User talk:Tnorsen/Sandbox/Bell's Theorem

Bell's inequality
Hi Tnorsen,

for the actual derivation of the inequality, I would propose to include three approaches (I can help with the writing if you find it boring):

1) The argument that appears in Bell's first article on the subject:

locality ==> pre-determined values for A(a), B(b) ==> inequality of the form |C(a,b)-C(a,c)| <= 1+C(b,c).

2) The argument that goes directly from locality to the CHSH inequality |C(a,b)-C(a,b')+C(a',b)+C(a',b')|<=2 and assumes no perfect correlations. This is the argument that Bell uses on most of his articles on the subject.

3) "Bell's theorem for children", i.e., the 1/4+1/4+1/4>=1 argument. It can be presented in the form of a game in which two people play a questions/answers game, they are allowed to meet and discuss a strategy before they receive the questions and are not allowed to communicate after they receive the questions (and, during their meeting, they don't have any information about the questions that are going to be asked later on, so that their strategy cannot depend on the questions they are going to be asked). There are three possible questions, only yes/no answers are allowed, they have to give the same answer when the same question is asked and give the same answer 1/4 of the time when a fixed pair of different questions is used. You can convince a 9 year old child that they cannot win the game, using no algebra, just an argument of the form: "since they must give the same answers when the same question is asked, they must agree about the answers in advance (by choosing a triple YYY, YYN, YNY, ... of answers). 1/4 of the time, we have answer1=answer2, 1/4 of the time, we have answer1=answer3 and 1/4 of the time we have answer2=answer3 and 1/4+1/4+1/4<1, so no strategy for choosing the triples can work".

This argument appears in this article by Dürr et al. (not in the form of a game between people) and also in this article by Jean Bricmont (in the form of a game), along with some nice sociological speculations about why physicists did not understand the argument. The idea of explaining the argument in the form of a game appears in Tim's book, though I don't recall seeing the 1/4+1/4+1/4>=1 argument there.Dvtausk (talk) 03:24, 23 March 2009 (UTC)

Yes, I definitely agree it'd be good to present roughly three different derivations. I like the idea of including the "original" Bell inequality (sort of for historical interest and to preserve the focus of keeping the article faithful to Bell's own work/ideas). And including the "direct" argument from local causality to the CHHS inequality would be good for obvious reasons. I'll refresh my memory on the "1/4 + ..." argument before commenting on that. This maybe sounds stupid, but I think we should also think about seeing what's on the "bell's theorem" page currently and including that same derivation (whatever it is) just because the more familiar we can make the new article to the people (read: bastards!) who currently "watch" this page, the more likely we will be to successfully change everything for the better. So something to think about.

As to who is going to do this, I don't find writing this stuff boring at all, but it takes time and I'm just trying to squeeze this in between other things. So I'll work on it whenever I find some spare time. But you (Dvtausk) or whoever else wants to, should certainly feel free to work on this if you want. I want as many hands on this as possible, and don't at all think of it like "my baby" such that I'll be upset if anybody adds/subtracts/changes. Just the opposite. So whoever is reading this should jump in and do some work on this article. Tnorsen (talk) 18:21, 23 March 2009 (UTC)

Bell's theorem without inequalities
Personally, I don't think the so called "Bell's theorem without inequalities" is particularly important (the versions with inequalities --- in particular the one that does not assume perfect correlations --- is the more robust one, since it is stable by small perturbations of the quantum predictions, while the versions that need perfect correlations are not). But, most people think that "Bell's theorem without inequalities" is very important, so it will probably have to be included in the Bell's theorem page.

The most popular version involves the so called GHZ state. The Hilbert space is 8-dimensional (three quantum systems, each described by a 2-dimensional Hilbert space), the GHZ state is an eigenstate of the four (mutually commuting) self-adjoint operators:

$$\sigma^1_x\sigma^2_x\sigma^3_x$$, $$\sigma^1_x\sigma^2_y\sigma^3_y$$, $$\sigma^1_y\sigma^2_x\sigma^3_y$$, $$\sigma^1_y\sigma^2_y\sigma^3_x$$.

Assuming locality, you can prove the existence of (non contextual) pre-determined values for all operators $$\sigma^i_x$$, $$\sigma^i_y$$, i=1,2,3, and then this leads to a contradiction of the type 1=-1. Obviously, the authors do not understand the "locality ==> hidden variables" part of the argument and they think that the argument merely proves the impossibility of "local hidden variables".

There is a larger class of "Bell's theorem without inequalities" that works like this: you use Schrödinger's enhanced version of the EPR argument, that allows you to (from locality and perfect correlations) prove the existence of non contextual hidden variables for any finite-dimensional Hilbert space. Then, any proof of impossibility of non contextual hidden variables can be used as a proof of non locality. Details appear in Doug Hemmick's dissertation.Dvtausk (talk) 14:42, 24 March 2009 (UTC)

Good idea. I'll add a section "Bell's Theorem Without Inequalities" to the main page, and somebody can write it at some point! Tnorsen (talk) 12:34, 25 March 2009 (UTC)