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The derivation of the PBR theorem makes an assumption that "systems that are prepared independently have independent physical states". However, it is known that it is possible to entangle two particles that have never interacted using a procedure known as entanglement swapping. This is a situation where two particles that have been prepared independently do not have independent physical states. What are the implications of this fact for the validity and scope of the PBR theorem?

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The PBR theorem is a proof by contradiction that (if its assumptions are true) you can't always have a psi-epistemic interpretation of quantum experiments. If you change the experiment, and entangle the two initial states via entanglement swapping, that doesn't alter the original argument in which you instead prepare individual states. And if there is one case that proves that a psi-epistemic interpretation impossible, that's enough for the PBR argument to go through.

Now, you may have noticed that the "^" geometry in spacetime is the same for entanglement swapping experiments as it is for the PBR theorem. (Two separate particles come together for a joint measurement.) Some people interpret entanglement-swapping experiments as indication that the eventual joint measurement must have some retrocausal influence on the initial state. If that were true, and this retrocausality also held for the similar PBR geometry, then one would doubt the validity of the assumption that the initial states were prepared independently. (The future measurement would serve to correlate those initial states.) And once you broke one of the assumptions of the PBR theorem in this manner (or in any manner), the usual PBR conclusions would no longer follow. That's one reason why physicists who like psi-epistemic explanations (including Pusey himself , the "P" in "PBR") are often intrigued by retrocausal explanations of quantum phenomena.

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"Prepared independently" means that the systems are prepared in separate labs which are isolated from the outside world (including each other). It's impossible to produce entangled systems under that constraint, whether you use entanglement swapping or not.

In quantum-circuit terms, the preparation process has to split into two independent circuits with no lines running between them. If you look at entanglement swapping circuits, such as the ones here, you'll see that they don't have that property.

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  • $\begingroup$ Impossible? Well, you could make two entangled particles in one lab, and two entangled particles in another isolated lab, independently. Then send one particle from each lab to a single joint measurement, which creates entanglement between the remaining two particles (via entanglement swapping). Those two particles can then be sent into a PBR experiment. You could say that these labs aren't isolated, but if they stayed so isolated that you couldn't do entanglement swapping then you couldn't do the PBR experiment either. $\endgroup$ Aug 28, 2023 at 3:45
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    $\begingroup$ @KenWharton "send[ing] one particle from each lab to a single joint measurement" violates independence. Also, you can't create entanglement that way—if A and B are a Bell pair and C and D are another Bell pair, there's nothing you can do to B and C alone that will make A and D into a Bell pair. That would allow for FTL communication. You have to operate on A or D using information obtained from B and C to complete the entanglement swapping protocol. $\endgroup$
    – benrg
    Aug 28, 2023 at 5:02
  • $\begingroup$ Okay, yes, depending on what precisely you mean by "entanglement swapping" (ES), that's right. You can't use ES to make one particular certain entangled state between A and D without breaking the independence between the labs. But even without that final step of the ES protocol, you can indeed make A and D "entangled", in one of 4 possible entangled states, which one determined by the outcome of the external B-C joint measurement. You can do that without breaking the independence between labs (except in the way already required for PBR). $\endgroup$ Aug 28, 2023 at 16:43

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