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How exactly is the Many-Worlds Interpretation able to predict the Information-theoretic uncertainty principle?

+ 3 like - 0 dislike
630 views

I've recently learnt (see e.g. here) that the Many-Worlds Interpretation of Quantum Mechanics is in fact able to make predictions such as the Information-theoretic Uncertainty Principle.

This is rather surprising to me, as I've been under the impression that interpretations of quantum mechanics are merely philosophical "interpretations" (as the names suggest, too). So my questions are:

  • Am I wrong about this?
    • If I am, what is the right way to think about quantum interpretations?
    • If I am not, how is it possible for a mere philosophical idea, even if mathematically supported, to make concrete predictions about physics?
  • Do the "variants" of the Many-Worlds Interpretation, like consistent histories, "many minds", etc., make the same predictions about quantum mechanics as the MWI?
  • What are some other examples of quantum interpretations that make real predictions? Bohmian mechanics doesn't count.
asked Jul 16, 2014 in Theoretical Physics by dimension10 (1,950 points) [ no revision ]

2 Answers

+ 3 like - 0 dislike

It's not that the many-worlds interpretation makes a prediction, it's just that in giving the idea mathematical form, Everett was lead to consider information in a quantum mechanical context, and in this context, the natural way to state the uncertainty principle is different, and information theoretic. The inequality is

$$I(x) + I(p) > ln(2/e) $$

Where

$$ I(x) = - \int |\psi(x)|^2 ln(|\psi(x)|^2) dx $$

$$ I(x) = - \int |\psi(p)|^2 ln(|\psi(p)|^2) dp $$

The inequality is true for all quantum wavefunctions, it was proven rigorously by Beckner in 1975. But this idea was clearly inspired by the application of information theoretic ideas to quantum wavefunctions found in Everett.

The main question in Everett's thesis is how to embed classical information into a quantum mechanical wavefunction evolving deterministically in time. His conclusion is that the classical information emerges as a branch-label for the various branches in the wavefunction's time evolution, and his basic thesis is that the information in our observations of the world correspond precisely to this classical information of branch-labels.

The only question of philosophy is whether you are to take the picture are meaning that the other branches are "real" or "unreal", a question without any significance, which Everett does not even consider, and which is completely irrelevant as far as the embedding of classical information into quantum mechanics is concerned.

answered Jul 16, 2014 by Ron Maimon (7,535 points) [ revision history ]
edited Jul 17, 2014 by Ron Maimon

A copy of the Beckner paper of 1975 mentioned by Ron can be found (as of July 16th) at http://www.cmc.edu/pages/faculty/MONeill/Math%20138/papers138/Beckner.pdf, or on JSTOR, http://dx.doi.org/10.2307/1970980.

Ron, it would seem that your second line should read \(I(p)=...\).

Beckner's equation (19) there, \(\int|f|^2\ln{(|f|^2)}\mbox{d}x+\int|\mathcal{F}f|^2\ln{(|\mathcal{F}f|^2)}\mbox{d}p\le\ln{2}-1\), for a normalized function \(\|f\|_2=\|\mathcal{F}f\|_2=1\), seems to be different from yours?

Yes, I was copying from Everett's thesis, using his conventions for normalization, whatever they were, and dropped the minus sign in the definition of information. Fixed.

Thanks for the clarification! 

I think that what @PeterMorgan meant was that the second equation is the definition of I(p), not I(x). 

Ron, I think there's still a difference. Reversing signs,\(\int|f|^2\ln{(|f|^2)}\mbox{d}x+\int|\mathcal{F}f|^2\ln{(|\mathcal{F}f|^2)}\mbox{d}p\le\ln{2}-1\) becomes \(-\int|f|^2\ln{(|f|^2)}\mbox{d}x-\int|\mathcal{F}f|^2\ln{(|\mathcal{F}f|^2)}\mbox{d}p\ge-\ln{2}+1=\ln{(e/2)}\). The equality is satisfied if \(f(x)\) is a Gaussian.

+ 2 like - 0 dislike

The many worlds interpretation holds that physical reality is described by the formalism of quantum mechanics without the collapse postulate. The theory includes a mapping between the terms invoked in equations and the sort of thing it is possible for us to observe. Once you have put this in place and applied information theory to quantum mechanics, the information theoretic uncertainty principle follows. There is nothing particularly unusual about this.

There is something very unusual about the discussion over the interpretation of quantum mechanics as David Deutsch has noted here and here. The problem is the following. Any scientific theory is not just a bunch of predictions. Rather, it is an explanation of how the world works and the whole idea of making predictions makes no sense unless that is the case. The way you do any measurement is set up a physical system that you hope will record the results of doing some particular thing to some physical system. There has be something happening in the real world to make the results of the measurement have a correspondence to what you are trying to measure. And attempts to improve measurements have to consist in inventing better explanations of how to produce such a correspondence. Predictions and explanations are not independent. As a result you can't divorce predictions from the picture that your theory gives of how the world works.

Quantum theory without collapse predicts the existence of multiple versions of the system being measured. This is not an optional extra. It is not something you can toss on a whim if it happens to suit your mood. You ask how an interpretation can make predictions not made by the other interpretations. Let's look at some of the alternatives. The Copenhagen interpretation amounts to sometimes quantum mechanics applies and sometimes it doesn't but we're not going to provide a precise explanation of why and how: this makes no predictions at all. Hidden variable theories: quantum mechanics is false all observables are always single valued but this happens to reproduce the predictions of quantum mechanics in some completely unspecified way. Since this doesn't explain how and where the hidden variables differ from quantum mechanics, it also makes no predictions. So the MWI makes more predictions than both of those theories.

Note also that the denial of the existence of the multiverse has helped lead to bottomless pits of confusion about quantum mechanics, such as the idea that quantum mechanics is non-local. In EPR experiments after the measurement is done the measuring instrument and the systems that carry the measurement results exist in multiple versions. Those versions only become correlated after they are compared and this process is entirely local. This happens because the measuring instrument and any other system that carries the measurement results contain locally inaccessible information: their observables depend on what measurement was conducted, but the expectation values of those observables don't have that dependence:

http://arxiv.org/abs/1109.6223

http://xxx.lanl.gov/abs/quant-ph/9906007.

There is no way even to express this explanation if you are not clear about whether or not "classical" systems, that is decoherent systems, are described by quantum observables and whether they exist in multiple versions.

Variants of the MWI may or may not make the same predictions. Some of them are not clear about what exists in reality. For example, consistent histories people often fudge the issue of the existence of the multiverse and I find it very difficult to tell what happens in reality in Zurek's existential interpretation. The many minds interpretation claims that the slicing of the multiverse into parallel universes is dependent on the existence of minds somehow. I don't think this is the case since decoherence can take place without minds and it suppresses interference. There are cases, such as the EPR experiment, where the parallel universe approximation breaks down for macroscopic objects, but this seems to happen whether or not people are present, so I think the many minds theories are false.

The best elaborated versions of the MWI are the Heisenberg picture based version by David Deutsch

http://xxx.lanl.gov/abs/quant-ph/0104033

and the spacetime  state theory by Wallace and Timpson

http://users.ox.ac.uk/~mert0130/papers-qm.shtml#ssr

You might also argue that the Montevideo interpretation, which takes account of decoherence caused by clocks, is a version of the MWI:

http://lanl.arxiv.org/abs/1406.4351.

As far as these theories are concerned I think we're beginning to get into research areas where it is not clear to anybody how to work out the correct predictions so we don't know how things will turn out.

As for other interpretations making predictions, the ones that do make predictions are alternative physical theories that are not the same as quantum mechanics, such as the GRW theory.

answered Jul 18, 2014 by alanforr (20 points) [ no revision ]

Welcome to Physics Overflow.

"Any scientific theory ... is an explanation of how the world works and the whole idea of making predictions makes no sense unless that is the case." There are alternatives, at the very least something akin to Lakatos's approach, that a theory is a systematic collection of bridge principles that relate fragments of the world to fragments of our models. Not that this is perfect, but it does make some sense, and does not require explanation.

As much as I agree that the Copenhagen interpretation is not great, and physicists largely paid lip service to it during its supremacy rather than following its precepts (such as they were) absolutely, still it seems too much to say that it "makes no predictions at all" (my emphasis), insofar as everyone was clear that the Hydrogen atom length scale is more in the regime of quantum theory than in the regime of classical electrodynamics.

This is an interesting reading. However I personally do not need any quantum interpretation, as for me it is enough that quantum mechanics just works (the probabilistic predictions that can be calculated are nicely in agreement with experiment).

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