Can the Born Rule Be Derived in the Many Worlds Interpretation?

[gill1109]

Nowadays there is a neat way to combine everything into one object, called an "instrument with settings". It's a black box which takes a classical input (the experimenter can turn a dial or press a button) and a quantum input (which is described by a density matrix). It has a classical output and a quantum output. We axiomatically state that a probabilistic mixture of quantum inputs is equivalent to an input of the corresponding mixture of density matrices. We assume that independent quantum inputs can be combined using tensor product formalism. We argue that any instrument must be linear, normalized, and totally positive. It follows by Naimark theorem that it has to have the Kraus representation form. Preparations are instruments with no inputs. Measurements are instruments with quantum input and classical output only. It's a theorem (called the dilation theorem) that every instrument can be realized by adding an independent auxiliary quantum input and then combining in turn unitary evolution, measurement of an observable with transformation of the state according to the Lüders - von Neumann collapse postulate, and finally possible discarding of (some parts of) the outputs.

So one can build the most general kinds of black boxes allowed by a few fundamental principles from "elementary boxes" for unitary evolution and von Neumann measurement, as long as one can also bring in auxiliary quantum systems.

This means that there are three equivalent ways to describe a quantum instrument
(1) by its properties of linearity, total positivity, normed
(2) by Kraus representation (collection of matrices ...)
(3) as combination of adding an ancillary Q system, do unitary transformation on composite system, do von Neumann measurement, possibly discard some parts of quantum or classical output

 

[bhobba]

Yes that is true.

And the dilation theorem is important in the way I tend to look at things - its another way you get POVM's from resolutions of the identity.

However I think the culmination of that way of thinking, ie probabilistically in terms of instrumentalist, reached a fairly definitive expression in the work of Hardy:
http://arxiv.org/pdf/quantph/0101012.pdf

Basically QM is the most reasonable probabilistic model that allows continuous transformations between pure states, forbidden in ordinary probability theory.

The argument goes something like this. Suppose we have a system in 2 states represented by the vectors [0,1] and [1,0]. These states are called pure. These can be randomly presented for observation and you get the vector [p1, p2] where p1 and p2 give the probabilities of observing the pure state. Such states are called mixed. Now consider the matrix A that say after 1 second transforms one pure state to another with rows [0, 1] and [1, 0]. But what happens when A is applied for half a second. Well that would be a matrix U^2 = A. You can work this out and low and behold U is complex. Apply it to a pure state and you get a complex vector. This is something new. Its not a mixed state - but you are forced to it if you want continuous transformations between pure states.

QM is basically the theory that makes sense out of pure states that are complex numbers. There is really only one reasonable way to do it - by the Born rule (you make the assumption of non contextuality - ie the probability is not basis dependant, plus a few other things no need to go into here) - as shown by Gleason's theorem.

Thanks
Bill

 

[stevendaryl]

Again, I'm to naive to understand the necessity of a collapse at all! Take a "classical" situation of througing a die. Without further knowledge about its properties, I use the maximum-entropy method to associate a probability distribution. Of course, the least-prejudice distribution in the Shannon-Jaynes information-theoretical sense is that the occurance of any certain value is $p_k=1/6$, $k \in \{1,2,3,4,5,6 \}$.

Now I through the die once and get "3". Is now my probability distribution collapsed somehow to $p_3=1$ and all other $p_k=0$? I don't think that anybody would argue in that way.

Right. But to push the analogy a little further, suppose that there were a pair of dice such that each die separately seemed to give a random number between 1 and 6, but when you compare the two results, you find that the numbers always add up to 7. I think the way people would reason about such a situation would be either to assume that the dice are not truly random---it's predetermined somehow what number will be rolled, even though we don't know how to calculate it---or to assume that there is a causal influence between the two dice. To use terminology from Bell, the first possibility would be a "hidden variables" assumption, while the second would be a "collapse" of the probability distribution. If one die stopped rolling with result 2, while the second die was still rolling, then you would no longer describe the second die as randomly selecting a number from 1-6, it's probability distribution would be collapsed to the single possibility, 5.

So classically these kinds of correlations imply either hidden variables or some kind of nonlocal influence between the two events. In the first case, there is a kind of collapse that is purely subjective due to gaining more information about the situation. In the second case, the collapse really represents something physical happening.

The weird thing about QM is not so much the adjustment of probability in light of new information--that happens classically, as well. But the weird thing about QM is that it seems that both of the explanations for nonlocal correlations seem to be ruled out (the first by Bell's theorem, and the second by relativity).

 

[gill1109]

The weird thing about QM is not so much the adjustment of probability in light of new information--that happens classically, as well. But the weird thing about QM is that it seems that both of the explanations for nonlocal correlations seem to be ruled out (the first by Bell's theorem, and the second by relativity).

Let's call it *wonderful*, not *weird*. We see from results like Hardy's that if we want various beautiful (and believable) things about nature to be true, and at the same time we admit that some things are intrinsically discrete, then the only way to make both happen at the same time is through quantum theory and thereby necessarily discarding some other cherished intuitions about the world. Note: quantum theory makes some things possible which classically would be impossible (like violating Bell inequalities, for instance) but it pays for this by making other things impossible which classically would be possible. In particular, since the theory is fundamentally stochastic there are lots of things which *can't* be done.

So quantum theory is *different* from classical physics, and therefore at first sight *weird*, but one can get used to it, and then it just becomes *wonderful*.

 

[stevendaryl]

Let's call it *wonderful*, not *weird*. We see from results like Hardy's that if we want various beautiful (and believable) things about nature to be true, and at the same time we admit that some things are intrinsically discrete, then the only way to make both happen at the same time is through quantum theory and thereby necessarily discarding some other cherished intuitions about the world. Note: quantum theory makes some things possible which classically would be impossible (like violating Bell inequalities, for instance) but it pays for this by making other things impossible which classically would be possible. In particular, since the theory is fundamentally stochastic there are lots of things which *can't* be done.

So quantum theory is *different* from classical physics, and therefore at first sight *weird*, but one can get used to it, and then it just becomes *wonderful*.

I'm not sure which result by Hardy you're talking about. You can certainly have discreteness without quantum mechanics (for example, cellular automata).

 

[bhobba]

So quantum theory is *different* from classical physics, and therefore at first sight *weird*, but one can get used to it, and then it just becomes *wonderful*.

That's pretty well it IMHO.

Its wonderful - but looked at the right way - very beautiful and actually reasonable after a fashion.

Thanks
Bill

 

[gill1109]

I'm not sure which result by Hardy you're talking about. You can certainly have discreteness without quantum mechanics (for example, cellular automata).

Discrete outcomes but symmetries under rotations implies it has to be stochastic because you can make discrete probabilities vary continuously ...

 

[TrickyDicky]

Right. But to push the analogy a little further, suppose that there were a pair of dice such that each die separately seemed to give a random number between 1 and 6, but when you compare the two results, you find that the numbers always add up to 7. I think the way people would reason about such a situation would be either to assume that the dice are not truly random---it's predetermined somehow what number will be rolled, even though we don't know how to calculate it---or to assume that there is a causal influence between the two dice. To use terminology from Bell, the first possibility would be a "hidden variables" assumption, while the second would be a "collapse" of the probability distribution. If one die stopped rolling with result 2, while the second die was still rolling, then you would no longer describe the second die as randomly selecting a number from 1-6, it's probability distribution would be collapsed to the single possibility, 5.

So classically these kinds of correlations imply either hidden variables or some kind of nonlocal influence between the two events. In the first case, there is a kind of collapse that is purely subjective due to gaining more information about the situation. In the second case, the collapse really represents something physical happening.

The weird thing about QM is not so much the adjustment of probability in light of new information--that happens classically, as well. But the weird thing about QM is that it seems that both of the explanations for nonlocal correlations seem to be ruled out (the first by Bell's theorem, and the second by relativity).

But the only thing Bell's theorem says is that nonlocal correlations can't be explained with a local hidden variables theory, it is agnostic about other explanations or hidden variables theories.So the first explanation(dice nontruly random, their correlation being due to a common nonlocal cause for instance) of nonlocal correlations is not ruled out as long as it doesn't involve locality.

 

[stevendaryl]

But the only thing Bell's theorem says is that nonlocal correlations can't be explained with a local hidden variables theory, it is agnostic about other explanations or hidden variables theories.So the first explanation(dice nontruly random, their correlation being due to a common nonlocal cause for instance) of nonlocal correlations is not ruled out as long as it doesn't involve locality.

But if the outcome is pre-determined, then there is no problem with locality. You only need nonlocality if the results are NOT predetermined.

 

[TrickyDicky]

But if the outcome is pre-determined, then there is no problem with locality. You only need nonlocality if the results are NOT predetermined.

That's where Bell enters to tell you there is indeed problem with locality so in the end you need nonlocality.

 

[Quantumental]

That's where Bell enters to tell you there is indeed problem with locality so in the end you need nonlocality.

Not if everything is predetermined/superdeterministic. Bell himself commented on this loophole several times.

 

[TrickyDicky]

Not if everything is predetermined/superdeterministic. Bell himself commented on this loophole several times.

Right, let me clarify, I didn't mean that kind of hard determinism when I used the example of a common cause. Common causation doesn't even necesarilly always imply determinism.
In any case it was the first example that came to mind, I wasn't implying that it was the explanation of nonlocality.

 

[gill1109]

That's where Bell enters to tell you there is indeed problem with locality so in the end you need nonlocality.

Bell told us in "Bertlmann's socks" that there were four possible *alternative* diagnoses of the "problem" and only one of them was "locality". Moreover his list wasn't even exhaustive. There is also "Bell's fifth position" namely that no-one will ever be able to create the initial conditions to create a successful loophole-free Bell-CHSH type experiment - because of quantum uncertainty principles ie because of QM.
http://arxiv.org/abs/quant-ph/0301059
OK other people have other opinions. I am just reporting John Bell's own logical analysis. He made clear that from a logical point of view, all options were wide open.

He had his own personal preference or opinion - ie he wanted a local hidden variables theory, he did accept QM predictions as being pretty damn near the truth, he did not buy into conspiracy (super-determinism) and he therefore went for non-locality, but he freely admitted that that was just a matter of taste.
BTW my personal preference, today, is to reject *realism*. But my opinion can change if I see new evidence e.g. experimental, mathematical, pointing in other ways. Or if someone explains that I was seeing things which I already thought I knew in a wrong way. It's wise (scientific?) to keep an open mind. Have an opinion, to be sure, but be prepared for it to be changed. Distinguish facts from opinions. Don't believe everything you read just because famous or respectable people wrote it.

 

[bhobba]

BTW my personal preference, today, is to reject *realism*.

Mine is to reject both, but as you mentioned in another thread locality is dependant on your definition of locality. My definition is the cluster decomposition property - locality applies only to uncorrelated systems.

Have an opinion, to be sure, but be prepared for it to be changed. Distinguish facts from opinions. Don't believe everything you read just because famous or respectable people wrote it.

If you are interested in the foundations of QM, in my experience its vital to think through it yourself and reach your own conclusions. There is a lot of misinformation out there, as, answering questions on the forum has taught me only too well.

My view has changed considerably over time. Perhaps the biggest shift was simply realising an observation doesn't have to involve a conscious organic observer.

Thanks
Bill

 

[gill1109]

My view has changed considerably over time.

Mine too! I believe that there are not any easy answers and people who tell you it is all very simple, have over-simplified or misunderstood or are fooling themselves.

BTW (discovered from the thread https://www.physicsforums.com/showthread.php?t=758324 on Bell's theorem) Bell says "I cannot say action at a distance is needed. I can say that you can't say it is not needed". This is like Buddha talking about self. He is actually saying that our usual categories of thought are *wrong*. Because of the words in our vocabulary and our narrow interpretation of what they mean, we ask stupid questions, and hence get stupid answers.



"The one place where the Buddha was asked point-blank whether or not there was a self, he refused to answer. When later asked why, he said that to hold either that there is a self or that there is no self is to fall into extreme forms of wrong view that make the path of Buddhist practice impossible. Thus the question should be put aside." http://www.accesstoinsight.org/lib/authors/thanissaro/notself2.html
Sorry for bringing esoteric philosophy to this forum, but it is pretty clear to me that John Bell himself was referring to exactly this same famous question. We know that Schrödinger, Bohr and other founding fathers of QM were drawn to Eastern philosophy - precisely because it emphasizes that usual categories of thinking may be a straight-jacket.