Bohmian loophole in PBR-like theorems

Question

I'm reviving and expanding this question, because of the new paper today, by Aaronson et al. The more general question is:

How does quantum-potential Bohmian mechanics relate to no-go theorems for psi-epistemic theories?

According to Bohmian mechanics, there is a wavefunction as in conventional quantum mechanics, which evolves according to a Schrödinger equation, and then a set of classical degrees of freedom, which evolve according to the gradient of the phase of this wavefunction. A Born-rule probability distribution for the classical degrees of freedom is preserved by this law of motion, so Bohmian mechanics provides a deterministic hidden-variables theory, albeit a nonlocal one.

However, the equations of motion for the classical variables can be rewritten in terms of two potentials, the local potential appearing in the Schrödinger equation, and a nonlocal "quantum potential" which depends on the form of the wavefunction. This means that Bohmian mechanics for any given wavefunction can be replaced by a nonlocal deterministic theory in which there is no wavefunction. (Note that the quantum potential will be time-dependent unless the wavefunction was an energy eigenstate; thanks to Tatiana Seletskaia for emphasizing this point.)

Meanwhile, there has recently been theoretical interest in ruling out so-called "psi-epistemic" hidden-variables theories, in which the quantum wavefunction is not part of the ontology. Obviously, "quantum-potential Bohmian mechanics" is a psi-epistemic theory. It ought to be relevant to this enterprise. But it may require some mental gymnastics to bring it into contact with the PBR-like theorems, because the mapping from quantum mechanics to these quantum-potential theories is one-to-many: scenarios with the same quantum equations of motion, but different initial conditions for the wavefunction, correspond to different equations of motion in a quantum-potential theory.

NOTE 1: Empirically we only have one world to account for, so a real-life quantum-potential theorist in principle shouldn't even need their formalism to match the whole of some quantum theory. Quantum cosmology only needs one wavefunction of the universe, and so such a theorist would only need to consider one "cosmic quantum potential" in order to define their theory.

NOTE 2: Here is the original version of this question, that I asked in January:

The "PBR theorem" (Pusey-Barrett-Rudolph) purports to show that you can't reproduce the predictions of quantum mechanics without supposing that wavefunctions are real. But it always seemed obvious to me that this was wrong, because you can rewrite Bohmian mechanics so that there's no "pilot wave" - just rewrite the equation of motion for the Bohmian particles, so that the influence coming from the pilot wave is reproduced by a nonlocal potential. Can someone explain how it is that PBR's deduction overlooks this possibility?

Answer 1

I once watched an online seminar by Robert Spekkens, who said something along the lines that no-go theorems are interesting, because they put constraints on what an epistemic interpretation of quantum mechanics can look like. Any no-go theorem makes a certain set of assumptions, and if the theorem is correct, we know that we must avoid at least one of those assumptions if we're to make a successful theory.

The Pusey-Barrett-Rudolph paper spells out some of its assumptions. (They do it most explicitly in the concluding paragraphs.) There may well be additional unmentioned assumptions (e.g. causality), but the ones they specifically mention are:

• There is an objective physical state $\lambda$ for any quantum system

• There exists some $\lambda'$ that can be shared between some pair of distinct quantum states $\psi_1$ and $\psi_2$. That is, $p(\lambda=\lambda' | \psi=\psi_1)$ and $p(\lambda=\lambda' | \psi=\psi_2)$ are both non-zero. (This is what it means for an interpretation to be epistemic, according to Spekkens' definition.)

• The outcomes of measurements depend only on $\lambda$ and the settings of the measurement apparatus (though there can be stochasticity as well)

• Spatially separated systems prepared independently have separate and independent $\lambda$'s.

It is from these that they derive a contradiction. Any theory that fails to make all of these same assumptions is unaffected by their result.

I'm fairly sure that under the standard formulation of Bohmian mechanics violates the second one, i.e. Bohmian mechanics is not an epistemic theory in the sense of Spekkens. This is because in Bohmian mechanics the physical state $\lambda$ consists of both the particle's real position and the "quantum potential", and the latter is in a one-to-one relationship with the quantum state.

If I understand correctly, you're suggesting that you could instead think of the physical state, $\lambda$, as consisting only of the positions and velocities of the particles, with the non-local potential being considered part of the equations of motion instead. But in this case, the third assumption above is violated, because you still need to know the potential, in addition to $\lambda$, in order to predict the outcomes of measurements. Since this formation would violate the assumptions of Pusey, Barrett and Rudolph's argument, their result would not apply to it.

In your updated question, you clarify that your suggestion is to fix the wavefunction to a particular value. In that case it's surely true that Bohmian mechanics reduces to particles moving according to deterministic but non-local equations of motion. But then you have only a partial model of quantum mechanics, because you can no longer say anything about what happens if you change the wavefunction. My strong suspicion is that if you take this approach, you will end up with an epistemic model, but it will be a model of only a restricted subset of quantum mechanics, and this will result in Pusey et al.'s result not being applicable.

Answer 2

In mathematical terms, the PBR theorem shows that the relation between the states of a more fundamental theory and the states of quantum mechanics (wave functions) is functional (i.e., to every state of the more fundamental theory, there belongs at most one wave function; the relation does of course not have to be onto or one-to-one).

If this is enough for you to say "wavefunctions are real" (your quote), then so be it. In general, however, this conclusion cannot "be obtained" (by arguments) from the result mentioned in the first paragraph. This is why one uses a language (psi-epistemic, etc.) which codifies exactly the functional dependency as mentioned above. In my view it is a mistake to take the exclusion of a non-functional relation between the states of a more fundamental theory and wave functions to be a sign that "wavefunctions are real".

Finally, let me just note that a state in Bohmian mechanics consists of the pair (wave-function, positions of particles). Therefore, the relation between the states in this theory and the states in quantum mechanics is certainly functional (projection on the first component if you wish). - As expected no problem and no implications here.

Best, J.K.

Ps: Please point out if there is an error here! Pps: Put in the above way, the PBR theorem is almost "trivially true".

Answer 3

Unfortunately I don't have the points to leave comments on other answers, as Nathaniel is correct, but since no answer has been accepted, here is the answer to your question with a citation to back it up:

You've made an incorrect assumption.

Meanwhile, there has recently been theoretical interest in ruling out so-called "psi-epistemic" hidden-variables theories, in which the quantum wavefunction is not part of the ontology. Obviously, "quantum-potential Bohmian mechanics" is a psi-epistemic theory. It ought to be relevant to this enterprise.

Bohmian mechanics is not ψ-epistemic, it is ψ-supplemented (a subset of ψ-ontic). See http://dx.doi.org/10.1007/s10701-009-9347-0