Quantum Probability, Bell's Theorem, and one of the best articles I've ever seen on QM (thanks to Scholarpedia)

2013 May 23
by Daniel Lakeland

Gelman's blog, which I read daily and often comment on, recently had a discussion about the use of "quantum probability" in everyday modeling. ie. the idea arises that perhaps it's not sufficient to use standard probability theory.

I am not generally inclined to believe that QM implies a different type of probability theory than the one we normally use. But I am also open minded as I don't really feel qualified to say what QM means, since I have a very basic understanding of it.

One area that I have read a little about is non-locality and Bell's Theorem, which is related to the idea that quantum mechanical results can not be the result of local hidden variables attached to the particles being measured. It's interesting that apparently people have been misinterpreting Bell's theorem pretty much since the very first moment it was published.

Scholarpedia has a really excellent article on Bell's theorem, it's full of historical analysis of Bell's own words and motivations, as well as people's different approaches to the problem. Towards the end it has a description of how one might interpret the differences between "classical" and "quantum" probability (under the heading "Classical versus quantum probability (and logic)" which apparently I can't link to directly).

One reason Bell worked on his research was that he was impressed with David Bohm's "pilot wave" version of QM. In this explanation, as I understand it, the QM state function is itself a real nonlocal thing. Interactions between a detector apparatus and one of an entangled pair of particles causes the other entangled particle to have the opposite property via nonlocal causation. ie. there is "spooky action at a distance" quite literally. By throwing out the idea that QM is localized in space, we gain the idea that electrons and photons and their spin or polarization are real objects that have a definite state and location which is mediated via interactions in this not-very-observable "state space" of the quantum world.

The reason I'm writing this article though is to point out how well written and clearly thought out the Scholarpedia article is, and how I really think you should go over there and read it, if you have any interest at all in QM.

4 Responses leave one →
  1. konrad permalink
    May 23, 2013

    Thanks for linking to this - the section you mentioned answers the question I was asking on that thread. And the rest also looks worth reading.

  2. Manoel Galdino permalink
    May 24, 2013

    I really like this post by Michael Nielsen...
    http://michaelnielsen.org/blog/why-the-world-needs-quantum-mechanics/

    I didn't have time to take a look at the link you provided, so I don't know how we can compare both texts. But I do believe that Michael Nielsen's post is a good place to start understanding these non-locality issues.

  3. Daniel Lakeland
    May 24, 2013

    Manoel: That post by Nielsen unfortunately jumps from "Q = 2.8" to "objects have no intrinsic reality". reading further in the article I linked to will point out that the correct statement is something more like "either objects have no intrinsic reality, or quantum mechanics is nonlocal and "simultaneous" interactions at space-like separation happen."

    the idea that Bohm had, and which Bell seemed to be very interested in, is that perhaps the wave function is real, and exists independent of the 3D world we experience. I think one explanation of this could be something like the world is really 3N dimensional where N is the number of particles in the universe, but we experience it as 3 dimensional in sort of the same way that we experience temperature. It's a property that comes out of a sort of average over many particles.

    Photons and other quantum particles can't themselves travel through the 3D space faster than light, but the wave function exists in this abstract space in which there is no restriction on "speed of travel". Like an elastic wave traveling along a rubber band at tens or hundreds of times faster than the individual atoms that make up the wave, the quantum (and only the quantum) "information" can get from one particle to another even if they are separated by distances in 3D that are space like (ie. can not communicate via light).

  4. May 25, 2013

    I address the instantaneous non-local interactions possibility in a footnote to my essay, noting merely that it's not the mainstream conclusion; it is, of course, an interesting possibility.

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