Measurement problem and computer-like functions

[Simon Phoenix]

Suppose we define the measurement of an observable A by v(A) v being an 'algorithm giving out one of the eigenvalues each time it is called' (we accept the axiom of choice)

Sorry, I'm a bit late to this thread and there have been many good answers, but I was struck by your initial question. Actually I think it's a good question because it really brings out an essential difference between classical and quantum thinking.

There is a difference between a measurement that 'chooses' one out of a set of pre-existing values, and a measurement that 'generates' a value that is a member of a set.

In standard QM there's no pre-existing value to 'choose' from.

In the usual Bell set up we have Alice and Bob, and at least conceptually we can imagine Alice to be on Earth and Bob to be on Pluto. One of the particles is winging its way to Bob who has set up his apparatus to measure some property. Now we could suppose that the particle is somehow carrying the set of possible values with it and all the measurement is going to do is to pick one of them. But what if Bob changes his mind about what to measure at the very last moment? Is the particle also carrying the new set of possible values with it in some pre-existing sense?

It's this kind of question that the Bell set up really tackles very beautifully. It asks what are the limitations on what we measure if we do assume that in some appropriate sense these properties have some kind of 'pre-existence'.

One of the things that took me a little while to appreciate when I first tried to understand Bell's arguments was the assumption that the result of Alice's measurement cannot depend on the setting of Bob's measurement device (and vice versa). It's so obvious - and it's also true in QM too. The only way we could have a dependence (assuming Alice and Bob are actually free to choose the settings) is if some information about Bob's setting reached Alice and affected the result she obtained.

Put this together with the assumption that there are some real properties that are orchestrating things (our so-called hidden variables) and one consequence of this is the Bell inequality.

OK - some of that is a little vague and imprecise, but I'm trying to highlight the essential components in an intuitive way. I don't think anyone would question too much the assumption that local results can't depend on distant settings but this question of whether physical properties pre-exist before measurement in some appropriate sense is really the mind-bender, for me at least.

 

[jk22]

Yess^the locality should hold, but it is maybe the question if it is in principle possible to determine completely with lambda the result, or if we should let the door open for an indeterminacy that would be determined afterwards.

However stebendaryl showed that if we can predict with certainty in some cases, then there is no place for indeterminacy, so that the parameters : angles and lambda should determine the result. Then the correlation is classical, it can have no bumps, or else it is a saw curve.