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This posting is directly related to the issue in The System and the Measuring Gadget.

The QM expectation is given by:

$$\langle\sigma_{1}.\vec{a}{\;}\sigma_{2}.\vec b\rangle=-\vec a.\vec b$$

In the above relation we are considering the measured value of spin which is the outcome between the value of some property of the system itself and the measuring gadget

The "classical" formula for evaluating the expectation with the hidden variable is as follows:

$$P(\vec a.\vec b)=\int d\lambda\,\rho (\lambda) A(\vec a)B(\vec b)$$

Now some property of the system may depend on the value of $\lambda$ and the probability distribution $\rho (\lambda)$. Is the effect of measurement being fully accounted for by the the hidden variable $\lambda$ and the pdf $\rho (\lambda)$, especially in view of the fact that the process of measurement modifies the wave function itself.

Would it be possible remove the contradiction between QM and commonsense intuition,expressed through Bell's Inequality, by considering the above factors?

Reference for Bell's Original Paper.

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Could somebody help me with getting bold face or arrows with the vectors a and b?Thanks in advance. –  Anamitra Palit Aug 7 '12 at 15:17
    
Try \vec{a} for $\vec{a}$ –  John Rennie Aug 7 '12 at 15:21
    
Thanks for your suggestion –  Anamitra Palit Aug 7 '12 at 15:34
    
I don't understand your question. Common sense intuition -- whatever that means but you seem to say Bell's inequality is what it means-- tells us that QM is wrong. Experiment tells us that common sense intuition is wrong. Science tells us that experiment is the arbiter here. –  Raskolnikov Aug 7 '12 at 15:37
    
Classicalintuition accepts the idea of physical properties of a system independent of experimental observations/interactions.QM emphasizes that the the measured values of the observables are dependent on the measurement itself. In the derivation of Bell's theorem[if you look at the second formula in the original posting],lambda and its pdf is used to understand/demonstrate the dependence of the $measured {\;}values$ A and B to calculate the expectation. In fact during measurement the wave function gets modified due to the involvement of the gadget.How does the hidden variable take care of it? –  Anamitra Palit Aug 7 '12 at 15:50
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1 Answer

This complicated formal argument is why Bell's original paper spawned countless attempts to circumvent it, and this is why to convince yourself that it can't be circumvented, you should use a case where intuition is firm. A clear argument was presented by Mermin (although Bell notes this too to in his paper some extent, when he states in a one-sentence aside that the small angle limit is intuitive, but unfortunately he doesn't go into detail).

The quantum correlation is such that if you make tiny angle between a and b, the correlation is proportional to $\cos(\theta) = 1 - {\theta^2\over 2}$. If you make the same angle between b and c, the correlation is $\cos(2\theta) = 1 - {2\theta^2}$ (since the angle is doubled).

So if a and b are such that they are 99% correlated, and b and c are 99% correlated, then a and c are 96% correlated in quantum mechanics.

The thing is, if you have predetermined yes-no answers which you know you are going to get, and these answers are carried by the electrons in secret crib-sheets, then you can bound the correlations. If a and b are 99% correlated, so a and b agree 99% of the time, and b and c agree 99% of the time, then a and c agree at least 98% of the time. The number of mismatches between a and b plus the number of mismatches between b and c is the maximum number of mismatches between a and c.

So you don't need to write down complicated averaging formulas, or do any mathematical manipulations--- the Bell inequality is that a and c have to be correlated more than 98%, and in quantum mechanics, it's 96%. That's it. There is nothing you can do. It can't be fixed.

Your idea is that maybe the measuring device state can depend on which answer the electron was going to give, so that the measuring devices are somehow secretly depending on the electron variables. This is called the "superdeterminism loophole", and it isn't a loophole as much as a conspiracy theory.

In order for this to be true, of each of the countless ways to measure the spin of an electron, there must be secret biases which lead the measuring device to respond in a biased way, no matter when or how it detects the electron, which electron it detects, or what you do to the device. This type of conspiratorial physics is ridiculous--- it demands that our devices are secretly coordinating with each other and with all the electrons they could possibly measure. It cannot be ruled out logically, but only in the same way that you can't rule out that we are living in the Matrix, and space-aliens are projecting thoughts into our heads.

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