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TobyBartels comments on Probability is in the Mind - Less Wrong

60 Post author: Eliezer_Yudkowsky 12 March 2008 04:08AM

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Comment author: TobyBartels 23 July 2011 09:22:50AM *  1 point [-]

I wrote:

The classic example is a spin-1/2 particle that you believe to be spin-up with 50% probability and spin-down with 50% probability.

I've begun to think that this is probably not a good example.

It's mathematically simple, so it is good for working out an example explicitly to see how the formalism works. (You may also want to consider a system with two spin-1/2 particles; but that's about as complicated as you need to get.) However, it's not good philosophically, essentially since the universe consists of more than just one particle!

Mathematically, it is a fact that, if a spin-1/2 particle is entangled with anything else in the universe, then the state of the particle is mixed, even if the state of the entire universe is pure. So a mixed state for a single particle suggests nothing philosphically, since we can still believe that the universe is in a pure state, which causes no problems for MWI. Indeed, endoself immediately looks at situations where the particle is so entangled! I should have taken this as a sign that my example was not doing its job.

I still stand by my responses to endoself, as far as they go. One of the minor attractions of the Bayesian interpretation for me is that it treats the entire universe and single particles in the same way; you don't have to constantly remind yourself that the system of interest is entangled with other systems that you'd prefer to ignore, in order to correctly interpret statements about the system. But it doesn't get at the real point.

The real point is that the entire universe is in a mixed state; I need to establish this. In the Bayesian interpretation, this is certainly true (since I don't have maximal information about the universe). According to MWI, the universe is in a pure state, but we don't know which. (I assume that you, the reader, don't know which; if you do, then please tell me!) So let's suppose that |psi> and |phi> are two states that the universe might conceivably be in (and assume that they're orthogonal to keep the math simple). Then if you believe that the real state of the universe is |psi> with 50% chance and |phi> with 50% chance, then this is a very different belief than the belief that it's (|psi> + |phi>)/sqrt(2) with 50% chance and (|psi> - |phi>)/sqrt(2) with 50% chance. Yet these two different beliefs lead to identical predictions, so you're drawing a map with extra irrelevant detail. In contrast, in the fully Bayesian interpretation, these are just two different ways of describing the same map, which is completely specified upon giving the density matrix (|psi><psi| + |phi><phi|)/2.

Edit: I changed uses of ‘world’ to ‘universe’; the former should be reserved for its technical sense in the MWI.