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Understanding bell's theorem: why hidden variables imply a linear relationship? 
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#1
Mar2412, 12:55 PM

P: 915

(Part of) The proof/logic of Bell's theorem goes thus:
With the measurements oriented at intermediate angles between these basic cases, the existence of local hidden variables would imply a linear variation in the correlation. However, according to quantum mechanical theory, the correlation varies as the cosine of the angle. Experimental results match the [cosine] curve predicted by quantum mechanics. Question: why do hidden variables need to imply a linear variation? we have many cases in physics/sciences/management/electrical where the relationship can be other than linear.....(exponential, cosine, sine, log, square, cube, quad, polynomial etc) 


#2
Mar2512, 10:21 PM

P: 1,414




#3
Mar2612, 01:22 AM

P: 1,583

But let me spell out the logic in greater detail. The example I'll discuss comes from here. We start with the experimental prediction of quantum mechanics that when you send two entangled photons into polarizers that are oriented at the same angle, the photons do identical things: they either both go through or they both don't. If you believe in local hidden variables, then you can conclude from this that right when the two photons were created, when they were presumably in sthe same place (that's the locality assumption), they decided in advance what polarizer orientations they should go through and which ones they shouldn't go through. So they basically have a list of "good angles" and "bad angles". If, for instance, one of the photons encounters a 15 degreeoriented polarizer, it will check whether 15 degrees is good or bad, and if it's good then it will go through. If x is the angle a polarizer is oriented, let's say P(x)=1 if x is a good angle, and P(x)=0 if x is a bad angle. Now Bell's theorem is concerned with the probability that the two photons behave differently if the polarizers are turned to different orientations. But since, as we said, the photons are just deciding to go through or not go through based on a previously agreed upon decision about what angles are good and bad, all we're talking is the probability that P(θ1)≠P(θ2), where θ1 is the angle of the first polarizer and θ2 is the angle of the second polarizer. In the Herbert proof I linked to, the specific case we're talking about is the probability that P(30)≠P(30), i.e. the probability that if you turn one polarizer at 30 degrees and the other one at 30 degrees, you get a mismatch. Now under what conditions is the statement P(30)≠P(30) true? Well, it can only be true if either P(30)≠P(0) OR P(0)≠P(30) (because if both of these were false we would have P(30)=P(0)=P(30)). The word "OR" is the crucial part, because one of the basic rules of probability is that the probability of A OR B is less than or equal to the probability of A plus the probability of B. So the probability that P(30)≠P(30) is less than or equal to the probability that P(30)≠P(0) plus the probability that P(0)≠P(30)  and Bingo, we've derived a Bell inequality! And note the crucial role counterfactual definiteness played in the proof: we are assuming it makes sense to talk about P(0), even though we only measured P(30) and P(30). In other words, the assumption is that measurements that we did not make still have well defined answers as to what would have happened if you made them. Does that make sense? The form of Bell's inequality, A+B≤C, fundamentally comes from the fact that probabilities are (sub)additive. 


#4
Mar2612, 02:13 AM

P: 1,414

Understanding bell's theorem: why hidden variables imply a linear relationship?



#5
Mar2612, 03:04 AM

P: 1,583

1. Entangled photons behave identically at identical polarizer settings. 2. The believer in local hidden variables says that the polarizer angles the photons will and won't go through are agreed upon in advanced by the two entangled photons. 3. In order for the agreedupon instructions (to go through or not go through) at 30 and 30 to be different, either the instructions at 30 and 0 are different or the instructions at 0 and 30 are different. 4. The probability for the instructions at 30 and 30 to be different is less than or equal to the probability for the instruction at 30 and 0 to be different plus the probability for the instructions at 0 and 30 to be different. 


#6
Mar2612, 03:47 AM

P: 1,414

If you want to understand entanglement in optical Bell tests, then you don't focus on the detection attributes. You focus on the design of the experiments, the known behavior of light, and the apparent fact that the variable that determines individual detection is irrelevant wrt coincidental detection. Thus arriving at the conclusion that coincidental detection is determined by an underlying parameter that isn't varying from pair to pair. It's determined by the relationship between entangled photons. A constant. Now, how would you model that? 


#7
Mar2612, 09:48 AM

P: 1,583




#8
Mar2612, 11:14 AM

Sci Advisor
PF Gold
P: 5,299




#9
Mar2612, 11:31 AM

Sci Advisor
PF Gold
P: 5,299

http://departments.colgate.edu/physi...nglement09.PDF See the equations leading up to (14), which is the result which is mathematically identical to Malus, but as you can see is obtained completely independently. 


#10
Mar2612, 12:22 PM

P: 915

1. between the angle and probabilities or 2. between the two entangled particles 


#11
Mar2612, 12:31 PM

P: 1,583




#12
Mar2612, 12:33 PM

P: 29




#13
Mar2612, 01:33 PM

Sci Advisor
PF Gold
P: 5,299




#14
Mar2612, 05:20 PM

P: 915

Are we saying that: for hidden variable hypothesis  the two photons would be expect to have perfect correlation (or anticorrelation) at ALL angle settings? where angle = angle between the axis of the two polarizers? and then we further conclude that: since the perfect correlation (or anti correlation) occurs only at angle 0 or 90 (and is a cosine relation at other angles) therefore the hidden variable hypothesis is weak/rejected 


#15
Mar2612, 05:38 PM

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#16
Mar2612, 08:46 PM

P: 915

is the correlation between quantum entangled particles not linear then? 


#17
Mar2612, 08:52 PM

P: 1,583




#18
Mar2712, 08:58 AM

P: 273

Here cosine curve is QM prediction and linear function is the maximum correlation achievable with local realistic model, obtained by replacing ≥ with = in Bells inequality. This linear function actually corresponds to a very simple thought experiment by Bohm: At the source two atoms get random but opposite spins. During the measurement the spin of each atom is projected onto the corresponding direction of measurement and the sign of this projection becomes the outcome. In Bell's terms A(a,λ) = sign cos (aλ) = { 1 when aλ ≤ π/2 otherwise 1 } Now, interesting question is what kind of functions are allowed by Bell's inequalities. The impression one gets is that that no LR model can get above the straight line. This is not so, its a bit more subtle. For example, consider roulette wheel with 10 alternating red/black sectors, where the outcome of a measurement is determined by the color at angle a when the wheel stops: A(a,λ) = sign cos 5(aλ) = { 1 when aλ mod π/5 ≤ π/10 else 1 }. It reproduces the usual values at 0°, 90°, 180° and 270° but jumps all over the place in between: from 1 at 0° to 1 at 36° back to 1 at 72°, crosses 0 at 90°, 1 at 108° etc., all that in perfect agreement with Bell. On the other hand, QM predicts 0.7 at 45° where no LR model can get above 0.5. 


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