# The Born Rule in Many-Worlds

1. Jul 24, 2014

### stevendaryl

Staff Emeritus
I'm not 100% sure that this is the right forum to post this, but I think that the people who read the Quantum Physics forum might be interested:

Sean Carroll has written a paper explaining how it is possible to derive the Born Rule in the Many Worlds Interpretation of Quantum Mechanics. I'm not sure that it's the final word on the subject, but it does grapple with the big question of how it makes sense to use probabilities to describe a universe that evolves deterministically.

http://www.preposterousuniverse.com...hanics-is-given-by-the-wave-function-squared/

2. Jul 24, 2014

### Jilang

Very nice. Note to self: Need to read this many times.

3. Jul 24, 2014

### e.bar.goum

I'm working through the paper http://arxiv.org/pdf/1405.7907v2.pdf at the moment, and I'm finding it a bit more understandable than the blog post. So if you're comfortable with the formalism of quantum mechanics, but uncomfortable with the detail in the blog, I suggest the paper!

4. Jul 24, 2014

### Staff: Mentor

I have had a look at these proofs of the Born rule in MW before. Wallace gives a lot of detail in his book on it which I forked out for (tried to find it the other day but couldn't - it's in the land of the lost right now ):
http://users.ox.ac.uk/~mert0130/books-emergent.shtml [Broken]

Anyway when I read it I did carefully go through his proof and its possible objections. For me the bottom line was the waging strategy he used amounted to basis independence which is in fact the key to Gleason's theorem. If the measure doesn't depend on the basis it belongs to (ie non contextuality) then Gleason applies and you get Born.

Sean also seems to agree:
'There is a result called Gleason’s Theorem, which says roughly that the Born Rule is the only consistent probability rule you can conceivably have that depends on the wave function alone. So the real question is not “Why squared?”, it’s “Whence probability?”'

Where I would differ from Sean and maybe Wallace is 'Whence probability'. From my applied math background studying things like statistical modelling etc I view determinism as simply a special case of probability - but the only probabilities allowed are 0 and 1. So if you ask the question what probability measure can be defined and apply Gleason you are not assuming probability. It just turns out that the measure you do come up with cant have 0 and 1 only defined on it - that's a simple corollary of Gleason - but for some reason it's not usually presented that way - and you have this colouring proof of Kochen-Sprecker - which always struck me as rather strange - maybe they don't want to get into the difficult math of Gleason's usual proof - but the new one based on POVM's is dead simple.

To me this proved QM is inherently probabilistic.

That said, I have discussed this before and others think I am assuming probability to begin with so cant draw that conclusion. Don't see it myself, but it's their view.

Thanks
Bill

Last edited by a moderator: May 6, 2017
5. Jul 25, 2014

### Fredrik

Staff Emeritus
I'm not a fan of these "derivations". I think they're based on some pretty major misconceptions. First of all, they all use the axiom that the Hilbert space of a composite system is the tensor product of the Hilbert spaces of the subsystems. How did that axiom find its way into the foundations of QM in the first place? The answer is that it ensures that probabilities assigned by the Born rule follow this standard rule of probability theory: P(A & B)=P(A)P(B) when A and B are independent.

A derivation of the Born rule (or the fact that the theory should assign probabilities) that relies on the Born rule is of course circular. You can argue that the tensor product stuff has also been derived by Aerts and Daubechies, and that they didn't directly use the Born rule. But their approach is based on another set of axioms, and I think they too can (and should) be justified using the Born rule. So I still think these derivations are circular.

Second, I think it's very naive to think that QM must contain a description of what's actually happening just because it makes excellent predictions about the result of experiments. I think the easiest way to explain what I mean is to consider one of the simplest probability theories.

Let $X=\{1,2,3,4,5,6\}$. Let $\Sigma$ be the set of all subsets of $X$. For each finite set $S$, we will use the notation $|S|$ for the cardinality of $S$, i.e. the number of distinct elements of $S$. Define $P:\Sigma\to[0,1]$ by $P(E)=|E|/|X|$ for all $E\subseteq X$.

So far we have only talked about the purely mathematical part of the theory. We turn this into a theory of applied mathematics, by adding a correspondence rule, i.e. an assumption about how something in the mathematics corresponds to something in the real world: We assume that for each $E\subseteq X$, $P(E)$ is the frequency with which an ordinary six-sided die will have a result in the set $E$, in a long sequence of identical experiments.

Now we have a falsifiable theory, and it makes excellent predictions about the results of experiments. But it tells us nothing about what's happening to the die, and no one in their right mind would think that they can find out what's happening to it only by reinterpreting this theory.

There's no doubt that QM is a far better theory than this one, but it's still a (generalized) probability theory. So why would anyone think that it contains all the information about what's actually going on? I see no reason to think that it does.

I cringe when I see statements like "quantum mechanics is an embarrassment". No, it's not. It's an amazing achievement in science and mathematics, with many awesome applications in technology. Carroll's statement about how it's an embarrassment is however an embarrassment.

6. Jul 25, 2014

### atyy

If I understand correctly, in Wallace's version, the worlds can interfere, causing lost books.

But perhaps it's natural in the context of MWI, compared to Copenhagen? In Copenhagen, noncontextuality seems poorly motivated to me, since different measurements should give different results. Or can contextuality also be imported into MWI, since decoherence picks a preferred basis?

Last edited by a moderator: May 6, 2017
7. Jul 25, 2014

### atyy

I don't quite see how the Born rule implies the tensor product structure.

Even then, it doesn't mean that the tensor product structure implies the Born rule.

8. Jul 25, 2014

### Fredrik

Staff Emeritus
I don't claim to be able to prove this myself. It looks very difficult actually. Aerts and Daubechies use a pretty sophisticated argument based on the quantum logic approach to QM to derive the tensor product stuff. I'm just saying that when I read parts of their paper years ago, it seemed to me that the entire approach could and should be justified using the Born rule. Unfortunately I don't think I can take the time to try to prove that I'm right in this the thread.

The part of my post that you're quoting is however not about this very difficult topic. It's about something much easier: I'm saying that the tensor product stuff plus the Born rule ensures that the probability rule "P(A & B)=P(A)P(B) when A and B are independent" holds. The argument for this is just a simple calculation like
$$P(a)P(b) =|\langle a|\psi\rangle|^2|\langle b|\phi\rangle|^2 =\left|\big(\langle a|\otimes\langle b|\big)\big(|\psi\rangle\otimes|\phi\rangle\big)\right|^2 =P(a\, \&\, b).$$ (I'm not sure what notational convention for tensor products of bras is the most popular. Maybe <b|should be to the left of <a| above). It seems very likely that this observation was the original motivation for the inclusion of the tensor product stuff in the axioms of QM. The fact that the Born rule is part of the motivation makes derivations of the Born rule that rely on tensor products pretty suspicious. So the advocates of these derivations would probably like to argue that the tensor product stuff is forced upon us by things that have nothing to do with the Born rule. Aerts & Daubechies appear to have made such an argument, but it seems to me that their proof relies on assumptions that can and should be justified by the Born rule. Again, I don't think I can take the time to try to prove that here.

Right. But these derivations are all about how the Born rule follows from the rest of QM (which includes the tensor product stuff), and they rely heavily on the tensor product stuff.

9. Jul 25, 2014

### stevendaryl

Staff Emeritus
I don't think you have to choose--quantum mechanics can be BOTH an amazing achievement in science and mathematics, and an embarrassment.

I think it's an embarrassment in the sense that we can't really come up with a completely consistent way of understanding what QM is saying about the world.

You can take the operational approach, and just say that it's a calculational tool for predicting probabilities of outcomes of measurements. And that's fine for most purposes, but presumably, a measurement is a particular kind of physical interaction, so what is special about measurement?

It seems to me that either you say there is something special about measurement, that measurements have definite outcomes, while other sorts of interactions only have probability amplitudes, or else you say that there is nothing special about measurements, that they don't have definite outcomes, either, or you say that everything has definite outcomes. Either measurement is different, which is weird, or it's not different, in which case it's not clear what the probabilities in QM are probabilities OF. If everything happens, then what does it mean to say that some things happen with higher probability?

I think it's a mess, conceptually. And it's kind of embarrassing that it's not much clearer 80 years (or however long it was) after it was developed.

10. Jul 25, 2014

### stevendaryl

Staff Emeritus
I don't see that. Tensor products are a natural way to make a composite vector space from two component vector spaces. Maybe the motivation for using a vector space might be indirectly motivated by their probabilistic interpretation, but it's more general than that.

11. Jul 25, 2014

### atyy

But doesn't the definition of A and B being independent hold regardless of the tensor product structure and the Born rule? The equations you wrote seem to say something else: the Born rule and the tensor product structure mean that product states are independent.

Could a rule that is contextual lead to observables that factor being correlated on a product state?

12. Jul 25, 2014

### Staff: Mentor

In MW what Wallice does is very natural. We know we will only experience one world - but which one? A betting strategy based on decision theory seems quite reasonable to determine it. But maybe that's just me and my applied math background - decision theory is rather an important area.

In Wallices book he has all sorts of theorems proving this and that about such strategies, but for my money, and if I recall correctly, he even states it explicitly in one of his theorems saying its equivalent to basis independence, its merely repackaged Gleason.

I love the elegance of MW - really its beauty incarnate - but for me unconvincing. This exponential dilution of the wave-function at a massive rate I simply cant swallow.

Thanks
Bill

Last edited: Jul 25, 2014
13. Jul 25, 2014

### stevendaryl

Staff Emeritus
I don't understand what's hard to swallow about that. That's sort of normal for probability distributions, isn't it? They might start off highly peaked at one possibility, but with time, they spread out in all directions, becoming more and more "diluted". Why does becoming exponentially diluted seem implausible?

14. Jul 25, 2014

### Fredrik

Staff Emeritus
I think it's fine for all purposes. It would be nice if QM had something more to tell us, but that's not a reason to think that it does.

The only thing that's special about measurements is that they are the interactions that we use to test the accuracy of the theory's predictions.

To me it seems sufficient that the final state of the measuring device is for practical purposes indistinguishable from a "definite" state.

I do. I mean, I think they follow the same laws, because I find the alternative extremely absurd. But "measurements" is still a proper subset of "interactions", so in a very restricted sense, they're "special".

Last edited: Jul 25, 2014
15. Jul 25, 2014

### Fredrik

Staff Emeritus
Independence is the result P(a & b)=P(a)P(b). This result follows from the tensor product stuff. Perhaps it also follows from something else. This is why this argument doesn't prove that we need to use tensor products, and that's why the Aerts & Daubechies argument is so appealing. It does prove that (given some assumption that are rather difficult to understand), we have to use tensor products.

Sure, but doesn't "the tensor product stuff" include the assumption that if |S> is the state of system A, |T> is the state of system B, and A and B haven't yet interacted in any way, then the state of the composite system is $|S\rangle\otimes|T\rangle$? Aren't we saying essentially the same thing?

I haven't really thought about that.

16. Jul 25, 2014

### stevendaryl

Staff Emeritus
If there is nothing special about measurements, and surely there isn't, then it's hard to understand how measurements have definite outcomes, when other sorts of interactions don't, according to QM. You could say that measurements don't, either, but that leads to Many Worlds, and it's kind of understand how probabilities work if everything happens.

We can pretend, for the purpose of getting on with doing science, but the theory itself shows that it's just pretending. Either the outcome is definite, or it's not. The one choice leads to weird wave function collapse, the other choice leads to weird Many Worlds. Conceptually, it seems a mess to me.

17. Jul 25, 2014

### Staff: Mentor

Yea - but in MW a wavefunction is not like probabilities which is an abstract thing - its considered very real indeed. That's why its dilution seems very implausible to me - real things from everyday experience cant be infinitely diluted - which is basically what is required.

Of course that doesn't disprove it - I just find it hard to take seriously.

Thanks
Bill

18. Jul 25, 2014

### Quantumental

This confused me and made me want to understand this argument better. What is it about the dilution that seems implausible?

Could you give an example of something similar?

19. Jul 25, 2014

### Staff: Mentor

Look around you. Take anything that's considered to exist in a real sense - using the common-sense view of real. You cannot continually subdivide it forever. That a wavefunction is considered real and isn't like that just seems implausible to me. It doesn't prove anything - its simply an opinion - opinions are like bums - everyone has one - it doesn't make it correct. But its my view.

As an example take a glass water. Keep dividing it and you stop at subatomic particles - you can go no further - and it still be considered water. Yet a wavefunction, that is considered just as real, can keep on doing that indefinitely - well simply strikes me as implausible.

Thanks
Bill

Last edited: Jul 25, 2014
20. Jul 25, 2014

### atyy

Yes, that makes sense with this additional statement about systems that have not interacted.

Now I'm wondering whether the statement about systems not having interacted has to be added by hand, or whether it can be "derived" from the idea that composite systems have tensor product Hilbert spaces.

If by "non-interacting subsystems" one means that the subsystems are independent, then I think it can be derived, if one assumes the Born rule for each non-interacting subsystem. In which case, the tensor product structure seems to be a consequence of the Born rule, as you said.

But if by "non-interacting" one means that the Hamiltonian of the composite system has no product terms, then I don't immediately see how to exclude the possibility that the non-interacting subsystems should not be independent. Which I guess is a way of saying that the tensor product structure does not imply the born rule.

Last edited: Jul 25, 2014