Bell's Inequality - A misintepretation of probability?

[pat devine]

I recently attended a presentation on the fundamentals of quantum mechanics which focused on the most recent experimental tests on Bells Inequality. As part of the introduction the speaker derived Bells Inequality. The speaker made it sound very straightforward and it was, the proof was a piece of cake. However, when I examined the proof later there appeared to me to be what might be a subtle flaw in the logic. A seemingly innocuous assumption that may well be inappropriate is made during the course of the proof. For the sake of allowing me a good nights can someone put me right on this:The lecturer started by positing that when considering spin 1/2 particles Bell he was unable to derive any LHV (Local Hidden Variable) theorem which was consistent with the results as QM but there are so many possible LHV models to consider that he came to an impasse. He then went on to say: now Bell did something quite brilliant, he ignored quantum mechanics all together and came up with a statistical inequality that must be observed by any LHV system. He was then able to show that the predictions of QM where not consistent with this inequality.

For the next bit its important to understand the Gedankenexperiment:

Two spin one half particles are in a singlet state. Each of the pairs spins are measured by two distant apparatus. Each apparatus is such that it allows two possible measurements -
-A random measurement (each of the apparatus can make an independent spin measurement ie each different than the other) and
-A fixed measurement. ( both apparatus have a setting fixed at say 30 degree)The inequality is then constructed using the very crucial assumption that measurements made along the fixed axis are negatively correlated. ie when A measures up B measures down. In the non quantum world we can assume this to be correct even when no measurement is made. Hence we can essentially measure each particle twice, once along the random axis and then again along the fixed axis - but we don't actually measure along the fixed axis we just know what the outcome of the correlation will be. However we can not assume this to be true in the quantum world. Each particle can be measured only once by each apparatus, not twice and within the derivation it is assumed that this is possible. It is a subtle point that I haven't seen any discussion of. To sum up, that Bells inequality is violated by quantum experiments is merely due to the fact that it is not appropriate for quantum systems and therefore its violation reveals nothing more than the fact that two concurrent measurements of spin are not possible on the same particle at the same time. Rather than the current interpretation that it somehow invokes the reality of non locality. If the inequality is modified in such a manner as to allow for a distinct and separate measurement of a particle along the fixed axis/setting, then another pair of particles is required which dilutes the expectation values and would likely result in an inequality which is consistent with QM.
Regards
P

 

[atyy]

The Bell inequality does is only meant to rule out the classical conception of special relativistic causality, ie. the cause of any event is in its past light cone.

Whether there is another notion of locality that is appropriate is a matter of debate. The one that is widely agreed on is that the quantum theory although nonlocal with respect to classical special relativistic causality, it remains local in the different sense of forbidding faster than light transfer of classical information.

Also, it should be noted that the quantum state itself is nonlocal, as it is in Hilbert space. So at each time, the quantum state is assigned to an entire surface of simultaneity.

 

[DrChinese]

Each particle can be measured only once by each apparatus, not twice and within the derivation it is assumed that this is possible. It is a subtle point that I haven't seen any discussion of. To sum up, that Bells inequality is violated by quantum experiments is merely due to the fact that it is not appropriate for quantum systems and therefore its violation reveals nothing more than the fact that two concurrent measurements of spin are not possible on the same particle at the same time. Rather than the current interpretation that it somehow invokes the reality of non locality. If the inequality is modified in such a manner as to allow for a distinct and separate measurement of a particle along the fixed axis/setting, then another pair of particles is required which dilutes the expectation values and would likely result in an inequality which is consistent with QM.

If you provide a definition which assumes a measurement of Alice does not affect the outcome of a measurement of Bob, and vice versa: you cannot construct a data set of values for Alice and Bob which are consistent with QM. You don't need to measure Alice twice, you only need to assume that Alice had values at angles not being measured. That is "realism". If you deny this, you are in the company of many people - and you fully agree with the Bell result. That says there is no local realism (consistent with QM).

 

[morrobay]

I agree with all the above: While the formalism of QM is non local. Non locality may just be an artefact that appears in experiments when introducing into QM assumptions like the EPR elements of reality and counterfactual reasoning that are foreign to QM.
Also, there does not seem to be enough emphasis on relevant physical mechanisms.
Thats why I like papers by Accardi on the chameleon effect and the papers by Unnikrishnan:
Correlation functions,Bells inequality and the fundamental conservation laws

 

[zonde]

The inequality is then constructed using the very crucial assumption that measurements made along the fixed axis are negatively correlated. ie when A measures up B measures down. In the non quantum world we can assume this to be correct even when no measurement is made. Hence we can essentially measure each particle twice, once along the random axis and then again along the fixed axis - but we don't actually measure along the fixed axis we just know what the outcome of the correlation will be. However we can not assume this to be true in the quantum world. Each particle can be measured only once by each apparatus, not twice and within the derivation it is assumed that this is possible.

Assumption about perfect negative correlation is made using "what if" question. And "what if" question is valid in any world (except superdeterministic) given independence between source and detectors (no feedback).

 

[harrylin]

[..] we don't actually measure along the fixed axis we just know what the outcome of the correlation will be. However we can not assume this to be true in the quantum world. Each particle can be measured only once by each apparatus, not twice and within the derivation it is assumed that this is possible. It is a subtle point that I haven't seen any discussion of. To sum up, that Bells inequality is violated by quantum experiments is merely due to the fact that it is not appropriate for quantum systems and therefore its violation reveals nothing more than the fact that two concurrent measurements of spin are not possible on the same particle at the same time. Rather than the current interpretation that it somehow invokes the reality of non locality. If the inequality is modified in such a manner as to allow for a distinct and separate measurement of a particle along the fixed axis/setting, then another pair of particles is required which dilutes the expectation values and would likely result in an inequality which is consistent with QM.

Just now I found a paper by Unnikrishnan that seems related to your point: http://www.iisc.ernet.in/currsci/jul252000/UNNIKRISHNAN.pdf

I have only quickly looked through it and it's not clear to me where in the main text calculations or interpretations are original (it should be as the title is quite spectacular), but in the discussion section I noticed the following remark:

"Obtaining the correct correlations assuming locality [..] But till an actual measurement is made the companion particle does not acquire a definite state. There is direct proof for this from the fact that while an actual measurement of position on one of the particles disperses its momentum according to the uncertainty principle, this measurement and the resulting 100% predictability of the companion particle’s position do not cause corresponding dispersion in the momentum of the companion particle. This is the lesson from Popper’s experiment. [..]The paradox arises only from the necessity to assume reality for the outcomes before a measurement is made"