Abstract
In this chapter we turn to the paradox of Einstein, Podolsky and Rosen, and Bell’s Theorem.
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Notes
- 1.
- 2.
- 3.
See [17].
- 4.
- 5.
See footnote 1 above. Not all authors agree with this position. Some reject quantum nonlocality. See Jarrett [20] and also Evans, Price and Wharton in [21]. Others argue that the theorem constitutes a proof that realism is impossible in quantum physics. See Bethe [22], Gell-Mann [23], p. 172, and Wigner [19], p. 291.
- 6.
In particular, one cannot take the conclusion of the EPR paradox—the existence of noncontextual hidden variables—as received and final. This conclusion is based not only on quantum mechanical predictions, but also on the assumption of locality, which will ultimately be seen to fail. The status of EPR becomes clearer when one recognizes that the analysis is in fact equivalent to a theorem, as we demonstrate in Sect. 3.2.3.
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The Bohm spin singlet version and the original version of the EPR paradox differ essentially in the states and observables with which they are concerned. We shall consider the original EPR state more explicitly in Sect. 4.2.1.
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- 9.
Note that a term such as \(|a\rangle^{(1)}|b\rangle^{(2)}\) represents a tensor product of the vector \(|a\rangle^{(1)}\) of the Hilbert space associated with the first particle with the vector \(|b\rangle^{(2)}\) of the Hilbert space associated with the second. The formal way of writing such a quantity is as: \(|a\rangle^{(1)}\otimes |b\rangle^{(2)}.\) For simplicity of expression, we shall omit the symbol ‘\(\otimes\)’ here.
- 10.
If we multiply a wave function by any constant factor c, where \(c \neq 0\) the resulting wave function represents the same physical state. We multiply \(\psi_{ss}\) by \(-1\) to facilitate comparison with (3.2).
- 11.
Already we see a contrast with the point of view of quantum theory, which asserts that no physical property has meaning apart from a measurement procedure. One could at this point assert that the incompleteness of quantum mechanics has been established. It is interesting to note in this connection that Einstein [26, pp. 167–168] disliked that Podolsky and Rosen had formulated incompleteness by reference to both x and p when just one quantity would suffice.
- 12.
A few things may need to be clarified. First, no dependence on \(\psi\) need be included in this formulation since we are considering a fixed wave-function, namely that of the spin-singlet state. Second, writing such a mathematical function does not in any sense constitute an additional assumption. The existence of particular values which match measurement results perfectly assures that a value map of just this form must exist.
- 13.
Readers may object that on the relevant spin space each component of the spin-\(\frac{1} {2}\) observable is non-degenerate, so that the considerations of measurement procedure play very little role. However, an observable need not be a member of a commuting family in order for a choice of measurement procedures to come into play. Even a spin-\(\frac{1} {2}\) particle exhibits contextuality and dependence of measurement result upon the choice of experimental procedure. See Sect. 2.5.
- 14.
Among the set \(\{\sigma^{(1)}_{\theta,\,\phi}\}\) it is of course, trivial to locate a pair of incompatible observables, e.g., \(\sigma^{(1)}_{x}\) and \(\sigma^{(1)}_{y}.\)
- 15.
Once again, we emphasize that this conclusion ought not be taken as the final word on the spin singlet state. It follows only if one admits the assumption of locality, which axiom is to meet its doom, once we are aware of Bell’s Theorem and its implications. The death of locality is to to discussed in the final section of the chapter.
- 16.
As everyone knows, a paradox is a self-contradiction. However, this name is not descriptive of what the analysis offers. In the situation analyzed by EPR, it happens that the conclusion runs counter to quantum mechanics. Here, quantum mechanics is essentially being used to point the road to its own limitations.
- 17.
We will use the term “the Einstein–Podolsky–Rosen Theorem” in this section, in spite of the fact that it is not a term which appears in the literature. We do so for ease of discussion, to avoid using cumbersome phrases such as “the theorem implied by the Einstein–Podolsky–Rosen paradox.” The proof that the Einstein–Podolsky–Rosen paradox is equivalent to a theorem is presented in Sect. 3.2.3.
- 18.
The latter is due to the fact that what previously served as a premise of Bell—the existence of non-contextual hidden variables—turns out to be a nonbasic premise within the complex argument.
- 19.
The large majority of experiments have dealt with a two-photon system, rather than two spins. See for example, Freedman and Clauser [27], Fry and Thompson [28], and Aspect et al. [29–31]. For such quantum systems, the most definitive theoretical analysis was given by Clauser, Horne, Holt and Shimony [32]. Here the photon polarizations play the role of the spin components. The experiment of Lamehi-Rachti, and Mittig [33] involves a pair of protons described by the spin singlet state. For a discussion of quantum nonlocality experiments, see Bell in [2, 34] and Herbert in [35].
- 20.
Bell briefly confronts the concept in [36], p. 47. He discusses the interpretation that “the world is superdeterministic.” In another work, [37, 38] Bell treats the same concept in a somewhat more mathematical and detailed manner. See also Goldstein, Norsen, Tausk, Zanghi in [39], specifically their reference to the ‘no-conspiracies’ assumption. We find the Bell term ‘superdeterminism’ more conducive to the expostulation of the principle involved.
- 21.
See for example, G.C. Ghirardi [40], pp. 243–246.
- 22.
Note that the means of such disturbance requires an inventive imagination. It seems rather odd to suppose that a signal might pass from the spin-measuring apparatus on one side of the lab to the particle on the other. However, since the question at issue is one that “shakes the foundations” of physical science, it might be reasonable to account for even such unknown and unusual hypotheses.
- 23.
In a hidden variables context, this would mean that the quantum system has “preprogrammed answers” for the measurements which will be made by the two scientists. But Bell’s theorem is derived based upon a continuously infinite set of hidden variables, and it is not clear whether or how Bell’s conclusion—empirical disagreement between the different styles of theories—can be developed from a small and finite set.
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Or perhaps we could call it “pseudo-choice” if we take seriously the position of superdeterminism.
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- 26.
Although there are some obvious restrictions set down by social norms and by biological limitations.
- 27.
Indeed, such a “martial law” precisely fixing human actions in lockstep with quantum states invokes a scenario as bizarre as any science fiction. Does this not remove the meaning most people ascribe to all human activities not just scientific, but intellectual, cultural and even social?
- 28.
- 29.
As noted above, there is another assumption behind the logic here, namely what we called the ‘no-superdeterminism’. However, this does not constitute a refutation of the argument, but only an explicit recognition of the same assumption that underlies all scientific research.
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Hemmick, D.L., Shakur, A.M. (2012). The Einstein–Podolsky–Rosen Paradox, Bell’s Theorem and Nonlocality. In: Bell's Theorem and Quantum Realism. SpringerBriefs in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23468-2_3
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