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:

=====

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

 

[Ben vdP]

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.
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.

Indeed for large objects, but that is no longer the context.

There has been another assumption made. The assumption that the measurement equals to or shows the value for the particle or is a projection of it. That is too simple, it doesn't have to be like that, and a measurement is not a passive operation.
It is not just in the math of the theory. Making one measurement 0 may make the other measurement 1 meaningless. It is too easy to assume that a value for the other measurement exists (or should be independent). In other words, not something to take for granted.

This might also be what Roberto Pavani pointed at.


To summarize:

The defining moment is when Victor prepares and sends the pair of particles. This implicitly impacts the measurements for Alice and Bob. The distance travelled by the particles is irrelevant, it doesn't impact the outcome. Bell's theorem is not able to explain it on the basis of some mechanism with the assumptions that have been made. The assumptions have, at least to a degree, been falsified.

 

[PeterDonis]

The assumptions have, at least to a degree, been falsified.

Yes, that was Bell's point when he formulated the theorem: reality violates the Bell inequalities, so at least one of the assumptions that were used to derive them must be false.

 

[Ben vdP]

Indeed, one other assumption that may have been made is that the dynamic behaviour and nature of interactions should be similar to those of a large classical object. Spot the 10 differences.

I think this one has been answered.

 

[Ben vdP]

The previously provided feedback and references have been helpful and gave some directions to look further into.

It was a bit confusing to read about measurements A0, A1, B0, B1 without an explicit statement on what they actually stand for
or what relationships exists between them.
It makes a lot of difference if the context is velocity and position or if the context is angle settings in spin component measurements.

I was looking for a description that would have been much more accurate and complete than the Wiki link.
And maybe some ingredients could then be identified that would make it all a bit more understandable.


This issue has been a bit in a limbo, but in-between all the distractions I had some time for it and some progress has been made.
Rather than opening a new discussion, I am revisiting this one, because it is all related.

Still there can be a lot that can be looked further into, I haven't gotten much to Bell inequalities in more detail for example.



- Confusion starts already by the usage of the term "particle" like in Bell's article. It gives the wrong idea.
It is unfortunate, but part of the legacy or historical development.

It may be possible to recognize a particle aspect and a wave aspect, but a quantum is not a particle.
It is not realistic to expect that a quantum should behave or interact or possibly self-interact in a similar way as a classical particle.
Particularly regarding spin and spin interactions.

Alice and Bob might be receiving and measuring whirling (EM) pulses or pulses with complex characteristics or force fields, instead of particles.
Anything would be better.



- As Bell puts it, the original idea of hidden variables should make qm complete or deterministic.

>>THE paradox of Einstein, Podolsky and Rosen [1] was advanced as an argument that quantum mechanics could not be a complete theory but should be supplemented by additional variables. These additional variables were to restore to the theory causality and locality.

Bell, J. S. (1964). On the Einstein Podolsky Rosen paradox.
Physics 1 195 200.

It could be that determinism or completeness are not such issues anymore as a century or half a century ago,
and that focus has shifted towards issues of causality and locality.
Maybe hidden variables or some factor are more needed for explaining correlations rather than determinism.

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).

Also in the link in note 5 is put:

>>The remarkable discovery made by Bell is that in any
theory of physics that is both local (physical influences
do not propagate faster than light) and realistic (physi-
cal properties are defined prior to and independent of ob-
servation) these correlations are bounded more strongly
than in quantum theory.


- Why should this be: "properties are defined prior to and independent of observation"

An electron has a spin, but it does not have a direct property for spin component in this direction or spin component
in that direction, or anything that is modelled on top of that. It looks rather artificial.
Maybe I have taken it too literal and is ment anything that in principle can be observed.


A detour with a limited more classical example:

An object has some different colour patterns with e.g. purple and other colours.
There are limited tests available for, lets say, measuring appearance or shape of the object.

A measurement is made on how the object appears in blue light (or filter).

The object however does not have a property how "it appears in blue light".
An explicit operation needs to be performed to get the measurement result.
And the result demonstrates itself only when the measurement is performed.

Another measurement or observable could be how "it appears in red light".
And again the measurement result is the combination of the colour pattern with the measurement operation.
It is not just a property of the object.

And if both tests needs to be done at the same time, then that could get a bit confusing.



- Regarding "assigning values to unmeasured quantities"

I see now that you can do this, at a minimum on paper, and name them observables. The concern was about possible side effects.

Physical measurements are operations and are invasive, I do not see them as just assignments or just determining values.


Another detour with a bit of drama.

Suppose there is a tornado and there is someone measuring wind speed using a handheld device.
The device will cause some drag to the wind and some energy transfer with that. But the impact of the measurement on the tornado
has hardly any significance. The effect can be fully neglected.
But On quantum level the effect cannot be neglected.



_ Bell used the example of two electrons created in a single proces moving away in opposite directions with opposite spins.

Classically you can think of two repelling electrons having a combined EM field. The shape of the field changes the further away they move.
But in the end the superposition principle in EM makes clear that there is no special relationship between the two electrons.
They just happen to have opposite spins. And a third electron just adds to the EM field.


But how do you describe it all in quantum mechanics?

There needs to be a system of two electrons that are related and not two separate systems of an electron each.
In qm that means a description with a wave function of the combination of the two.
And OK, a pair of photons could be a cleaner example.

In QM normally there are descriptions involving interactions, couplings and forces. And it could be possible to describe the field of the other
by using a potential, but no potential for a historical relationship.

Bell does not mention there are any forces between the two, only the historical relationship.
It just looks rather awkward to describe that they have opposite spins instead of to describe some kind of spin-spin interaction.

For entanglement you end up with wave functions like:

$$ Ψ = \frac { |+〉 ⊗ |-〉 - |-〉 ⊗ |+〉 } {\sqrt{2}} $$

or

$$ Ψ = \frac { |+〉 ⊗ |+〉 + |-〉 ⊗ |-〉 } {\sqrt{2}} $$


This results into expectation values


$$ cos ^ 2 ( \frac { α-β } {2} ) $$ and $$ sin ^ 2 ( \frac { α-β } {2} ) $$


The exact formulas are dependant on choices made in the definitions of the relevant angles with spin up and spin down.



- When measuring spin of a single quantum at an angle θ rotated to the z-axis, then it can be calculated that the expectation value is:


$$ cos ^ 2 ( \frac { θ } {2} ) $$


In a model with hidden variables; the hidden variable or factor needs to determine how a specific instance of an electron or photon will behave in a measurement. When Alice measures this then Bob would measure that.
The two electrons or photons of a specific pair out of an ensemble needs to behave very similar wjth regards to measurements with probability 1. If the angles of measurements are changed in unison then that should not change the outcome.

If the setting of only one is changed then there is a deviation in angle settings.

The above formula then gives as a result:


$$ cos ^ 2 ( \frac { α-β } {2} ) $$


This is the same result as with the wave function mentioned earlier.


In this example it is not significant in which order the measurements are made.
in a measurement one pair out of an ensemble is found. And as one measures one of the pair, immediately it is clear how the partner in crime would measure.



There actually is a hidden factor here, the spin orientation of the electrons or photons.
It is not possible to directly measure the spin orientation for a wave function $Φ(x,y,z,ξ,t)$
in which ξ denotes the orientation of the spin (actually three angles).

It touches trying to find a quantum description of a classical model of an electron having no model for the internal rotation.
It could just as well look like ball lightning.


Instead the standard methode is to use the spin impulsmoment representation with
using operator $S_z$ and eigenvalue $s_z$ , $Φ(x,y,z,s_z,t)$


( My old lecture notes mentions:

- G.E.Uhlenbeck, S.Goudsmit in Naturwiss. 13 (1925) 953; Nature 117 (1926) 264
- Zeitschr. fur Physik 43 (1927) 601 of Handbuch der Physik, Band V, Teil 1.
Pauli )

In the experiment the electrons or photons of a pair ought to have matching spin orientations due to conservation laws.



- Finding large correlations at remote distances does not require non-locality nor causality, but it requires a common factor.

Almost a century ago it was believed that new cases of polio were caused by eating ice-cream.

https://www.pearson.com/channels/ma...ed-link-between-ice-cream-and-polio-freakonom

The common factor is the summer when it is warm and a lots of ice-cream gets eaten. But under sumner conditions polio also spreads quicker.
I think the example is not really interesting or relevant except as an illustration.


The common factor in the physical problem is the way the pair of electrons or photons have been formed, which is historical.

Quantum mechanically there are consequences for the correlations that goes back to this original creation of the pairs.
Retro causality has put that on it's head, it seems.


You could say that all interactions are local and that causality between observables is not a necessity.

 

[Ben vdP]

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

A minor note:

Unfortunately the article in Phys. Rev. Lett. mentioned in note 7 was not accessible through the link as it required credentials.

There are several other links available like https://faculty.washington.edu/afine/PRL1982.pdf
I do not know what would be the most proper one,

 

[Ben vdP]

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.

A minor note :

I do not actually get the metaphor in note 5 above .

What is actually the common factor?
I would expect that you get higher correlations due to for example having a common taste.

 

[Nugatory]

A minor note :

I do not actually get the metaphor in note 5 above .

What is actually the common factor?
I would expect that you get higher correlations due to for example having a common taste.

For the sake of the analogy, we're assuming that the couples have been preselected to meet the conditions that every couple consists of one man and one woman and every couple consists of one blue-eyed and one brown-eyed person.
If you prefer, we could state it as Victor selects two individuals of opposite gender and eye color and sends one to Alice and one to Bob, bith of whom know that this is how Victor is choosing.

Either way, the important point here is that in stating the conditions we are implicitly assuming counterfactual definiteness: it's clear that we could in principle do this experiment with a collection of real humans and Alive and Bob would be justified in asszigning a value to the properties that have not been measured.