Abstract
We explain first the meaning of nonlocality and then show that a combination of the 1935 argument of Einstein, Podolsky and Rosen with the 1964 argument of Bell establishes the existence of nonlocal causal connections in the world.
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Notes
- 1.
We will discuss some of them, as well as the historical controversies and misunderstandings about Einstein , Podolsky , and Rosen in Sect. 7.1.
- 2.
Such as the value of the momentum in the example given above, or the value of the spin in the examples discussed in Sect. 4.4. Of course, we know, because of the theorem in Sect. 2.5 that it is not easy to introduce hidden variables. But we will re-examine here the issue of hidden variables from a different perspective to the one in Chap. 2.
- 3.
Our goal in this book is to present ideas in their simplest form, and we will do that also in this chapter. But there are several caveats to be made: first, while we discuss below particles with spin, actual experiments are made with photons , with their polarization playing the role of spin. Moreover, we will base our arguments in Sect. 4.4 on some perfect correlations predicted by quantum mechanics. But the actual experiments do not test those correlations directly. Instead, they are based on the Clauser–Horne –Shimony –Holt (CHSH) inequality [95]. We refer to [229] for a discussion of various forms of the Bell and CHSH theorems and the relation between them. In any case, the logic of our arguments always takes for granted the empirical correctness of quantum mechanical predictions, which have been verified to great accuracy in all experiments, even if the tests were made in somewhat different circumstances than the ones considered here. See [19–21] for some of the original experiments and [229] for references to later ones. Several possible loopholes are closed in [262].
- 4.
We note that Bell made no assumption whatsoever about “realism ” or “determinism”. We mention this here because there is an enormous amount of confusion in the literature based on the supposition that Bell makes such assumptions. The reader unfamiliar with the literature on Bell’s theorem may simply ignore this remark. Besides, those confusions will be discussed later, in Sect. 7.5.
- 5.
For discussions of Bell’s results similar to the one in this chapter, see, e.g., [10, 211, 229, 319, 327, 355–358, 459, 460, 478].
- 6.
We base ourselves in this section on [355], where the description of the experiment is due to de Broglie [118, 119]. Einstein’s original idea was expressed in a letter to Schrödinger , written on 19 June 1935, soon after the EPR paper was published [186, p. 35]. However, Einstein formulated his “boxes” argument in terms of macroscopic objects (small balls) and then gave a slightly different and genuinely quantum mechanical example, but illustrating the same point as the one made here. Figure 4.1 is taken from [355]. The “boxes” argument is also mentioned by Deltete and Guy [120] in a discussion of Einstein’s objections to the quantum orthodoxy. A somewhat similar experiment to the one with the boxes was done by A. Ádám, L. Jánossy, P. Varga, in 1955 [1]; we will return to it in Sect. 7.1.5.
- 7.
The precise distribution does not matter, provided it is spread over \(B_1\) and \(B_2\) defined below; it could be distributed according to the square of the ground state wave function of a particle in the box B.
- 8.
This is of course a thought experiment. We assume that we can cut the box in two without affecting the particle. But similar experiments have been made with photons and were already suggested by Heisenberg in 1929 (see Sect. 7.1.5).
- 9.
If we refer to the four possible positions mentioned in Sect. 2.5 concerning the status of the quantum state, then viewing the quantum state as reflecting our knowledge of the system is part of the third reaction, while thinking of the quantum state as being physical and of the position of the particle as being created when one measures it is part of the second reaction. In Chap. 2, we showed that the third reaction is untenable if it applies to all “observables ”, but not necessarily if it applies only to particle positions (in Chap. 5, we will see how a position can be attributed to particles independently of measurements and in a consistent way). We also showed that the second reaction is untenable because of the linearity of Schrödinger’s equation , unless one appeals to a non-physical agent.
- 10.
This example is given by Bell in his interview with Jeremy Bernstein [56, p. 63].
- 11.
We should stress (for the experts) that our notion of locality is not the same as what is sometimes called locality or local commutativity in quantum field theory . See [48, 229] for a discussion of this point.
- 12.
See Sect. 6.2 for examples of theories, different from the de Broglie–Bohm theory, in which there is indeed such a nonlocal creation of particles.
- 13.
Of course, instantaneity is not a relativistic notion, so let us say, instantaneous in the reference frame where both boxes are at rest. We will discuss relativity in Sect. 5.2.2 (when we speak of relativity in this book, we always refer to the special theory, unless we mention the general one). For the conflict between relativity and ordinary quantum mechanics in the example of the boxes, see [355, Note 58].
- 14.
Newton thought that gravitation was mediated by particles moving at a finite speed, so that the effect of gravitation could not be instantaneous . See [327] for more details.
- 15.
This is related to the phenomenon of decoherence , to be discussed in Sect. 5.1.6.
- 16.
- 17.
Readers who prefer to see the real physical situation directly can proceed to the next section.
- 18.
In the reference frame in which the experiment takes place.
- 19.
Someone who certainly thinks that the answers are predetermined is Robert Griffiths , who offers the following analogy to illustrate the situation described here:
Colored slips of paper, one red and one green, are placed in two opaque envelopes, which are then mailed to scientists in Atlanta and Boston. The scientist who opens the envelope in Atlanta and finds a red slip of paper can immediately infer, given the experimental protocol, the color of the slip of paper contained in the envelope in Boston, whether or not it has already been opened.
Robert B. Griffiths [249]
We will discuss Griffiths’ views about quantum mechanics in Sect. 6.3.
- 20.
One of the weaknesses of the original EPR paper was that they considered two quantities, position and momentum , instead of one, which would have been sufficient for their argument to work: if one can predict the momentum of particle A by measuring that of particle B, far away from A, and if that measurement does not affect particle A (by assumption of locality), then particle A must have had a well defined momentum all along. The same argument holds for the position: if the two particles have opposite momenta and start from the same place, then measuring the position of one particle allows us to infer the position of the other and therefore, assuming once again no effect on A from the measurement on B, this means that particle A had a position all along. But by considering both position and momentum , Einstein , Podolsky , and Rosen may have given the impression that they were trying to prove that one could measure these two quantities simultaneously , which was not their point, at least not Einstein’s point. We will discuss this further in Sect. 7.1.
- 21.
The argument given here is taken from [149]. In the original paper by Bell [35], the proof, although fundamentally the same, was more complicated. There has also been quite some discussion in the literature about the difference between outcome independence and parameter independence, but our approach bypasses that distinction (see [229, 319] for a critical discussion of these notions).
- 22.
In [332], Mermin provides nice idealized illustrations of possible results of experiments showing the violation of Bell’s inequality, in the version given here.
- 23.
This formulation of the EPR argument, with spin variables instead of position and momentum , is due to Bohm [61] and was used by Bell in [35] and later. However, there exists an unpublished note by Einstein discussing the problem in terms of spin variables [432].
- 24.
See also Appendix 2.B for more details.
- 25.
This follows from the rotational invariance of the state (4.4.2.1) in spin space.
- 26.
This is somewhat similar, but not identical, to the contradiction derived in Appendix 2.F. See [240], [335, Sect. VIII], or [70, p. 146], for more details.
- 27.
For a general proof of the impossibility of using EPR-type experiments to send messages , see [37, 158, 208], [70, p. 139], or [319, Chap. 4].
- 28.
Another suggestion sometimes made is that the ordinary rules of probability do not apply in the EPR –Bell situation. But, as pointed out by Tumulka [478], since the reasoning here relies only on frequencies of results of experiments, and since the latter obviously do satisfy the ordinary rules of probability, this attempt to “save locality”, by trying to deny the implications of the EPR–Bell argument, does not work.
- 29.
Another reaction which, in a sense, also avoids the problem, is the one proposed by Gisin in [214]: he does emphasize that Nature is nonlocal, but he attributes the nonlocal correlations to “pure chance” and thinks that this proves the non-deterministic nature of the Universe . But since there exists a nonlocal deterministic theory, the de Broglie–Bohm theory, that accounts for these nonlocal effects, one cannot use them to prove that a deterministic account is impossible (see Chap. 5 and [313].). One may “like” the de Broglie–Bohm theory or not, but one cannot deny its existence.
- 30.
See the book on relativity and nonlocality by Maudlin [319] for a detailed discussion of the differences between messages and information and of what exactly is compatible or not with relativity .
- 31.
One can show that if a spy tries to intercept the particles being transmitted from the source to Alice and Bob , in order to know which results Alice and Bob will get when they do their measurements, then because of the collapse rule, that interception will necessarily have effects on those results such that Alice and Bob can detect the presence of the spy. In that sense, quantum cryptography is foolproof.
- 32.
See, e.g., Nielsen and Chuang [352] or Preskill [404].
- 33.
See, for example, the quote by Pauli in Sect. 1.5.
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Appendices
Appendix
4.A The Frequency of “Answers” to Different “Questions”
Here we derive the number 1/4 for the frequency of answers to different questions, for an appropriate choice of the directions 1, 2, 3 . Compute first \( \mathbf{E}_{ \mathbf{a}, \mathbf{b} } \equiv \langle \Psi |\sigma ^A_\mathbf{a} \otimes \sigma ^B_\mathbf{b}|\Psi \rangle \), where \(\mathbf{a}\), \(\mathbf b\) are unit vectors in the directions (1, 2, or 3, specified below) in which the spin is measured at A or B, and \(\sigma ^A_\mathbf{a} \otimes \sigma ^B_\mathbf{b}\) is a tensor product of matrices , each one acting on the A or B part of the quantum state (with \(\sigma _\mathbf{a}= a_1 \sigma _1+ a_2 \sigma _2 +a_3 \sigma _3\), where, for \(i= 1, 2, 3\), \(a_i\) are the components of \(\mathbf{a}\) and \(\sigma _i\) the usual Pauli matrices introduced in Appendix 2.F). The quantity \( \mathbf{E}_{ \mathbf{a}, \mathbf{b} }\) is bilinear in \(\mathbf{a}\), \(\mathbf b\) and rotation invariant, so it must be of the form \(\lambda \mathbf{a} \cdot \mathbf b\), for some \(\lambda \in \mathbf R\).
For \(\mathbf{a} = \mathbf b\), the result must be \(-1\), because of the anti-correlations (if the spin is up at A, it must be down at B and vice versa). So \(\lambda =-1\), and thus \( \mathbf{E}_{ \mathbf{a}, \mathbf{b} }= -\cos \theta \), where \(\theta \) is the angle between the directions \(\mathbf{a}\) and \(\mathbf b\). We know that \(v_A (\mathbf{a}), v_B (\mathbf{b} )= \pm 1\). Thus,
and
One then chooses the directions
Since \(\cos 120=\cos 240= -1/2\), we get \(P\big (v_A (\mathbf{a})=- v_B(\mathbf{b})\big )=1/4\). Thus we have perfect anticorrelations only 1/4 of the time when \(\mathbf{a}\) and \(\mathbf{b}\) are different. With our convention, this means that one gets the same answer when one asks different questions on both sides only 1/4 of the time.
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Bricmont, J. (2016). The Second Mystery: Nonlocality. In: Making Sense of Quantum Mechanics. Springer, Cham. https://doi.org/10.1007/978-3-319-25889-8_4
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