Measurement problem: Origin of probabilities in Many-Worlds Interpretation

As far as I can tell there appears to be an active group of academics (including the likes of Sean Carrol) who believe in the Many-Worlds Interpretation of quantum mechanics, but feel that the origin of probability within the framework needs explanation. Having read some of the papers which use game theory ideas, or arguments about self locating probabilities etc. I am not entirely convinced and feel that they are often slightly circular (although I probably don't fully understand them).

All the descriptions of Many worlds I have seen are essentially the following:

We consider a microscopic system with say 2 states e.g. a spin which can be spin up $$|\uparrow\rangle$$ or down $$|\downarrow\rangle$$. We then interact this spin with a large measurement device which reads $$1$$ for spin up ($$|1\rangle$$) and 0 for spin down ($$|0\rangle$$) and we say is in some premeasurement state $$|\text{ready}\rangle$$. We can describe the measurement procedure for the coherent state $$|\uparrow\rangle+|\downarrow\rangle$$ as

$$|\text{ready}\rangle(|\uparrow\rangle+|\downarrow\rangle)\to|1\rangle|\uparrow\rangle+|0\rangle|\downarrow\rangle.$$

But then this entanglement spreads rapidly across the environment and we go from

$$|\text{environment}\rangle(|1\rangle|\uparrow\rangle+|0\rangle|\downarrow\rangle)\to|\text{environment }\alpha\rangle|1\rangle|\uparrow\rangle+|\text{universe }\beta\rangle|0\rangle|\downarrow\rangle$$

The MWI is to say that both parts of this state are equally real and that the stochastic appearance of this process to us is simply that the Hamiltonian governing the evolution of this system will never lead these two parts of the state to interfere with each other. Hence if we consider ourselves to be part of the measurement device, then having become entangled with the up part of the state we will never become mixed with the other part of the state ever again and so from our perspective the world has split into many parts.

I believe the question that these academics are addressing at this point is, given the world splits, why do we see it split with probabilities given by the Born rule.

It seems to me that this problem only arrises because of the way that the above argument is presented. In particular I am not entirely sure what the basic postulates someone like Sean Carrol think are necessary (please give reference to any good sources on this matter).

The question: What is wrong with the following argument? (If this is a pre-existing argument I would appreciate links to counter arguments)

The state of a pure system is described by a mathematical object $$W$$ (usually called a density operator) which evolves linearly in time i.e.

$$\dot W(t) = LW(t)$$

and the system can be characterised by its associated physical properties such as positions, momenta etc. which are given by an inner product in this space

$$a(t) = (A|W(t)),$$

such that W(t) is fully described by the associated complete set of physical properties $$\{a(t),b(t),\dots\}$$.

Now let us imagine that we are unsure whether the system is initially in state $$W_1$$ or state $$W_2$$ and that we are able to assign each associated probabilities $$p_1$$ and $$p_2$$. Then we can describe the state of the system as

$$W(t)=p_1W_1(t)+p_2W_2(t)$$

and we have (in perfect agreement with what we mean by probabilities) that

$$a(t) = (A|W(t)) = p_1(A|W_1(t))+p_2(A|W_2(t)) = p_1a_1(t)+p_2a_2(t).$$

Now let us consider our MWI problem, we have a system in some pure state $$W_s(0)$$ which interacts with some measurement device in a pure state $$W_m(0)$$ and the environment which all evolve as before into a new state

$$W(0)= W_e(0) \otimes W_m(0) \otimes W_s(0)\to W(t).$$

But now if an external observer considers the state of the measurement device and spin system by considering the complete set of physical properties which describe them $$\{a(t),b(t),\dots\}$$ it appears to be identical to a state for which the observer is not sure of the pure state, in which there is a 50% probability that the measurement device says $$1$$ and then spin is up and a 50% probability that the measurement device says $$0$$ and then spin is down. So I don't see what the issue is, any comments from within the many worlds perspective as to why this doesn't explain the origin of probabilities in measurement would be much appreciated

• Aren't the probabilities the same in all interpretations of QM? If so, why would proponents of MWI feel especially compelled to defend them? – D. Halsey Mar 24 at 0:19
• In particular I am not entirely sure what the basic postulates someone like Sean Carrol think are necessary (please give reference to any good sources on this matter). See preposterousuniverse.com/blog/2014/07/24/… . Basically it's just a Hilbert space and the Schrodinger equation. – Ben Crowell Mar 24 at 3:07
• I'm unclear on why you write $W=p_1W_1+p_2W_2$, where $p_i$ are the probabilities. This doesn't look consistent with the Born rule. Are your $W$'s wavefunctions? Density matrices? – Ben Crowell Mar 24 at 3:18
• @BenCrowell The $W(t)$ are what we would usually call density operators, I will edit to make this clearer. – J.L. Mar 24 at 7:13
• @ChiralAnomaly In the proposed solution I am describing what happens in precisely the "typical case" in the density operator perspective, you end up with a mixed state when only considering properties of the measurement device + spin system – J.L. Mar 24 at 7:20