Does the Many Worlds interpretation trivialize the Born rule?

Question

I'm thinking that the Many worlds interpretation trivializes the Born rule. Suppose we take an electron's spin's state vector such that the probability of spin up is 0.6 and that of spin down is 0.4. We take 100 such identical systems and perform measurements on them one after one.

According to the Many worlds interpretation, a total of $2^{100}$ worlds will be created, each corresponding to the decohered branches of the wavefunction. We can draw it like a tree where there are $2^{100}$ child nodes. Each of these nodes has a unique history leading up to the parent.

But, which of these histories follow the Born rule? There would inevitably be some histories for which the number of Spin up measurements was $60$, and the number of Spin down measurements was $40$. The number of these histories will be $C(100, 60)$

These particular histories trivially just happen to follow the Born rule. There is no preference given to the Born rule in the wavefunction evolution. This would mean that the Born rule is not really a law of physics. We just happen to belong to one of the histories which has followed the Born rule so far. Does the Many worlds interpretation really say this?

Answer 1

"According to the Many worlds interpretation, a total of $2^{100}$ worlds will be created, each corresponding to the decohered branches of the wavefunction."

That's a simplification. In the Many Worlds interpretation, when the observer and the electron interact, they both evolve jointly into a superposition of orthogonal states. So you don't just have to count all the possible outcomes for the electron state, but also all the possible outcome states for all the $>10^{23}$ atoms of the observer. So say each observation results in a split into $10^{23}$ joint electron-observer states, in about 60% of them the electron spin is 'up', and 40% of them the electron spin is 'down'. Then subsequent observations split the cases further, into $(10^{23})^{100}$ cases. But the distribution of the total number of 'up' observations among those states has a peak around 60. The vast majority of worlds follow the Born rule, only a much smaller number (but still very big) don't.

When we decompose a state $s$ into a sum of orthogonal states $s_1+s_2+s_3+\ldots$, their magnitudes combine like orthogonal vectors. $|s|^2=|s_1|^2+|s_2|^2+|s_3|^2+\ldots$. If each individual substate is assumed to have roughly equal magnitude, then the number of joint substates divides in proportion to the squared magnitudes of the states being observed.

Followers of MWI believe that, unlike other interpretations, the Born rule can be explained from more basic principles, rather than having to be assumed as an axiom. But the details of exactly how it works are widely considered to be still a work in progress. Proposals are somewhat fuzzy and hand-wavy, and still controversial.

Everett argued in his thesis that it had been explained, although his argument amounts to saying that because squared magnitudes were the only measure on state space that could conserve probability, it was the appropriate one to use. (To comment on what MWI says or doesn't say, it's well worth reading his thesis first.) You can argue about whether or not he was right, but MWI certainly doesn't propose a trivial "we just happen to belong to one of the histories which has followed the Born rule so far" explanation of the Born rule.

Answer 2

The Born rule states that if you have a state $|\psi\rangle$ and you measure an observable with eigenstates $|i\rangle$, the probability of getting the $j$th state is $|\langle j|\psi\rangle|^2$. Regardless of what interpretation of quantum theory you adopt, that statement is entirely consistent with getting a set of observations whose relative frequencies don't match the Born rule probability, just as it is possible to flip a fair coin and get 100 heads in a row.

In most intepretations of quantum theory, the Born rule is treated as a postulate. In the MWI the Born rule has been explained in a couple of different ways.

One explanation uses decision theory. The probability of an outcome is a function of the state that satisfies particular properties required by decision theory, like if a person uses those probabilities you can't make him take a series of bets that will make him lower the expectation value of his winnings, the probability of an outcome can't be changed by a later measurement and some other assumptions:

https://arxiv.org/abs/0906.2718

https://arxiv.org/abs/quant-ph/0303050

https://arxiv.org/abs/quant-ph/9906015

https://arxiv.org/abs/1508.02048

There is another approach to deriving the probability rule called envariance, which uses properties of the state of a system that remain unchanged after interaction with the environment:

https://arxiv.org/abs/quant-ph/0405161

Answer 3

The Born rule relates quantum amplitudes (complex numbers in the closed unit disc) to classical probabilities (real numbers in $[0,1]$). That doesn't actually help with interpretation, because the meaning of classical probabilities is no clearer than the meaning of quantum amplitudes.

What does it mean to say that a coin has a 0.6 probability of coming up heads? It doesn't mean that if you flip it 100 times you'll get 60 heads. According to classical probability calculus, the chance of getting 60 heads in that experiment is just $0.6^{60} 0.4^{40} C(100,60) \approx 0.08$. Incidentally, what does that mean? It doesn't mean that if you repeat the 100-flip experiment 100 times, you'll get 60 heads eight times.

You can prove that in the limit of infinitely many tosses, the ratio of heads to tosses approaches a limit of 0.6 – or more precisely, that the probability that it doesn't approach that limit is exactly 0. You can prove the same thing about amplitudes of worlds in the many-worlds picture, without any reference to classical probabilities or the Born rule. And that's about the best you can do.

In practice, we deal with this by assuming that sufficiently unlikely scenarios don't happen – for example, that the very large amount of evidence we've gathered in favor of the existence of electrons is due to electrons actually existing, and not to a statistical fluke, even though the chance it's a fluke isn't precisely zero. For that purpose, you can equally well assume that the classical probability isn't within $\epsilon$ of zero or that the quantum amplitude isn't within $\sqrt\epsilon$ of zero.

I'm not suggesting that this is an acceptable state of affairs, only that the Born rule doesn't make it better. There's no evidence that classical probability underlies quantum mechanics any more than Newtonian mechanics underlies special relativity. Older ideas about the nature of the world aren't more sensible, just more familiar.