User:MRFS/Paradox

Introduction
Back in 1935 a debate raged between Einstein and Neils Bohr about the behaviour of electrons. Erwin Schrödinger had noticed an odd feature that he christened "entanglement" about a small system consisting of two electrons. Basically this meant that many of the properties of one, in particular its spin, were exactly mirrored in the other. Einstein took the view, as did most scientists, that entanglement was a short lived process such as a collision that produced two independent electrons. Bohr however argued that the entangled state persisted until an event happened to one of the electrons to break their bond such as another collision or a measurement. He further claimed that this event would instantaneously affect the other electron even were it on the other side of the universe. Einstein hated this idea. He called it "Spooky action at a distance". His own ideas were described as "Local Realism" but as no test was available to rule out either of these competing ideas the matter lay unresolved for almost 30 years.

Bell's Test (1964)
Enter a brilliant scientist, John Bell, who devised just such a test. Once a means had been found to split a stream of entangled electrons in two the method was quite straightforward. One of the resultant streams would go to Alice and the other would go to Bob. They would have two detectors each which would record the spin of each electron as the stream passed through. There was a small catch in that their detectors could only make binary measurements. An electron's spin is akin to a tiny North/South bar magnet and the best the detectors could do was to record each spin as N/S or S/N. Once the experiment was over Alice and Bob would send the four recordings to a boffin who would calculate a statistic |S|. Any value of |S| exceeding 2 would be incompatible with Local Realism. In contrast, Quantum Theory predicted values of |S| up to 2√2 (called Tsirelson's bound).

The Calculation
Bell's instructions to the boffin were quite explicit. He would receive 4 bitstreams of equal length n which were labelled A, A', B, B' according to the detectors they came from. First he had to compare streams A and B entry by entry and create a further stream of length n by writing "A" (for Agree) if the entries matched and "D" (for Disagree) if they didn't. He was then required to divide the number of A's less the number of D's by n, and label the result E1. He then had to do similar comparisons for A and B' labelling the result E2, for A' and B labelling the result E3, and for A' and B' labelling the result E4. Finally he had to calculate S=E1-E2+E3+E4 and report the outcome |S|.

Consternation
Bell's paper sat on the shelf until 1969 when Clauser, Horne, Shimony, and Holt proposed an experiment to test it. They also simplified and generalised Bell's paper, and created the CHSH inequality which confirmed that |S|≤2. The first experiment (Freedman & Clauser) took place in 1972 and a further one took place in 1982 (Aspect). In 1998 two experiments were run (Tittel and Weihs). Since the millennium the experiment has been performed over a dozen times. It came as a major surprise when the results were announced. |S| was bigger than 2! Einstein was wrong! Many scientists refused to believe this and began to look for "loopholes" in the way the tests were being performed. However as more and more organisations ran their own tests the results increasingly favoured Quantum Theory. Eventually most theoretical physicists accepted that "Spooky action at a distance" was actually happening despite the apparent craziness of the idea.

Paradox 1
Assume that A, A', B, B' are four bitstreams of equal length n. Let A(i), A'(i), B(i), B'(i) denote the ith entry in each stream. Make four new streams F1, F2, F3, F4 of length n as follows :-
 * F1(i) = 1 if A(i) = B(i); otherwise F1(i) = –1.
 * F2(i) = 1 if A(i) ≠ B'(i); otherwise F2(i) = –1.
 * F3(i) = 1 if A'(i) = B(i); otherwise F3(i) = –1.
 * F4(i) = 1 if A'(i) = B'(i); otherwise F4(i) = –1.

For each i let F(i)=F1(i)+F2(i)+F3(i)+F4(i). Obviously these F's are closely connected to Bell's E's. In fact ∑F1(i)=nE1, ∑F2(i)=−nE2, ∑F3(i)=nE3, ∑F4(i)=nE4 and hence ∑F(i)=nS. But for any given i there are just 16 possible combinations for F1(i), F2(i), F3(i), F4(i) and for each of these 16 combinations it turns out that F(i)=±2. Therefore |∑F(i)|≤2n and so |S|≤2.

Paradox 2
Quantum theory places a limit on the maximum value |S| can reach. It is called the Tsirelson Bound and it is equal to 2√2. Therefore experimental results yielding an |S| between 2 and 2√2 were said to vindicate Bohr's theory as opposed to local realism. However the previous paragraph has shown that despite experimental claims |S| cannot exceed 2, and it is reasonable to wonder how such an obvious fact could have escaped the theoretical physics community for nigh on 50 years. The answer may lie in a major flaw in the underlying analysis which predicts that if A, A', B, B' are at certain angles (0°, 90°, 135°, 225°) to the electron stream then |S| will actually reach 2√2. To see where this erroneous figure comes from let ν = 1/√2 or 0.7 in round figures. Suppose the length of each bitstream is 100. The analysis shows that E1 = ν, E2 = -ν, E3 = ν, E4 = ν whence E1 - E2 + E3 + E4 = 4ν, the Tsirelson Bound. Then
 * E1 = ν implies detectors A and B have 15 disagreements;
 * E3 = ν implies detectors A' and B have 15 disagreements, so A and A' have at most 30 disagreements;
 * E4 = ν implies detectors A' and B' have 15 disagreements, so A and B' have at most 45 disagreements;
 * but E2 = -ν implies detectors A and B' have 85 disagreements.

Therefore 85 ≤ 45. QED

Conclusion
Unlike Bell's inequality the first paradox has made absolutely no assumptions about the contents of the 4 bitstreams, yet it has arrived at precisely the same conclusion |S|≤2. Thus Bell's concern that |S| might exceed 2 was unnecessary as it is a mathematical impossibility. No matter how inaccurate the detection equipment might be there is simply no way that |S| can ever exceed 2. Where does this leave those organisations that have reported an |S| > 2? The plain answer is that there must be miscounting somewhere in their processes. Most likely they have not adhered to the instructions given to the boffin. Should anyone doubt that let them take the 4 bitstreams generated in their experiment and repeat the calculation to derive |S|. They will inevitably find an error, and if a computer is involved it isn't impossible the error will be found in the code. As for Quantum Theory the implications are profound. Many books have been written about it, and at a conservative estimate at least 90% of the literature contains highly misleading information. In summary, this theory as it's currently understood and taught has been on completely the wrong track for the past 40 years, and it's surely time for a major rethink. Local realists who spend their time seeking loopholes in the Bell tests can stop looking because the results from those tests are utterly discredited by the paradoxes above. Local realism is firmly back on the table, and now it is Quantum Theory that has serious questions to resolve because its 20+ different interpretations indicate a very obvious state of chaos.