Bell's Inequality Theorem

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I read the following article (I think the author writes on this forum) and thought I understood the reasoning (at least a 0.333 chance of a match, whatever the setting, quantum mechanics for a 120 degree difference in angle predicts a 0.25 chance of a match, measurement shows it to be around 0.25 chance of a match).

http://www.drchinese.com/David/Bell_Theorem_Easy_Math.htm

But I was assuming that the result at a given angle would be the same if the same photon were tested again at that angle, and was was thinking that if further tests on the *same* particle could be made (in a thought experiment at least) then wouldn't you end up with the result for all three settings, and why couldn't it have been that at the beginning (as the combinations would be what was measured, and would be at the 0.25 probability of a match that they were measured to be)? I accept that it would be strange, because it would seem as though the combinations were contrived to score the 0.25 match given what we measured, and how could a physical reality react like that to the angles we choose to measure? I'm assuming I am missing something here. Is it perhaps that it is only when the first measurement is taken, and the particle materialises at a certain position that there is the 0.25 chance of a match, maybe after that the measurements are 0.333+ of a match, and that this is used for quantum encryption to check if something has already been observed. Just not sure. Any help appreciated.
 

DrChinese

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But I was assuming that the result at a given angle would be the same if the same photon were tested again at that angle, and was was thinking that if further tests on the *same* particle could be made (in a thought experiment at least) then wouldn't you end up with the result for all three settings, and why couldn't it have been that at the beginning (as the combinations would be what was measured, and would be at the 0.25 probability of a match that they were measured to be)? I accept that it would be strange, because it would seem as though the combinations were contrived to score the 0.25 match given what we measured, and how could a physical reality react like that to the angles we choose to measure? I'm assuming I am missing something here. Is it perhaps that it is only when the first measurement is taken, and the particle materialises at a certain position that there is the 0.25 chance of a match, maybe after that the measurements are 0.333+ of a match, and that this is used for quantum encryption to check if something has already been observed. Just not sure. Any help appreciated.
Welcome to PhysicsForums, name123!

You can do multiple tests of polarization on a photon, by putting it through a series of polarizers. But you won't be demonstrating much.

Assume you have angles X, Y and Z:
Measure Alice's entangled photon at X, Y and then Z.
Measure Bob's entangled photon at Y, Z and then X.

You only have a chance that you will see the same outcomes of X, Y and Z for both.

On the other hand: any photon polarized at X can be checked at X again and the result will be the same.
 
444
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Thank you Dr Chinese for the welcome, reply, and your website :) Though unfortunately, I'm still not quite clear on what the scientific results are, and still need just a little more help please.

If I measure Alice's entangled photon at X, Y, and Z then I presumably know the results for Bob's for X, Y, and Z because they are the opposite because they are entangled. Now what I am unsure of is whether if I took the three measurements you talked of; Alice's at X and Bob's and Y, then Alice's at Y and Bob's at Z and then Alice's at Z and Bob's at X: would the likelihood of a match only be 0.25 for the first measurement ( Alice's X and Bob's Y) which causes the photon to materialise, and then the further two measurements have the 0.333 chance of a match as would be expected given the X, Y and Z values Alice's photon turns out to have? Or, and this would seem really weird, but is the chance of a match 0.25 for all three measurements (surely after all three are taken, and the results remain the same, the chance of a match would return to 0.333 if you were to repeat the tests randomly testing two of the tree angles (since it has those values before these measurements))?
 

Nugatory

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If I measure Alice's entangled photon at X, Y, and Z then I presumably know the results for Bob's for X, Y, and Z because they are the opposite because they are entangled.
The first measurement you make on each particle ends the entanglement. When you measure Bob's X you learn what the value of Alice's X would be if it is measured and when you measure Alice's Y you learn what the value of Bob's Y would be if it is measured. After you have those two measurements, the subsequent behavior of both particles will be that of of two independent particles evolving from initial states determined by their respective first measurements. Thus, only the first measurement Bob makes gives him any information about Alice's results, and vice versa.
 

DrChinese

Science Advisor
Gold Member
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Thank you Dr Chinese for the welcome, reply, and your website :) Though unfortunately, I'm still not quite clear on what the scientific results are, and still need just a little more help please.

If I measure Alice's entangled photon at X, Y, and Z then I presumably know the results for Bob's for X, Y, and Z because they are the opposite because they are entangled. Now what I am unsure of is whether if I took the three measurements you talked of; Alice's at X and Bob's and Y, then Alice's at Y and Bob's at Z and then Alice's at Z and Bob's at X: would the likelihood of a match only be 0.25 for the first measurement ( Alice's X and Bob's Y) which causes the photon to materialise, and then the further two measurements have the 0.333 chance of a match as would be expected given the X, Y and Z values Alice's photon turns out to have? Or, and this would seem really weird, but is the chance of a match 0.25 for all three measurements (surely after all three are taken, and the results remain the same, the chance of a match would return to 0.333 if you were to repeat the tests randomly testing two of the tree angles (since it has those values before these measurements))?
Actually, if you measure Alice's entangled photon at X, Y, and Z then you ONLY know the results for Bob for X! The others will have an element of randomness relative to Alice. And vice versa. You would use the Malus law - cos^2(theta) - to calculate that.

Please be aware that the .333 chance is somewhat hypothetical. This is actually a lower limit for a realistic assumption more than anything. Since local mechanisms give the wrong results, no one really is putting forth any alternative predictions.

Nugatory weighed in as I was writhing this... see his comment about entanglement ending upon measurement.
 
444
3
Thank you both for your help, that has sorted it out (the 0.25 is only for the first measurement, as then the entanglement ends).

Thanks & Happy New Year :)
 

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