photon pair: (0,0), (0,1),

(1,0), or (1,1). Every photon pair is prepared in such a way that if the crystals are aligned in parallel, only the outcomes (0,1) and (1,0), for short U (for “Unequal”), will ever result

(perfect anti-correlation). If the crystals are at an angle δ = ( b – a) relative to each other

8 (rotated around the x-axis), the outcomes depend on δ as expected from the usual optics

at polarization filters: The outcomes (0,0) and (1,1), for short E (for “Equal”), occur in

the proportion sin 2(δ).

Every experiment starts with the preparation of a pair of photons. When the photons are

maybe about half way on their path to the crystals, Alice randomly rotates her crystal

either to let a equal φ0 = 0º or φ1 = 3 π/8 = 67.5º. Bob adjusts his crystal similarly to b being either φ1 or φ2 = π/8 = 22.5º.

A few important didactical points: No other angles may be considered in order to ensure

C2 of the simple Randi-type challenge! The magnitudes of δ are multiples of 22.5º, but

one should label with the absolute angles to keep locality explicit. Restricting to exactly

NTotal = 800 photon pairs renders the numbers small and cheating difficult. We completely avoid probabilities and consider only expected counts expressed in small integers N.

Alice and Bob have each only two different angles to choose from, so there are four equally likely combined choices: Out of 800 runs, the angles are about Na,b ≈ 200 times in

each of the four configurations (φa, φb) with a œ {0,1} and b œ {1,2}. The outcomes of all

runs are counted by the 16 numbers Na,b (A,B), where a, A and B all take the values 0 or 1; only b is 1 or 2. Perfect anti-correlation leads to N1,1 (E) = 0 and N1,1 (U) ≈ 200. Generally, it holds that

2 2 NNab,,(E) ≈ ab sin(),δ NN ab ,,( U) ≈ ab cos() δ . (1)

Apart from N1,1 (E) = 0, only three of these are important: The sum of N0,1 (U) ≈ 200 *

2 2 cos (3 π/8) ≈ 30 and N1,2 (E) ≈ 200 * sin (–π/4) ≈ 100 is expected to be by 40 occurrences

less than the third number, namely N0,2 (U) ≈ 170 alone.

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3.3 Local Realistic Model with Hidden Variables

Let us try to model the experiment described with help of hidden variables. A pair of balls is prepared, say instructions are written on them, and then split. Before the balls arrive, Alice and Bob randomly select angles. Each ball results in a measurement 0 or 1 according to the angle it encounters and the hidden variables it carries. We will never assume anything about the complexity of the hidden variables, which may be as complex as desired, except that the anti-correlation at equal angles (a = b = 1) must be ensured.

Local realism means here that each ball is a real object having all necessary information locally with it. No measurement depends on angles selected far away. This models the fact that photons travel at the speed of light. Nothing travels faster than light, so the photons must know any hidden variables already when they are created and they must take this information with them on their way.

Assume the instructions somehow prescribe “If a = 1, then A = 0”, or short “ A1 = 0”.

The ball at Bob’s place cannot know which angle Alice has just chosen. She might have

picked a = 1, and if so, Bob’s measurement cannot be 0 if he also picks b = 1. Thus, the

hidden variables, however complex they may be, must prescribe the complementary

information “ B1 = 1.” Furthermore, A0 and B2 must be somehow prescribed by the hidden variables, otherwise the occurrences Na,b (A,B) cannot reproduce the sin( d) dependence. In

summary, the hidden variables may be an infinite table or a complex formula, but they

must at least effectively contain the prescription (A0, B1, B2) with A1 = 1 – B1. According to these three degrees of freedom, each pair of balls falls into one and only one of 23 = 8

different classes, which one may index by i = 4(1–A0) + 2 B1 + B2, with the total number

10 7 of pairs being N i = 800 again ( i is an index, not a power). For example, there will ∑i=0

be N6 occurrences of (0, 1, 0).

Every pair encounters one of the four possible configurations of angles, hence for

5 5 5 5 5 example N = N 0,1 + N 0,2 + N 1,1 + N 1,2 . All choices of angles occur about equally often and the hidden variables cannot bias the choice (they have not arrived yet when the

i i angles are chosen). Hence, all N a,b are expected to be roughly equal to N /4, which seems

trivial enough but is a most important step, the one that enforces Einstein-locality:

i i Na, b ≈ N 4 (2)

0 1 1 2 4 4 5 All the cases counted by N a,b , N 0,1 , N 1,1 , N 0,2 , N 1,1 , N 1,2 , and N 1,1 lead to

2 4 4 5 6 6 measurement outcome ( A,B) = (1,0). Equivalently, N 1,2 , N 0,1 , N 0,2 , N 0,1 , N 0,2 , and N 1,2

1 1 2 3 3 5 2 correspond to (0,0), while N 0,2 , N 1,2 , N 0,1 , N 0,1 , N 0,2 , and N 1,2 to (1,1). Finally, N 1,1 ,

3 3 5 6 6 7 N 1,1 , N 1,2 , N 0,2 , N 0,1 , N 1,1 , and the four N a,b correspond to (0,1). This enumerates all

i the 32 possible N a,b exhaustively. The reader should check that reproducing the 16

2 counters Na,b(A,B) of Section 3.2 leads, for example, to N1,1 (E) = 0 and N1,1 (0,1) = N 1,1 +

3 6 7 0 1 N 1,1 + N 1,1 + N 1,1 . Important are the following three equalities: N0,1 (U) = N 0,1 + N 0,1 +

6 7 0 1 6 7 0 2 5 7 N 0,1 + N 0,1 ≈ ( N +N +N +N )/4, N 0,2 (U) ≈ ( N +N +N +N )/4, and N 1,2 (E) ≈

(N2+N6+N1+N5)/4. Bell’s inequality is here the mathematically trivial statement that

N0+N2+N5+N7 is by 2( N1+N6) smaller than N0+N1+N6+N7 and N 2+N6+N1+N5 added together. In other words, it is expected that:

N0,2(U) ≤ N 0,1( U) + N 1,2 ( E ) (3)

Even if the hidden variables are deliberately chosen in cunning ways, the inequality is expected to hold true, because it derives from the randomness of the measurement angles leading to Eq.(2). Therefore, the quantum experiment described in Section 3.2, where

11 N0,2 (U) alone is larger than the right hand sum by 40, cannot be described by any LR model.

Simply not preparing any i = 1 or 6 pairs sets N1 and N6 equal to zero and ensures that the equals sign in Eq.(3) holds. This would violate the Bell inequality every second run. It is crucial to stress that quantum mechanics violates the inequality almost every time . LR models that violate it “often” and are presented as an advance toward a revolutionary discovery should be immediately rejected by pointing out that a choice of hidden variables which violates Bell 50% of the time has been presented here already, and is thus uninteresting. The challenge to the anti-Bellist is to reproduce quantum mechanics and violate the inequality almost every time .

3.4 Computer Realization

Basic computer realizations of the discussed EPR setup with hidden variables are very simple indeed. Implemented in Mathematica™, a core algorithm that assumes random hidden variables consists of only nine vital lines of code (See Fig. 1 and Supplemental

Material after references for the analysis part of the program). Constructing the hidden variables [in the program conveniently (1–A0, B1, B2) instead of ( A0, B1, B2)], choosing

the angles, and calculating the measurements and the Bell inequality, are all

accomplished in under one second. Constructing the hidden variables (H) is for example

accomplished by the line:

Table[H[i, j] = If[Random[] < 0.5, 0, 1], {i, 800}, {j, 3}]

A typical result is the output (See also Fig. 1):

12 “{94, 106, 100} The Bell inequality predicts that the sum of the first two numbers is larger than the third. It holds in all local realistic (LR) models. On principle, all LR models can be realized by modifying this computer realization. Quantum mechanics is confirmed by experimental observation. It violates the Bell inequality (almost every time!) and thus disproves Einstein-local naïvely classical realisms.”

In the rare event that the inequality is violated by an unlikely coincidence between hidden variables and random angles, the program asks to play again, explaining that quantum mechanics violates the inequality almost every time, not merely once in a while.

Fig. 1: The simplest hidden variable local realistic model with remarks on how it would have to be turned into a multi-player game in order to be convincing as a physical realization rather than simulation. Below that (below the green rectangle) is the output of a typical run.

The task of a challenger (anti-Bellist) who claims she has a LR model that can give rise to quantum physical behavior is to merely modify the program according to her model.

13 Any LR model whatever needs modification of only three lines of the code, namely the construction of the hidden variables and the parts where the measurement is accomplished in Alice’s and Bob’s place. The rest, like the random choice of angles, must stay unaltered. For example, if the challenger claims hidden photon polarizations, a random angle r œ [0, 2 p] may replace the previous hidden variables:

Table[H[i] = 2 π Random[], {i, 800}]

Mathematica represents these angles to six digits behind the decimal point, which is a much finer resolution than any EPR experiment has ever achieved. Note that if the LR model assumes photons to live in a hidden, higher than three dimensional space, several angles may be entered. The hidden variables may reflect for example the topological double covering of the SU(2) group by angles periodic in 4 p instead of 2 π. Randomness

may be abandoned in favor of a table describing 800 objects.

The QRC has much didactic potential through this ‘gaming’. If, for example, the

quantum measurement outcomes are mistakenly believed to be only due to the

probabilities as they are known from single photons at polarized filters ( classical

indeterminism), a random selection with probability cos 2(a – r) may modify the measurement part of the program. An example for a thus modified program (Fig. 2) leads to the output

“Less than 90% Anti-Correlation when Alice and Bob happen to measure with the same angle. … Model fails to describe the Anti-Correlation observed in quantum experiments.”

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Fig. 2: An example for how a challenger who believes the photons to carry fixed polarization vectors r may modify the given, simplest example above. Note that the model violates the Bell inequality one out of 30 times on average, but the analysis rejects it even those times for its poor anti-correlation. Notice also how simple the program modification is although the LR model now includes vectors (in real coordinate space) as hidden variables instead of just numbers.

The simplicity of the program derives from the simplicity of the experimental setup, i.e. the restricted choice of angles, not from enforcing simplicity of the LR model, which may have any complexity. Anybody who has come up with a novel LR model would be able to modify the program according to it – a much easier task than thinking up LR models with surprising statistical correlations. If the modified program then indeed violates the

Bell inequality, it would attract a great deal of attention, so much so that many people would turn it into an online multiplayer game as described in the remarks inside the program. Already available entertainment multiplayer games are much more complex than the one envisioned here and a whole industry exists to program them and educate programmers. Such a game, distributed over three different computers (namely: the host

15 server, Alice, and Bob) and which cuts internet communication at appropriate times to ensure no cheating via artificial non-locality, would constitute a classical physical system that violates Bell’s inequality! It would become known worldwide in a matter of days.

This makes the here published program a true Randi-type challenge.

3.5 Preempting Remarks

The following remarks aim to preempt the most likely from a wide spectrum of possible criticisms. Experts may require the QRC to be watertight against the detection loophole, pointing out that using computers to simulate entanglement has been discussed before, even in form of a challenge (Gill 2003) 19 against dubious claims. However, a Randi-type challenge must be understood by many educators. Therefore, undetected photons or a lot of missed anti-correlation at equal angles are not permissible. The CHSH inequality, which cannot be explained as intuitively on the lay level in our opinion, is thus unnecessary. It is only added as an option to the computer code. This supports the view that the detection loophole is a technicality that can be narrowed further by improving detectors, while the communication loophole is crucial: The photon pair creation event E can “know” the detectors’ angles simply by being in the same “world-branch” as those settings (speaking in the language of many-world interpretation), if only the random setting decision’s branching had sufficient time to arrive at E.

At the other end of the spectrum are those who will criticize the QRC because it threatens their pseudoscientific claims. They question the equivalence of classical physics, especially theirs, and classical computation. We counter this objection as follows: All experimental observations have finite resolution due to experimental

16 errors/accuracy. Therefore, the practically limitless finiteness of today’s computer memory does not present a credible loophole. There is no difference for a computer whether it calculates relations applicable to our usual Euclidian three dimensional space or something else. Many strange geometries and topologies (e.g. black holes and worm holes and the SU(2) double covering that Fermions are susceptible to) have been simulated in virtual reality. The words “simulation” and “virtual reality” do not provide loopholes, because the described multi-player game computer setup constitutes a classical physical system; computers are physical! In short: if the computer setup could violate

Bell’s inequality, that very computer network would be a classical physical system that violates the Bell inequality and such an engineering feat would in fact deserve a Nobel

Prize.

4 Acknowledgements

I thank Richard D. Gill from the Mathematics Department of the University of Leiden,

Netherlands, for a careful reading of the manuscript and suggesting to present the QRC entirely independent of certain interpretations of quantum mechanics.

5 References

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5 Aspect, A., Grangier, P., Roger, G. (1982) “Experimental realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: a new violation of Bell's inequalities.” Phys. Rev. Lett. 48 , 91-94 (1982) 6 Bell, J. S.: “On the problem of hidden variables in quantum mechanics.” Rev. Mod. Phys. 38 , 447–452 (1966) 7 Clauser, J.F., Horne, M.A., Shimony, A., Holt, R.A.: “Proposed Experiment to Test Local Hidden- Variables Theories.” Phys. Rev. Lett. 23 , 880-884 (1969) 8 G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, A. Zeilinger: “Violation of Bell’s inequality under strict Einstein locality condition.” Phys. Rev. Lett. 81 , 5039-5043 (1998) 9 G. Kirchmair, F. Zähringer, R. Gerritsma, M. Kleinmann, O. Gühne, A. Cabello, R. Blatt, and C. F. Roos: “State-independent experimental test of quantum contextuality.” Nature 460 , 494-497 (2009) 10 Barrett, J., Hardy, L., Kent, A.: “No Signaling and Quantum Key Distribution.” Phys. Rev. Lett. 95 , 010503 (2005) 11 Acin, A., Gisin, N., Masanes, L.: “From Bell’s Theorem to Secure Quantum Key Distribution.” Phys. Rev. Lett. 97 , 120405 (2006) 12 Everett, Hugh: “‘Relative State’ Formulation of Quantum Mechanics.” Rev Mod Phys 29 , 454-462 (1957), reprinted in B. DeWitt and N. Graham (eds.), The Many-Worlds Interpretation of Quantum Mechanics , Princeton University Press (1973) 13 B. DeWitt: “The Many-Universes Interpretation of Quantum Mechanics.” in DeWitt, B. S., Graham, N. (eds.), The Many-Worlds Interpretation of Quantum Mechanics, Princeton University Press, Princeton NJ (1973) 14 Deutsch, David: “The Fabric of Reality.” Allen Lane: New York (1997) 15 Lewis, David Kellogg: “On the Plurality of Worlds.” Blackwell (1986) 16 D. Z. Albert, B. Loewer: “Interpreting the many-worlds interpretation." Synthese 77 , 195-213 (1988) 17 M. Lockwood: ‘ “Many minds" interpretations of quantum mechanics.’ Brit. J. Phil. Sci. 47 (2), 159-188 (1996) 18 S. Vongehr: “Official Quantum Randi Challenge” www.science20.com/alpha_meme/official_quantum_randi_challenge-80168 (2011) 19 Richard D. Gill, J.A. Larsson: “Accardi contra Bell (cum mundi): The impossible coupling.” In Mathematical Statistics and Applications: Festschrift for Constance van Eeden . Eds: M. Moore, S. Froda and C. L'eger, pp. 133-154. IMS Lecture Notes -- Monograph Series 42 , Institute of Mathematical Statistics, Beachwood, Ohio (2003) 20 Grangier, Philippe: ‘“Disproof of Bell’s Theorem”: more critics’ http://lanl.arxiv.org/abs/0707.2223 (2007) 21 Moldoveanu, Florin: ‘Disproof of Joy Christian’s “Disproof of Bell’s theorem”’ http://lanl.arxiv.org/abs/1109.0535 (2011)

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Supplemental Material

Figure S1 : The Analysis module

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