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It has been demonstrated (physically and mathematically) that Bohmian mechanics (pilot waves) produce the same statistical results for the following phenomena:

Double slit banding

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Tunneling probability on thin barriers

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Particle location probability distribution

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It is said, however, that entanglement cannot be explained with pilot waves (https://www.youtube.com/watch?v=r0plv_nIzsQ).

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I fail to see why this is so? Entangled particles have to originate at the same proximity at some point. They might separated from each other , but that also takes time and within bounds of the speed of light limit. What experiment result has proved that pairs of entangled particles cannot already have predetermined states as they take leave of one another? After all, there is no way to turn back time and repeat the exact same measurement many times to prove beyond doubt that the result is random but correlated over very large distances?

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  • $\begingroup$ Nice question. I have a few comments. First i essentially disagree with the claim (see however the next point). I'm afraid a YouTube video is not the best source of science. My point is due to the fact that pilot wave theory is the same as quantum theory for say, positional degrees of freedom. It should however be said that it's been hard to extend pilotwave theory to spin degrees of freedom. Which is the language which is usually used to speak about entanglement (though of course not necessarily) $\endgroup$
    – lcv
    Commented Nov 2, 2022 at 19:45
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    $\begingroup$ The videos is about simulating entanglement with pilot waves in water, not about entanglement with pilot waves in QM. $\endgroup$
    – Mauricio
    Commented Nov 2, 2022 at 20:02
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    $\begingroup$ In Bohmian mechanics, the wave is a wave in configuration space, not real space. This is a key point where it parts ways from the fluid analogy and this is also the reason it can exhibit entanglement. (Actually, even in ordinary QM the wavefunction is a function on configuration space as well.) $\endgroup$
    – Maximal Ideal
    Commented Nov 2, 2022 at 21:06

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I think you might have misinterpreted the video. Towards the end, it says that the water droplet experiments which (many say) resemble Bohmian Mechanics cannot demonstrate entanglement. It wasn't talking about Bohmian Mechanics itself.

Bohmian Mechanics has no issue explaining entanglement. One simple way to see this - it is a short proof that position measurements in Bohmian Mechanics will always agree with standard QM. And to measure spin, using e.g. a Stern Gerlach apparatus, the spin is inferred from the position of the particle - spin up/down correspond to if the trajectory skewed up or down. So it is clear that any such experiments would match regular QM.

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  • $\begingroup$ So, as Bohmian Mechanics is essentially deterministic, what would be the reasoning for entangled particles here? That unbeknownest to us, the correlated positional difference between the pair of particles have all along determined the measured results, meaning that the complementary spin has existed all along from the moment the pair of particles are created? $\endgroup$
    – James
    Commented Nov 2, 2022 at 20:13
  • $\begingroup$ In Bohmian Mechanics there is explicit nonlocal influence. The position of each particle gets plugged into the equation for the velocity of every other particle, regardless of the separation between their coordinates. This makes it very much possible to have correlations between measurement results across large distances over small times. $\endgroup$
    – don't train ai on me
    Commented Nov 2, 2022 at 20:59
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Consider two electrons in an entangled state such as this one: $$|\psi\rangle = \frac{1}{\sqrt{2}}(|p_x,\uparrow\rangle |-p_x,\downarrow\rangle - |p_x,\downarrow\rangle |-p_x,\uparrow\rangle)$$ That is, the electrons leave a source with opposite momenta $p_x,-p_x$, and when the particle traveling "left" (with negative momentum in the x direction $-p_x$) has spin up ($\uparrow$), then the particle travelling "right" (with positive x-momentum $p_x$) has spin down ($\downarrow$). Similarly when the particle traveling left has spin down, the particle traveling right has spin up. Easy peasy, right?

Now imagine you put two Stern-Gerlach magnets in oposite directions in the way of the beams of electrons emerging from the experiment. Stern-Gerlach magnets scatter the electron up if the spin is up, and down if the spin is down. In quantum mechanics this is straightforward, this is a measurement that will yield probabilities given by the Born rule. We just need to know one particle goes up, the other one goes down. It turns out like this no matter in which order the particles enter the magnets and undergo the interaction/measurement. This applies always and is a tested prediction of quantum mechanics.

However, in Bohmian mechanics there is an issue. There one tries to construct a piloting wave that interacts with the Stern-Gerlach magnet and then guides the particle. However, it turns out that one cannot make the guiding wave only a function of the position of just one of the two electrons!

To understand this, imagine the Stern-Gerlach magnet on the left is a bit further so that the electron traveling with momentum $p_x$ enters the Stern-Gerlach device on the right earlier. It goes "up" or "down" as a random process as determined by the potential generated by the guiding wave. But what about the electron on the left? It has to go in the opposite direction once entering the Stern-Gerlach magnet! Its guiding wave has been determined by the fate of the electron on the right. This turns out to be an instantaneous effect that travels faster than light.

Without going into technical details, this is the property that breaks the whole idea of Bohmian mechanics of constructing a nice local potential that guides your particle through nice local interactions. When entanglement is introduced, there necessarily has to be "spooky action at a distance" in the equations of Bohmian mechanics that is about as weird as quantum mechanics on its own. (This is also the reason why the droplet analogue of Bohmian mechanics will never be able to describe entanglement.)

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  • $\begingroup$ "However, it turns out that one cannot make the guiding wave only a function of the position of just one of the two electrons!" Yes, this is correct, but it is always like this in BM. Generally in an entangled state the velocity of each particle is a function of the positions of each of the others. This allows for non-local influence. But nonetheless the theory predicts the correct experimental results, so OP's question is answered in the positive. If your criteria for a plausible theory has "locality" as a must, then you can cross BM off the list. But nonetheless explain experiments it does. $\endgroup$
    – don't train ai on me
    Commented Nov 2, 2022 at 21:06
  • $\begingroup$ @doublefelix Yes, I agree that a formal extension of Bohmian mechanics can be constructed for entangled particles. However, the original idea of a "pilot wave" that emerges from local interactions and then guides the particles through local potential-like interactions is simply not realizable when entanglement comes into the picture. I argue that to answer OP's question, as it stands, one has to be clear on this point! $\endgroup$
    – Void
    Commented Nov 2, 2022 at 21:27
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    $\begingroup$ @Void How can we be certain that the spin is not decided all along, when the paired particles are created in the beginning? In Bohmian mechanics, our incomplete knowledge of the precise position of the particle causes the banding pattern observed (due to extreme sensitivity of final deterministic path to its initial location). Are there experiment results that have proved that the spin of entangled particles can't have been decided in the beginning, and we just have incomplete knowledge of it? After all, there is no way to turn back time and repeat the exact measurement many times to prove it. $\endgroup$
    – James
    Commented Nov 3, 2022 at 2:43
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    $\begingroup$ @James The essential assumption is that one is able to repeat experiments over and over while preparing the same initial state/setup over and over. One can then decide to switch the measurement devices or perform series of experiments and collect the statistics. Ultimately, if the spins had been decided at creation, the statistics of the series of measurements would have to fulfill some of the Bell inequalities. But it has been shown robustly that nature violates these inequalities (see this years Nobel prize in Physics :]). $\endgroup$
    – Void
    Commented Nov 3, 2022 at 11:37
  • $\begingroup$ @Void i posted a follow up question on this physics.stackexchange.com/questions/735097/… $\endgroup$
    – James
    Commented Nov 3, 2022 at 17:59

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