Several years ago now a much discussed theorem in the foundations of physics was discovered by three physicists, Matthew F. Pusey, Jonathan Barrett and Terry Rudolph. I had meant to discuss their paper some time ago, and realized I couldn't find anywhere, where anyone was talking about the theorem in the context of the Consistent Histories approach. So I've decided now to correct that.

The theorem states that under certain conditions a model which affirms both that (a) there are underlying ontological states of a system and (b) the wave function is given an epistemic interpretation, cannot be consistent with quantum theory, and so must be discarded.

Right off the bat, we know the Consistent Histories approach affirms (b), so the most straight forward response could be to reject (a) and some Consistent Historians are perfectly willing to do that, e.g. Lubos Motl and Roland Omnes. As for (b) in the CH approach, the wave function is treated as "pre-probability" in the sense that, it's used to express an observers subjective knowledge of an underlying state, which they then use to calculate probability.

As far as the CH goes, (b) is non-negotiable and we know this for two reasons. The wave function describes non-commuting operators, which in this approach means it describes physical properties of a system which are incompatible, like definite momentum and position. The second is that, while in quantum theory you may obtain the wave function by integrating the Schrodinger equation, you can obtain the same probabilities using a method that makes no mention of that wave function. By integrating in a different time direction (a different wave function is obtained).

With that in mind, here's a more interesting question. Can one be a Consistent Historian and affirm both (a) and (b) without any inconsistency? In other words, can one be both a Consistent Historian and a scientific realist? I think the answer is, obviously, a yes, but we'll need to learn a bit more about the PBR theorem first.

In quantum theory having underlying ontic states but an epistemic interpretation of the wave function means you have a probability distribution over those states, which may overlap over some of those states.

The PBR result shows that for a tensor product corresponding to say, two elections, being prepared in any state, any possible outcome of measurement should have a zero probability of occurring (see the original paper for technical details). Which is clearly false. So there's a tension created between (a) and (b).

I already stated that there were conditions of the PBR theorem that needed to be assumed in order to get the argument to work, so we're going to have deny one or more of these assumptions if we want to maintain both (a) and (b). As it turns out, we don't need to postulate anything new, because the CH approach already, denies the "separability" assumption made by PBR (it was also assumed in Bell's theorem), that ontic properties are localized in spacetime.

That is to say that the properties of a quantum system may be correlated with another, very far away system. When two systems are considered, along with a measuring device there is going to be a certain correlation between them. What happens to the measured properties of P is not irrelevant if a statement about the properties of P' has already been made.

This tells us that systems with no direct interaction can still be correlated, even if they're separated at distances greater than what the speed of light can reach (in a given time). This is only true in some special cases and is determined by our preparation device. This is hardly surprising to a Consistent Historian, in this approach there is only one wave function, the wave function of the universe, that never collapses, only parts of it decohere with respect to other pats. It makes sense to talk about the properties of a whole system rather than individual parts.

As one quantum theorist put it "a wave function for several particles is not a product of the wave function for individual particles".

This isn't the only possible way of 'getting around' the PBR theorem but its the one I find most intuitive and appealing. It's compatible with relativity arguments and doesn't postulate anything like retro-causality (though a CH may avail themselves of that too). If it isn't entirely clear, try not worry, hopefully this blog post will become more understandable when I talk about 'Consistent Histories and Bell's theorem' and it might be a good idea, to come back and re-read this post at a future time.

The theorem states that under certain conditions a model which affirms both that (a) there are underlying ontological states of a system and (b) the wave function is given an epistemic interpretation, cannot be consistent with quantum theory, and so must be discarded.

Right off the bat, we know the Consistent Histories approach affirms (b), so the most straight forward response could be to reject (a) and some Consistent Historians are perfectly willing to do that, e.g. Lubos Motl and Roland Omnes. As for (b) in the CH approach, the wave function is treated as "pre-probability" in the sense that, it's used to express an observers subjective knowledge of an underlying state, which they then use to calculate probability.

As far as the CH goes, (b) is non-negotiable and we know this for two reasons. The wave function describes non-commuting operators, which in this approach means it describes physical properties of a system which are incompatible, like definite momentum and position. The second is that, while in quantum theory you may obtain the wave function by integrating the Schrodinger equation, you can obtain the same probabilities using a method that makes no mention of that wave function. By integrating in a different time direction (a different wave function is obtained).

With that in mind, here's a more interesting question. Can one be a Consistent Historian and affirm both (a) and (b) without any inconsistency? In other words, can one be both a Consistent Historian and a scientific realist? I think the answer is, obviously, a yes, but we'll need to learn a bit more about the PBR theorem first.

In quantum theory having underlying ontic states but an epistemic interpretation of the wave function means you have a probability distribution over those states, which may overlap over some of those states.

The PBR result shows that for a tensor product corresponding to say, two elections, being prepared in any state, any possible outcome of measurement should have a zero probability of occurring (see the original paper for technical details). Which is clearly false. So there's a tension created between (a) and (b).

I already stated that there were conditions of the PBR theorem that needed to be assumed in order to get the argument to work, so we're going to have deny one or more of these assumptions if we want to maintain both (a) and (b). As it turns out, we don't need to postulate anything new, because the CH approach already, denies the "separability" assumption made by PBR (it was also assumed in Bell's theorem), that ontic properties are localized in spacetime.

That is to say that the properties of a quantum system may be correlated with another, very far away system. When two systems are considered, along with a measuring device there is going to be a certain correlation between them. What happens to the measured properties of P is not irrelevant if a statement about the properties of P' has already been made.

This tells us that systems with no direct interaction can still be correlated, even if they're separated at distances greater than what the speed of light can reach (in a given time). This is only true in some special cases and is determined by our preparation device. This is hardly surprising to a Consistent Historian, in this approach there is only one wave function, the wave function of the universe, that never collapses, only parts of it decohere with respect to other pats. It makes sense to talk about the properties of a whole system rather than individual parts.

As one quantum theorist put it "a wave function for several particles is not a product of the wave function for individual particles".

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