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The oil bath experiments of Couder and Fort have been able to reproduce various "pilot wave like" quantum behavior on a macroscopic scale. Particularly striking is the fact that the double-slit interference behavior could be reproduced. Immediately one wonders about the possibility of realizing entanglement phenomena using these oil bath experiments. The article linked to above contains a quote that it is impossible to realize entanglement phenomena in this sort of experiment because a higher dimensional system would be needed to exhibit these phenomena.

Question: Is it theoretically impossible to realize entanglement-like phenomena (e.g. non-local behavior or violation of some sort of Bell inequality) using a Couder-Fort experiment? What are the details of this impossibility claim?

Note that a recent paper further reinforces the claim that the oil bath experiments are closely analogous to quantum mechanics. Violation of Bell inequalities does not appear in this paper, though.

EDIT: To clear up any misunderstanding, I am trying hard here not to make the ridiculous claim that a classical system should violate the Bell inequalities. I am aware that looking at the phase space of a classical system as an underlying space we can only get classical correlations and these must obey the Bell inequalities. I suppose the sharper question I should ask is the following:

Refined Question: Where does the mathematical analogy between the DeBroglie-Bohm pilot wave theory and the mathematical model of the oil bath experiment break down?

If the analogy is perfect, then we should be able to interpret the oil bath experiment mathematically as a non-local hidden variable theory. Such a theory should violate some sort of analogue of Bell's theorem, shouldn't it? The original Bell inequality was perfectly equivalent to an inequality in classical probability, and so I don't see how this is exclusively tied to the dimension of the phase space.

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    $\begingroup$ The mathematical analogy is between a two dimensional Hilbert space and a two dimensional vibrating oil surface. It might be possible to instead make an analogy between the 2d oil surface and two entangled particles each moving in one dimension. This would then violate Bell's inequalities in just the same way as quantum physics. However, since it would be a different analogy, I've no idea if it's possible. $\endgroup$ – Nathaniel Feb 28 '14 at 15:10
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I think that 't Hooft's ideas about superdeterminism and Bell's theorem are relevant to this topic. If the universe is superdeterministic so that all experiments are determined by initial conditions, then the contra-factual arguments that lead to the Bell non-locality conclusions are ruled out. The universe only plays once - is how some have put this. In the early days, this was called "conspiracy against the experimenter", and it was categorically excluded from the discussions. It still remains the biggest loophole, and I don't think it can ever be eliminated. But it means that strict free will doesn't exist. The fact is that both Alice and Bob consist of a finite number of atoms each, and are certainly quantum mechanical systems themselves. Consider the bouncing drop experiments. It's surely a deterministic classical system to a high degree of accuracy. To mimic quantum mechanics, we must not only have bouncing drops that are entangled, we must also have measuring apparatus and observers which consist only of bouncing drops. We are super-observers of the drop motions, but we are not built up of bouncing drops. We are allowed to measure the drops without disturbing them, but "embedded observers" made up entirely of bouncing drops may not be able to measure things as finely as we can. To those embedded observers, the universe of the bouncing drops might appear as undeniably entangled. So there is hope for the bouncing drops as an example of 't Hooft's superdeterministic realization of quantum mechanics as an emergent phenomena. I have no idea whether the classical lagranians for the bouncing drops will allow this behavior, but I see it as the only hope for a description of entanglement in this classical framework.

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Couder and Fort's experiment is based on a mathematical analogy between the Hilbert space of a particle moving in two dimensions, and the two surface of a vibrating oil bath, which interacts with an oil droplet bouncing on top of it.

Naïvely, one might try to extend this analogy a two-particle system by having two oil droplets bouncing on a single two-dimensional oil surface. It would of course bt impossible to implement Bell inequality violation in this type of experiment, because of Bell's theorem. Bouncing oil droplets are a macroscopic, and therefore classical, phenomenon, and therefore the two oil droplets could only have classical correlations. (Of course one could try to use very small droplets, such that quantum effects become important, but then it becomes a different type of experiment.)

More specifically, this analogy would break down because the Hilbert space of two particles moving in two dimensions is four-dimensional. So the correct analogy for a two-dimensional two-particle system is not two droplets bouncing on a two-dimensional surface but a single droplet bouncing on a four-dimensional surface. This obviously couldn't be implemented in the laboratory.

However, it might be possible to emulate the behaviour of two entangled particles, each moving in one dimension, since the Hilbert space of such a system is two-dimensional. Then the oil drop's $x$ coordinate would correspond to the position of one particle, and its $y$ coordinate to the other. Interpreted this way, if this works, it should emulate a violation Bell's inequalities in just the same way that quantum mechanical systems do. (Note that of course it doesn't actually violate Bell's inequalities, which it can't do, because it's still a classical system.)

Since this would be a different analogy I've no idea whether it's possible. My suspicion is that either it would be exactly the same experiment but just interpreted differently, or else there would be some fundamental reason why it can't work. I don't know which of these is the case, but it's an interesting thing to think about.

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    $\begingroup$ "the Hilbert space of two particles moving in two dimensions is four-dimensional"...no. Even the Hilbert space of a single particle moving in one dimension is infinite-dimensional, since it is the space of square-integrable functions on the real line $L^2(\mathbb{R})$. What you probably mean is that the surface of the oil forms (a part of) $\mathbb{R}^2$, and so the wave-structures on it are in (a part of) $L^2(\mathbb{R}^2$. But this would be the case for every excitation of a surface, and your dimensions make no sense at the latest when you start to talk about the entangled particles. $\endgroup$ – ACuriousMind Apr 16 '15 at 17:53
  • $\begingroup$ @ACuriousMind you're right of course, I'm not sure what I was thinking. Well, I suppose what I meant was, if I have two $n$-state quantum systems I need an $n^2$-dimensional Hilbert space to describe it, not a $2n$-dimensional one, and if I take the infinite limit it kind of looks like going from 1D to 2D, or 2D to 4D. But you're right that "the Hilbert space of two particles moving in two dimensions is four-dimensional" is nonsense. I may or may not correct this answer at some point. $\endgroup$ – Nathaniel Feb 13 '18 at 12:57
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Concerning the question: "Is it theoretically impossible to realize entanglement-like phenomena (e.g. non-local behavior or violation of some sort of Bell inequality) using a Couder-Fort experiment?", I have recently discussed this with John Bush from MIT, one of the experts of these experiments. I believe it is possible that a Bell inequality can indeed be violated in such experiments (I think that John agrees, but I do not wish to speak in his name). The reason is that one of the premises of Bell's theorem (measurement independence or MI) is not valid in such experiments. See following article: http://arxiv.org/abs/1406.0901 (published in Found. of Physics, Volume 46, Issue 4, pp 458-472, 2016). If a background field is present (the pilot wave in the droplet experiments) MI can be violated, giving rise to nonlocal-looking correlations. In other words, background-based hidden-variable theories can, after all, violate a Bell inequality, even while being compatible with free will and locality. The background field such theories invoke is surely 'extended' as any field is, but it is NOT nonlocal in the important sense, namely 'superluminal'.

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  • $\begingroup$ Dear LouisV: Are you in any way associated with the author of the link? For your information, Physics.SE has a policy that it is OK to cite oneself, but it should be stated clearly and explicitly in the answer itself, not in attached links. $\endgroup$ – Qmechanic Sep 18 '16 at 16:26
  • $\begingroup$ Dear Qmechanic, yes, I am the author of the link. $\endgroup$ – LouisV Nov 18 '16 at 20:12

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