Entanglement and Bell Inequalities : No Correlation


 Claims that entanglement, or non-local interaction ("spooky action-at-a-distance), has been experimentally proven are false. The claims are based entirely on the fact that the results of experiments performed to test that issue (EPR-Bohm experiments) violate various theoretical inequalities (Bell Inequalities).The various Bell Inequalities used to interpret the results of EPR-Bohm experiments are shown to be inappropriate for such experiments. The inequalities do not correctly provide for the correlation between the particles in each pair, which is the essence of such experiments; and therefore the inequalities are irrelevant. Claims that the results of such experiments, because they violate the inequalities, require the conclusion that the measurement of one particle in an "entangled" pair can instantly affect the state of its partner, even at great distances, are untrue and need to be modified.

with the assumption that the wave function does give a complete description, they showed 23 that the two physical quantities of non-commuting operators can have simultaneous reality. 24 The negation of alternative (1) leads to the negation of alternative (2), and thus their stated 25 conclusion. They stressed the fact that by simultaneously real they are referring to quanti-26 ties that either one or the other, but not not both simultaneously, of the two quantities can 27 be predicted. They posed the locality condition that the process of measurement carried 28 out on the first system does not disturb the second system in any way. 29 It is difficult to carry out such experiments, but in 1957 Bohm and Aharonov [2] described 30 experiments that can be performed that test the concept of the EPR paradox. In one case, 31 two electrons in a singlet spin state are separated by a method that does not alter the 32 total spin and, when they are sufficiently far apart, measurements are made on the two 33 systems. In another case, two photons are created with opposite momentum and with each 34 photon in a polarization state that is orthogonal to the other. The experiments were still 35 difficult to carry out with the technology available at that time, but preliminary results 36 appeared to favor the view that measurements on one system affected measurements on the 37 correlated system that was spatially distant from the first. They did allow for the possibility 38 that quantum theory is not complete and may be an approximation arising from "a deeper 39 subquantum-mechanical level". 40 In 1964 Bell[3] derived a set of inequalities that he claimed must apply to any system, 41 even with hidden variables, if that system were to meet the requirements of locality (i.e., 42 measurements on one system can not affect, nor be affected by measurements on the other). 43 Using the example of of an experiment based on the Bohm-Aharonov experiment [2] (EPR- 44 Bohm experiments), and imposing his locality requirement, Bell developed inequalities that 45 were incompatible with the statistical predictions of quantum mechanics. Inequalities. This paper relies heavily on their review. In Section 3.3 of Clauser and Shimony 65 they discuss the locality concept and provide a very reasonable definition: 66 Suppose a pair of correlated systems, which have a joint state λ, separate. They 67 then continue to evolve perhaps in an inherently stochastic way, and given λ, 68 a and b, one can define probabilities for any particular outcome at either ap-69 paratus. We allow that, given λ, these two probabilities may each depend on 70 the associated (local) apparatus parameter, a or b respectively, and of course 71 upon λ, but we assume that these probabilities are otherwise independent of 72 each other.

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As discussed in Section A, above, it is not an acceptable locality condition.
It is clear that this suffers from the same factorization problem of Equation (1)  to a response α j in A and β j in B. Each term α j and β j is either +1 or -1. He defines 271 correlation by a statistical mean where the brackets around a quantity designates the statistical mean of that quantity over 273 j. C is equal to the fraction of events when α j and β j have the same signs minus the fraction 274 where they have opposite signs. There are two positions a (1) and a (2) of the knob a and two 275 positions, b (1) and b (2) for the knob b. He defines: (2) and then defines for event j: Each product α j β j is equal to ±1; so therefore and finally 282 C (1,1) + C (2,1) + C (1,2) − C (2,2) ≤ 2 (10) which is the famous CHSH[17] inequality.

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[It should be noted that the CHSH Inequality is obtained using a "correlation The statistical means used in the development by Eberhard also do not provide for 290 correlation between the particles, which is why he reaches the same inappropriate inequality. (i) The fate of photon a is independent of the value of β, i.e., is the same in an event of 308 the sequence corresponding to setup (α 1 , β 1 ) as in the event with the same event number 309 k for (α 1 , β 2 ); also same fate for a in (α 2 , β 1 ) and (α 2 , β 2 ); this is true for each k for these 310 sequences.

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These are perfectly reasonable locality conditions, but they only apply for each given k, 314 not for the sums over all k. Note that the relative orientations of the detectors for a given 315 k will not, in general, be the same as that for any other k's.

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Following an argument first used by Stapp[13], when four sequences are found satisfying and with the outcome o for photon b when α is α 1 and β is β 1 .

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[Note: This is the number of events with that outcome, not an event for a single 323 value of k, and that the concept of "conjugate event" does not really apply.

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Nevertheless, Eberhard proceeds to develop his inequality using these totals, 325 which for notational purposes he calls boxes.]

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He starts by considering n oo (α 1 , β 1 ). He then subtracts from that total some of the boxes 327 for which events, had they been for the same k, would have been conjugate to events that 328 go into the box n oo (α 1 , β 1 ).