A Gedanken Experiment Regarding Bell's Theorem and Nonlocality

[Gary Boothe]

A GEDANKEN EXPERIMENT REGARDING BELL'S THEOREM AND NONLOCALITYLet’s say I have four boxes with three compartments in each one, and each compartment contains either a white sock or a black sock. This is analogous to photons having spin components either clockwise or counterclockwise (black or white socks) around three perpendicular axes (the three compartments).

There are only four different sets of possible “entangled” boxes, analogous to entangled photons. I can represent the four sets of entangled boxes as:

Set 1 Set 2 Set 3 Set 4

B W B W B W B W

B W B W W B W B

B W W B B W W B

The boxes are "entangled" in that if a black sock is in the left top compartment of a box, the sock must be white in the right top compartment of the other box, and so on for each of the four possible sets of boxes.

Let’s say I keep one box in a set and mail the other one to you. For this box and every one I send to you, we open only one compartment at a time at random. After opening our compartments at random, we then compare the colors of our socks.

With the first possible set of boxes, If I open the top left box, you may open any of the three compartments to get a white sock, and the possible outcomes are BW,BW,BW, and of course this is repeated three times for a total of nine possible outcomes:

BW,BW,BW,BW,BW,BW,BW,BW,BW.

So, we get different colored socks 100% of the time.

For the second set of possible boxes, the possible outcomes when I open a compartment at random, and you open a compartment at random are:

BW,BW,BB,BW,BW,BB,WW,WW,WB.

So, we get different colored socks only 5 times out of 9, or 55.55% of the time.

For the third and fourth set of boxes, we also get different colored socks 55.55% of the time. So, if we randomly opened thousands of "entangled" boxes with predetermined colored socks, on average we will see different colored socks 66.67% of the time.

(100% + 55.55% + 55.55% + 55.55%)/4 = 66.67%

But what happens if the sock color is not predetermined, as quantum theory asserts? Would we get a different percentage of different colored socks? Yes, it is obvious that we would get different colored socks 100% of the time.

What I have shown above is that different results (66.67% and 100%) are obtained for entangled boxes when the colors of the socks are either predetermined (as in classical physics) or indeterminate (as in quantum theory). Isn't this the principle behind Bell's Theorem and the case for nonlocality, or am I completely off base? Any response would be greatly appreciated.

 

[DrChinese]

Gary,

There are a lot of issues with your question that make it impossible to answer in its current form. Usually for an example of this type, there are 2 entangled particles. So I don't follow what there is 4 of.

Also, electrons have spin along 3 non-commuting axes, but photons I would call it 2 instead. Probably we should shift to an example that we can all discuss, ok?