Bell's theorem and measurements not done

[Ben vdP]

TL;DR

Bell's theorem:
Can you create expressions that include outcomes of measurements that have not been performed.

I wanted to have a quick check on how Bell's theorem was formulated so consulted the wiki page:

https://en.wikipedia.org/wiki/Bell's_theorem

But then saw the following odd section:

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Hypothetical characters Alice and Bob stand in widely separated locations. Their colleague Victor prepares a pair of particles and sends one to Alice and the other to Bob. When Alice receives her particle, she chooses to perform one of two possible measurements (perhaps by flipping a coin to decide which). Denote these measurements by A0

and A1

. Both A0

and A1

are binary measurements: the result of A0

is either +1 or −1, and likewise for A1. When Bob receives his particle, he chooses one of two measurements, B0 and B1, which are also both binary.

Suppose that each measurement reveals a property that the particle already possessed. For instance, if Alice chooses to measure A0 and obtains the result +1, then the particle she received carried a value of +1 for a property a0.

Consider the combination a0b0+a0b1+a1b0−a1b1=(a0+a1)b0+(a0−a1)b1.

Because both a0 and a1 take the values ±1, then either a0=a1 or a0=−a1. In the former case, the quantity (a0−a1)b1 must equal 0, while in the latter case, (a0+a1)b0=0. So, one of the terms on the right-hand side of the above expression will vanish, and the other will equal ±2. Consequently, if the experiment is repeated over many trials, with Victor preparing new pairs of particles, the absolute value of the average of the combination a0b0+a0b1+a1b0−a1b1 across all the trials will be less than or equal to 2. No single trial can measure this quantity, because Alice and Bob can only choose one measurement each, but on the assumption that the underlying properties exist, the average value of the sum is just the sum of the averages for each term. Using angle brackets to denote averages |⟨A0B0⟩+⟨A0B1⟩+⟨A1B0⟩−⟨A1B1⟩|≤2. This is a Bell inequality, specifically, the CHSH inequality.

Its derivation here depends upon two assumptions: first, that the underlying physical properties a0,a1,b0, and b1 exist independently of being observed or measured (sometimes called the assumption of realism); and second, that Alice's choice of action cannot influence Bob's result or vice versa (often called the assumption of locality).

=====

Alice performs (each time) by choice one out of two measurements A0 or A1 but not both.

If she chooses A0 then the outcome of it is 1 or -1.
But the outcome of measurement A1 that has not been done is not +1 or -1 it is actually undefined.
So the values of all the expressions are also undefined.

Furthermore it is supposed that the measurements of A0 and A1 are independent; that does not have to be the case either.

So what would justify the reasoning in the section?

 

[Ben vdP]

I might have lost some of the formatting when trying to copy and paste from the wiki page.

 

[Nugatory]

So what would justify the reasoning in the section?

It’s a proof by contradiction argument. The calculation is correct if the realism and locality assumptions hold; we do the experiment and find that the result of the calculation is wrong; therefore at least one of those assumptions must be wrong.

The realism assumption is what justifies assigning values to unmeasured quantities: When Alice measures +1 on axis 0, realism implies that Bob would have measured -1 on axis 0 even if they chose to measure on axis 1 instead. (I prefer the term “counterfactual definiteness” myself).

Furthermore it is supposed that the measurements of A0 and A1 are independent; that does not have to be the case either.

Only if we assume that the results of Alice’s and Bob’s coin flips, used to choose the measurement axis, are not independent. In principle there is no way of completely refuting this “superdeterminism loophole” but in practice the design of the best current experiments makes it wildly implausible.

There is also the “fair sampling loophole: if the pairs detected and used in the calculation are somehow biased the inequality can be violated by a local realistic theory. Modern experiments have decisively eliminated that possibility.

 

[Ben vdP]

Ok, thanks for the reply.

I may have unconsciously rejected the realism assumption since I find it highly questionable that you can assign values to unmeasured quantities. Nor that measurements A0 and A1 are independent is just a case of coin flipping. There is a reason you can perform only one of the two measurements.

That Alice measures +1 on axis 0 implying that Bob measures -1 on axis 0 is OK.
But if Bob measures on axis 1 then I don't think you can say anything about any measurement on axis o for Bob or only limited for the pair of particles.

That is within an idealized experiment, supposing also that the measurements reflect actual values for the particles, not obvious, and there is no process that can distort it.

 

[Nugatory]

I may have unconsciously rejected the realism assumption since I find it highly questionable that you can assign values to unmeasured quantities.

It is clearly valid to assign values to unmeasured properties of large objects and it was something of a surprise that unmeasured properties of much smaller ones did not work that way.
One larger object example that I’ve seen (and borrowed): There is a room full of heterosexual couples. We know that they have been preselected such that one member of each couple is blue-eyed and one brown-eyed. Your Victor character chooses a couple, sends one member to Alice and the other to Bob who (by the rules of the game) are each allowed to measure only one property selected by coin-flip of their experimental subject.
Now when Alice and Bob meet afterwards, Alice says “My person was a woman” and Bob says “my person had blue eyes”. Alice’s person was a brown-eyed woman even though her eye color was not measured; Bob’s person was a blue-eyed man although his gender was not measured.

Nor that measurements A0 and A1 are independent is just a case of coin flipping. There is a reason you can perform only one of the two measurements.

I don’t understand this point. Yes, of course there is a reason Alice can only perform one of the two measurements (it’s in the math of the theory) but that doesn’t stop her from flipping a coin to choose which one she makes, nor that Bob can use his own independent coin for his choice.

 

[Nugatory]

That is within an idealized experiment, supposing also that the measurements reflect actual values for the particles, not obvious,

To evaluate that we have to look at the detailed construct of the real non-idealized experiment

and there is no process that can distort it.

This would be an example of the fair selection loophole, which as I said above has been decisively closed.

This experiment might be a good start: https://arxiv.org/pdf/1508.05949

 

[Roberto Pavani]

A side note: Fine (Phys. Rev. Lett. 48, 291, 1982) shows that the Bell inequalities hold if and only if a joint distribution for all observables exists. But Kolmogorov's linearity of expectation requires exactly that a common probability space.
So when the joint distribution doesn't exist, isn't the sum itself outside the domain of applicability of the formula?

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.48.291