A Does the Bell theorem assume reality?

[DarMM]

We already have Bell's theorem

Yes which removes local single-world causal hidden variable theories. Anything outside that isn't eliminated, so for remaining ideas like the Relational Block World, Many Worlds or QBism the theorem has no power.

but QBism seems immune to no-go theorems

Well any of the remaining interpretations seem immune to no-go theorems so far. Many have made this objection to Many-Worlds. In both cases the objection doesn't make sense to me, we simply don't currently have a no-go theorem against it.

I wouldn't even say it's wrong though — they've just restricted it's domain of applicability to single-user experiences. If one is interested in how nature works and doesn't take a solipsistic view, QBism doesn't have anything to say.

QBism isn't solipsistic though. It just says QM is a single-user Bayesian calculus, you can then ask why does the Bayesian calculus for single users have this form, e.g. why is the Law of Total Probability modified, what does that imply about the world? They do make specific claims about the external world several times in their papers, which they wouldn't if they were solipsistic.

This is something QBists refuse to do

Again I don't think so, since they make claims about the external world.

Also none of these objections relate to $\psi$-epistemic approaches more broadly which you seemed to be disagreeing with above by arguing for $\psi$ being ontic.

 

[akvadrako]

In both cases the objection doesn't make sense to me, we simply don't currently have a no-go theorem against it.

I mean it seems immune to all possible no-go theorems, not just the ones we have. Can you imagine the kind of no-go result we might expect about the reality that QBism proposes?

Again I don't think so, since they make claims about objective reality.

You're right, they do make assumptions like reality is local. To be more precise, they don't make any quantifiable predictions beyond single-user cases. Once you are trying to analyze a system with two users, it doesn't say how those two subjective systems interplay beyond the assumptions they've made about the shared reality.

Also none of these objections relate to $\psi$-epistemic approaches more broadly which you seemed to be disagreeing with above by arguing for $\psi$ being ontic.

Let's assume $\psi$ is epistemic. That doesn't show how to reconcile different wave functions in the two-user case. Assuming (a different) $\psi$ is objective is one way to do that — at least it goes some of the way and provides a framework to work in. Maybe that's the key point: it's an explanation one-level deeper than the epistemic-only approaches offer. They make claims about multi-user experience, but don't explain how they are achieved. If they did provide an alternative, we could be talking about that theory instead of the epistemic side.

In summary, I don't see anything wrong with epistemic theories, they just have limited scope. Where QBism goes beyond epistemic claims, it seems to be mostly assumptions and doesn't provide much explanatory power.

 

[stevendaryl]

But this is where I have the problem. Why I am seeking to undertstand the "reality" that Bell is using here.

Let me pick a particularly simple version of the EPR paradox. We have a source of anti-correlated electron-positron pairs. Alice has a device that measures spins along one of three possible axes:

• a: Along the y-axis.
• b: Along the line that makes a 120 degree angle with the y-axis in the x-y plane.
• c: Along the line that makes a 240 degree angle with the y-axis in the x-y plane.

We'll map "spin-up" to the result +1 and "spin-down" to the result -1.

Bob has the same three choices.

Alice's and Bob's results are anti-correlated, meaning if they both choose to measure the spin along the same axis, they always get opposite results.
So here is a model for what is happening:

1. Associated with each particle pair, there are three numbers, $A, B, C$. Each is either +1 or -1.
2. If Alice measures her particle along axis a, she will get result A. If she measures her particle along axis b, she will get B. If she measures along axis c, she will get C.
3. Bob always gets the opposite: If he measures along axis a, he will get result -A, etc.

The awkward situation is that even though this model has 3 numbers associated with each pair, $A, B, C$, Alice can only measure one of them, and Bob can only measure one. So at best, they can only measure 2 out of the 3 numbers. But the model assumes that there is a result for all three directions, even if you can only measure two of them.

Let's assume that Alice and Bob perform lots and lots of measurements on twin pairs. Let's define some statistical quantities:

• $E(a, b) = \frac{1}{N} \sum_n A_n B_n$ where $A_n$ is the value of $A$ for pair number $n$, and $N$ is the number of pairs produced.
• Similarly, $E(a, b)$ and $E(b, c)$

Here's where we use some pure mathematics to get some inequalities on these quantities.

1. $E(a,b) + E(a, c) = \frac{1}{N} \sum_n (A_n B_n + A_n C_n)$

2. Since $B_n = \pm 1$, $B_n B_n = 1$. So we can rewrite the right-hand side of equation 1 as
$\frac{1}{N} \sum_n (A_n B_n + A_n B_n B_n C_n)$
$= \frac{1}{N} \sum_n (A_n B_n (1 + B_n C_n))$

3. Taking absolute values, we get:
$\frac{1}{N} |\sum_n (A_n B_n (1 + B_n C_n))| \leq \frac{1}{N} \sum_n |A_n B_n| |1+B_n C_n|$

4. Since $|A_n B_n| = 1$, we get:
$\frac{1}{N} \sum_n |A_n B_n| |1+B_n C_n|$
$= \frac{1}{N} \sum_n |1+B_n C_n|$
$= \frac{1}{N} \sum_n (1+B_n C_n)$
$= 1 + \frac{1}{N} \sum_n B_n C_n$
$= 1 + E(b, c)$

5. So we conclude that:
$|E(a,b) + E(a, c) | \leq 1 + E(b,c)$

But experimentally, we find that:

• $E(a,b) = E(a, c) = -1/2$

Technically, we can prove that $E(a,b) = cos(120) = -1/2$

That violates the inequality:

$|E(a,b) + E(a,c)| = |-1/2 + -1/2| = 1$
$1 + E(b,c) = 1 + -1/2 = 1/2$

There is one technical assumption that may or may not be worrisome. Alice and Bob can't actually measure $E(a,b)$ for the entire run, because some of the runs, they will measure the spins along axes $a$ and $c$. For that run, they have no idea what the value of $B$ is. For other runs, they will have no idea what the value of $A$ or $C$ is. What's assumed (and I'm not sure if there is a name for this assumption) is that the correlation $E(a,b)$ computed using only those runs where $A$ and $B$ are measured gives the same result as if we had computed $E(a,b)$ using all the runs. That is, we're assuming that the statistics for unmeasured quantities is the same as for the quantities that were actually measured.

 

[Mentz114]

Let me pick a particularly simple version of the EPR paradox. We have a source of anti-correlated electron-positron pairs. Alice has a device that measures spins along one of three possible axes:

• a: Along the y-axis.
• b: Along the line that makes a 120 degree angle with the y-axis in the x-y plane.
• c: Along the line that makes a 240 degree angle with the y-axis in the x-y plane.

We'll map "spin-up" to the result +1 and "spin-down" to the result -1.
[snipped some]
There is one technical assumption that may or may not be worrisome. Alice and Bob can't actually measure $E(a,b)$ for the entire run, because some of the runs, they will measure the spins along axes $a$ and $c$. For that run, they have no idea what the value of $B$ is. For other runs, they will have no idea what the value of $A$ or $C$ is. What's assumed (and I'm not sure if there is a name for this assumption) is that the correlation $E(a,b)$ computed using only those runs where $A$ and $B$ are measured gives the same result as if we had computed $E(a,b)$ using all the runs. That is, we're assuming that the statistics for unmeasured quantities is the same as for the quantities that were actually measured.

Nice exposition. Is that not rotational symmetry ? Which we already assumed somewhere down the line I think.

 

[DarMM]

I mean it seems immune to all possible no-go theorems, not just the ones we have. Can you imagine the kind of no-go result we might expect about the reality that QBism proposes?

If I could imagine a no-go result, I'd be working on publishing it.
Couldn't you say the same about Many-Worlds?
Or the acausal explanations?

To be more precise, they don't make any quantifiable predictions beyond single-user cases. Once you are trying to analyze a system with two users, it doesn't say how those two subjective systems interplay beyond the assumptions they've made about the shared reality.

It makes the same quantifiable predictions as QM, the same as other interpretations claim. As for the two-user case, it resolves these in the usual de Finetti or subjective Bayesian way they are resolved, via a de Finetti type theorem or similar, I don't see any issues. It isn't saying the underlying reality is subjective after all. Although Rovelli's Relational Interpretation does.

Let's assume $\psi$ is epistemic. That doesn't show how to reconcile different wave functions in the two-user case. Assuming (a different) $\psi$ is objective is one way to do that — at least it goes some of the way and provides a framework to work in. Maybe that's the key point: it's an explanation one-level deeper than the epistemic-only approaches offer. They make claims about multi-user experience, but don't explain how they are achieved. If they did provide an alternative, we could be talking about that theory instead of the epistemic side.

This all applies equally to let's say the macrostate $\rho$ in Statistical Mechanics. I could equally say there is an issue with the two user case there, posit an objective $\rho$ to resolve it and say that this provides you one level deeper of an explanation and this makes it superior to viewing $\rho$ as epistemic and that $\rho$-epistemic views are limited in scope.

However for Statistical Mechanics, you'd be wrong. $\rho$ is an epistemic object.

I don't think the statement "$\psi$ being epistemic is more limited in scope" is a valid argument against $\psi$ being epistemic, it's just stating what its scope would be if it in fact were epistemic, but you can't use that to help decide if it is epistemic.

Where QBism goes beyond epistemic claims, it seems to be mostly assumptions and doesn't provide much explanatory power.

Well they would say seeing $\psi$ as epistemic explains several of its properties more naturally and provides simpler reasoning about many quantum mechanical features. See explanations of teleportation and no-cloning. It was also the motivation for the di Finetti theorem which is now used in Quantum Information and Engineering.

All any of the interpretations can currently offer is "motivation", up until the point were they're eliminated or make different predictions.

 

[DarMM]

Also I should say $\psi$-epistemic interpretations aren't limited in scope in general. For example retrocausal and acausal views come with a specification of what reality is like/the underlying physics. It just turns out $\psi$ isn't a part of that underlying physics, but simply something you use in certain epistemic situations like $\rho$ in statistical mechanics.

 

[stevendaryl]

Nice exposition. Is that not rotational symmetry ? Which we already assumed somewhere down the line I think.

No, it's not rotational symmetry. So the assumption is that there are three hidden variables $A_n, B_n, C_n$ for run number $n$. Alice and Bob only see 2 out 3 of those. So their record of the run of values looks like

$A_1 = +1, B_1 = ?, C_1 = +1$
$A_2 = ?, B_2 = +1, C_2 = -1$

etc.

The question marks represent the values not measured. So when you're trying to compute the correlation between variables A and B, for example, you have to skip run number 1, because on that run, you don't know the value of $B$.

So the assumption is that $\frac{1}{N} \sum_n A_n B_n \approx \frac{1}{N_{AB}} \sum'_n A_n B_n$. The first sum is over all runs, while the second sum is over those runs where you happened to have measured $A$ and $B$.

 

[stevendaryl]

Nice exposition. Is that not rotational symmetry ? Which we already assumed somewhere down the line I think.

Rotational symmetry would be the assumption that $E(a,b) = E(a,c) = E(b,c)$. That's not the same thing.

 

[Mentz114]

Rotational symmetry would be the assumption that $E(a,b) = E(a,c) = E(b,c)$. That's not the same thing.

Yes, I worked it out. The fact that we only have limited information is not affected by any rotation which makes rotation irrelevant.

This is notable, $E(a,b) = E(a, c) = -1/2$ because that distribution has maximum entropy reflecting the absent degree of freedom.

 

[N88]

Let me pick a particularly simple version of the EPR paradox. We have a source of anti-correlated electron-positron pairs. Alice has a device that measures spins along one of three possible axes:

• a: Along the y-axis.
• b: Along the line that makes a 120 degree angle with the y-axis in the x-y plane.
• c: Along the line that makes a 240 degree angle with the y-axis in the x-y plane.

We'll map "spin-up" to the result +1 and "spin-down" to the result -1.

Bob has the same three choices.

Alice's and Bob's results are anti-correlated, meaning if they both choose to measure the spin along the same axis, they always get opposite results.
So here is a model for what is happening:

1. Associated with each particle pair, there are three numbers, A,B,CA,B,CA, B, C. Each is either +1 or -1.
2. If Alice measures her particle along axis a, she will get result A. If she measures her particle along axis b, she will get B. If she measures along axis c, she will get C.
3. Bob always gets the opposite: If he measures along axis a, he will get result -A, etc.

The awkward situation is that even though this model has 3 numbers associated with each pair, A,B,CA,B,CA, B, C, Alice can only measure one of them, and Bob can only measure one. So at best, they can only measure 2 out of the 3 numbers. But the model assumes that there is a result for all three directions, even if you can only measure two of them.

Let's assume that Alice and Bob perform lots and lots of measurements on twin pairs. Let's define some statistical quantities:

• E(a,b)=1N∑nAnBnE(a,b)=1N∑nAnBnE(a, b) = \frac{1}{N} \sum_n A_n B_n where AnAnA_n is the value of AAA for pair number nnn, and NNN is the number of pairs produced.
• Similarly, E(a,b)E(a,b)E(a, b) and E(b,c)E(b,c)E(b, c)

Here's where we use some pure mathematics to get some inequalities on these quantities.

1. E(a,b)+E(a,c)=1N∑n(AnBn+AnCn)E(a,b)+E(a,c)=1N∑n(AnBn+AnCn)E(a,b) + E(a, c) = \frac{1}{N} \sum_n (A_n B_n + A_n C_n)

2. Since Bn=±1Bn=±1B_n = \pm 1, BnBn=1BnBn=1B_n B_n = 1. So we can rewrite the right-hand side of equation 1 as
1N∑n(AnBn+AnBnBnCn)1N∑n(AnBn+AnBnBnCn)\frac{1}{N} \sum_n (A_n B_n + A_n B_n B_n C_n)
=1N∑n(AnBn(1+BnCn))=1N∑n(AnBn(1+BnCn))= \frac{1}{N} \sum_n (A_n B_n (1 + B_n C_n))

3. Taking absolute values, we get:
1N|∑n(AnBn(1+BnCn))|≤1N∑n|AnBn||1+BnCn|1N|∑n(AnBn(1+BnCn))|≤1N∑n|AnBn||1+BnCn|\frac{1}{N} |\sum_n (A_n B_n (1 + B_n C_n))| \leq \frac{1}{N} \sum_n |A_n B_n| |1+B_n C_n|

4. Since |AnBn|=1|AnBn|=1|A_n B_n| = 1, we get:
1N∑n|AnBn||1+BnCn|1N∑n|AnBn||1+BnCn|\frac{1}{N} \sum_n |A_n B_n| |1+B_n C_n|
=1N∑n|1+BnCn|=1N∑n|1+BnCn|= \frac{1}{N} \sum_n |1+B_n C_n|
=1N∑n(1+BnCn)=1N∑n(1+BnCn)= \frac{1}{N} \sum_n (1+B_n C_n)
=1+1N∑nBnCn=1+1N∑nBnCn = 1 + \frac{1}{N} \sum_n B_n C_n
=1+E(b,c)=1+E(b,c)= 1 + E(b, c)

5. So we conclude that:
|E(a,b)+E(a,c)|≤1+E(b,c)|E(a,b)+E(a,c)|≤1+E(b,c)|E(a,b) + E(a, c) | \leq 1 + E(b,c)

But experimentally, we find that:

• E(a,b)=E(a,c)=−1/2E(a,b)=E(a,c)=−1/2E(a,b) = E(a, c) = -1/2

Technically, we can prove that E(a,b)=cos(120)=−1/2E(a,b)=cos(120)=−1/2E(a,b) = cos(120) = -1/2

That violates the inequality:

|E(a,b)+E(a,c)|=|−1/2+−1/2|=1|E(a,b)+E(a,c)|=|−1/2+−1/2|=1|E(a,b) + E(a,c)| = |-1/2 + -1/2| = 1
1+E(b,c)=1+−1/2=1/21+E(b,c)=1+−1/2=1/21 + E(b,c) = 1 + -1/2 = 1/2

There is one technical assumption that may or may not be worrisome. Alice and Bob can't actually measure E(a,b)E(a,b)E(a,b) for the entire run, because some of the runs, they will measure the spins along axes aaa and ccc. For that run, they have no idea what the value of BBB is. For other runs, they will have no idea what the value of AAA or CCC is. What's assumed (and I'm not sure if there is a name for this assumption) is that the correlation E(a,b)E(a,b)E(a,b) computed using only those runs where AAA and BBB are measured gives the same result as if we had computed E(a,b)E(a,b)E(a,b) using all the runs. That is, we're assuming that the statistics for unmeasured quantities is the same as for the quantities that were actually measured.

Thanks for this. It's very helpful and I'm trying to send a reply. BUT the PREVIEW seems to be muddling some of your equations. So some of these responses will be me testing how they come out when posted: that is, to see if the Posted Reply differs from what looks to be muddled in the Preview.

Bear with me: I'll post what I'm seeing to see if I'm doing something wrong.

EDIT-1: The posting looks wrong to me. As though the QUOTE function is not reproducing your equations properly. Does anyone else see that? Thanks.

EDIT-2: To fix the mix-up: Do I need to recode all the LaTeX?

 

[stevendaryl]

Thanks for this. It's very helpful and I'm trying to send a reply. BUT the PREVIEW seems to be muddling some of your equations. So some of these responses will be me testing how they come out when posted: that is, to see if the Posted Reply differs from what looks to be muddled in the Preview.

Bear with me: I'll post what I'm seeing to see if I'm doing something wrong.

EDIT-1: The posting looks wrong to me. As though the QUOTE function is not reproducing your equations properly. Does anyone else see that? Thanks.

EDIT-2: To fix the mix-up: Do I need to recode all the LaTeX?

I think if you just hit "reply" to my post, rather than "quote", it will display correctly.

 

[N88]

Let me pick a particularly simple version of the EPR paradox. We have a source of anti-correlated electron-positron pairs. Alice has a device that measures spins along one of three possible axes:

• a: Along the y-axis.
• b: Along the line that makes a 120 degree angle with the y-axis in the x-y plane.
• c: Along the line that makes a 240 degree angle with the y-axis in the x-y plane.

We'll map "spin-up" to the result +1 and "spin-down" to the result -1.

Bob has the same three choices.

Alice's and Bob's results are anti-correlated, meaning if they both choose to measure the spin along the same axis, they always get opposite results.
So here is a model for what is happening:

1. Associated with each particle pair, there are three numbers, $A, B, C$. Each is either +1 or -1.
2. If Alice measures her particle along axis a, she will get result A. If she measures her particle along axis b, she will get B. If she measures along axis c, she will get C.
3. Bob always gets the opposite: If he measures along axis a, he will get result -A, etc.

The awkward situation is that even though this model has 3 numbers associated with each pair, $A, B, C$, Alice can only measure one of them, and Bob can only measure one. So at best, they can only measure 2 out of the 3 numbers. But the model assumes that there is a result for all three directions, even if you can only measure two of them.

Let's assume that Alice and Bob perform lots and lots of measurements on twin pairs. Let's define some statistical quantities:

• $E(a, b) = \frac{1}{N} \sum_n A_n B_n$ where $A_n$ is the value of $A$ for pair number $n$, and $N$ is the number of pairs produced.
• Similarly, $E(a, b)$ and $E(b, c)$

Here's where we use some pure mathematics to get some inequalities on these quantities.

1. $E(a,b) + E(a, c) = \frac{1}{N} \sum_n (A_n B_n + A_n C_n)$

2. Since $B_n = \pm 1$, $B_n B_n = 1$. So we can rewrite the right-hand side of equation 1 as
$\frac{1}{N} \sum_n (A_n B_n + A_n B_n B_n C_n)$
$= \frac{1}{N} \sum_n (A_n B_n (1 + B_n C_n))$

3. Taking absolute values, we get:
$\frac{1}{N} |\sum_n (A_n B_n (1 + B_n C_n))| \leq \frac{1}{N} \sum_n |A_n B_n| |1+B_n C_n|$

4. Since $|A_n B_n| = 1$, we get:
$\frac{1}{N} \sum_n |A_n B_n| |1+B_n C_n|$
$= \frac{1}{N} \sum_n |1+B_n C_n|$
$= \frac{1}{N} \sum_n (1+B_n C_n)$
$= 1 + \frac{1}{N} \sum_n B_n C_n$
$= 1 + E(b, c)$

5. So we conclude that:
$|E(a,b) + E(a, c) | \leq 1 + E(b,c)$

But experimentally, we find that:

• $E(a,b) = E(a, c) = -1/2$

Technically, we can prove that $E(a,b) = cos(120) = -1/2$

That violates the inequality:

$|E(a,b) + E(a,c)| = |-1/2 + -1/2| = 1$
$1 + E(b,c) = 1 + -1/2 = 1/2$

There is one technical assumption that may or may not be worrisome. Alice and Bob can't actually measure $E(a,b)$ for the entire run, because some of the runs, they will measure the spins along axes $a$ and $c$. For that run, they have no idea what the value of $B$ is. For other runs, they will have no idea what the value of $A$ or $C$ is. What's assumed (and I'm not sure if there is a name for this assumption) is that the correlation $E(a,b)$ computed using only those runs where $A$ and $B$ are measured gives the same result as if we had computed $E(a,b)$ using all the runs. That is, we're assuming that the statistics for unmeasured quantities is the same as for the quantities that were actually measured.

Thanks again, that looks better. I used QUOTE because I wanted to annotate with some queries that help to clarify my problem; for the good thing is that you allude to them. I expect to be back later, hopefully by tomorrow.

 

[*now*]

(Snip)
It makes the same quantifiable predictions as QM, the same as other interpretations claim. As for the two-user case, it resolves these in the usual de Finetti or subjective Bayesian way they are resolved, via a de Finetti type theorem or similar, I don't see any issues. It isn't saying the underlying reality is subjective after all. Although Rovelli's Relational Interpretation does. (Snip)

Could you elaborate on the differences, please, contrasting the two interpretations in that way?

 

[DarMM]

Could you elaborate on the differences, please, contrasting the two interpretations in that way?

Contrasting Rovelli's Relational view and QBism you mean? When you say "that way" what way do you mean, i.e. what form do you want the contrast to take or what aspect do you want it to focus on?

 

[*now*]

Hi DarMM yes, regarding the contrasting of subjectivity or objectivity. There could be confusion and it could be a matter of semantics to some extent, because, for instance, Fuchs, 2017, wrote of Qbist forthright and obstinate holding to the “subjective factor”, and listed other interpretations such as Zeilinger’s on a scale, from more to less, with RQM the least so. However, there seemed little explanation for that. So, elaboration such as Qbist adoption of the subjective or personalist school of Bayesian probability or possibly adoption of internal or not external epistemology, or that some interpretations seem to give particular weight to consciousness, compared with the stances of RQM, could help a lot, thanks.

https://link.springer.com/chapter/10.1007/978-3-319-43760-6_7

 

[RUTA]

I've been thinking* and there's a possible link with QBism and these views. Bear with me, because I might be talking nonsense here and there's plenty of scare quotes because I'm not sure of the reality of various objects in these views.

In these views let's say you have a classical device $D_1$ the emitter and another classical device $D_2$ the detector, just as spacetime in Relativity is given a specific split into space and time by the given "context" of an inertial observer, in these views we have spacetimesource which is split into

1. Space
   
2. Time
3. A conserved quantity, $Q$

by the combined spatiotemporal contexts of those two devices.

That conserved quantity might be angular momentum, or it might be something else, depending on what $D_1$ and $D_2$ are. Then some amount of $Q$ is found at earlier times in $D_1$ and in later times in $D_2$, not because it's transmitted, simply that's the "history" that satisfies the 4D constraints.

Quantum particles and fields only come in as a way of evaluating the constraint via a path integral, they're sort of a dummy variable and don't fundamentally exist as such.

So ultimately we have two classical objects which define not only a reference frame but a contextual quantity they "exchange". This is quite interesting because it means if I have an electron gun and an z-axis angular momentum detector, then it was actually those two devices that define the z-axis angular momentum itself $J_z$ that they exchange, hence there is obviously no counterfactual:
"X-axis angular momentum $J_x$ I would have obtained had I measured it"
since that would have required a different device, thus a different decomposition of the spacetimesource and a completely different 4D scenario to constrain. Same with Energy and so on. $J_z$ also wasn't transmitted by an electron, it's simply that integrating over fermionic paths is a nice way to evaluate the constraint on $J_z$ defining the 4D history.

However and here is the possible link, zooming out the properties of the devices themselves are no different, they are simply contextually defined by other classical systems around them. "Everything" has the properties it is required to have by the surrounding context of its environment, which in turn is made of objects for which this is also true. In a sense the world is recursively defined. Also since an object is part of the context for its constituents, the world isn't reductive either, the part requires the whole to define its properties.

It seems to me that in such a world although you can mathematically describe certain fixed scenarios, it's not possible to obtain a mathematical description of everything in one go, due to the recursive, non-reductive nature of things. So possibly it could be the kind of ontology a QBist would like? Also 4D exchanges are fundamentally between the objects involved, perhaps the sort of non-objective view of measurements QBism wants.

Perhaps @RUTA can correct my butchering of things!

*Although this might be completely off as I've read the Relational Block World book and other papers on the view, as well as papers on Retrocausal views like the Transactional interpretation, but I feel they haven't clicked yet.

It looks like you have a good feel for our view! In a recursive explanation there is a "base case" upon which everything is built recursively. For us, if you want to think of it this way, the "base case" would be the existence of classical/diachronic/time-evolved objects (objects with worldlines in spacetime) defined by classical information that is then self-consistent per some 4D-global constraint (the "recursive relation"). For example, Einstein's equations provide a 4D constraint on spatiotemporal measurement, energy, mass, and momentum, but before you can apply Einstein's equations you need the worldlines of the classical objects you're dealing with in the spacetime manifold. So, we don't use the term "recursive," rather we use the term "self-consistent," but our view could be characterized as recursive in the sense I just explained.

I don't know that a relative-states view like QBism with its no collapse approach and rejection of objective reality would like our standard formalism view with its objective collapse and corresponding objective reality. That seems totally contrary to their view that the probabilities of QM are subjective (meant for individual observers). The whole point of "objective" is that everyone agrees with it, while you saw Healey's post in IJQF saying different observers' results could disagree (that's what is meant by "subjective"). Healey in particular complains that of our 4D-constraint-based adynamical explanation is "retrocausal" (it's not retrocausal because it's not causal). He wants a dynamical view of reality (a time-evolved story). To get a dynamical view in accord with relativity, he has given up objective reality. But maybe you're seeing something about their view that I'm missing?

 

[N88]

With underlined comments inserted by N88, with thanks for your detail. Thus:

Let us pick a particularly simple version of the EPR paradox. We have a source of anti-correlated electron-positron pairs. Alice has a device that measures spins along one of three possible axes:

• a: Along the y-axis.
• b: Along the line that makes a 120 degree angle with the y-axis in the x-y plane.
• c: Along the line that makes a 240 degree angle with the y-axis in the x-y plane.

We'll map "spin-up" to the measurement result +1 and "spin-down" to the measurement result -1.

Bob has the same three choices.

Alice's and Bob's measurement results are anti-correlated, meaning if they both choose to measure the spin along the same axis, they always get opposite results.
So here is a model for what is happening:

1. Associated with each particle pair, there are three numbers associated with measurements, $A, B, C$. Each is either +1 or -1.
2. If Alice measures her particle along axis a, she will get result A. If she measures her particle along axis b, she will get B. If she measures along axis c, she will get C.
3. Bob always gets the opposite: If he measures along axis a, he will get result -A, etc.

The awkward situation is that even though this model has 3 numbers associated with each pair, $A, B, C$, Alice can only measure one of them, and Bob can only measure one. So at best, they can only measure 2 out of the 3 numbers. But the model assumes that there is a measurement result for all three directions, even if you can only measure two of them.

But Bell is studying EPRB and seeking a more complete specification, one beyond quantum theory. And $A, B, C$ are measurement results. So, if there is no measurement how can there be a measurement result? So I agree with Peres' 1995 textbook (p.168): Our conclusion can be succinctly stated: "unperformed experiments have no results."

Thanks to your detail, my problem becomes clearer: Let's assume that Alice and Bob perform lots and lots of measurements on twin pairs. Let's define some statistical quantities:

• $E(a, b) = \frac{1}{N} \sum_n A_n B_n$ where $A_n$ is the value of $A$ for pair number $n$, and $N$ is the number of pairs produced. (X)
• Similarly, $E(a, b)$ and $E(b, c)$ (Y)

Here's where we use some pure mathematics to get some inequalities on these quantities.

1. $E(a,b) + E(a, c) = \frac{1}{N} \sum_n (A_n B_n + A_n C_n)$

Here is my problem: the n-numbered pairs under the $(a,b)$ setting have been measured, as in (X). So now you are measuring the new M m-numbered pairs under $(a,c)$, as in (Y).

So the physics under EPRB requires [with me fixing plus-sign to minus-sign as in Bell (1964)]:

1A. $E(a,b)-E(a,c)=\frac{1}{N}∑_n(A_nB_n)-\frac{1}{M}∑_m(A_mC_m).$
####

So it seems to me that you must be using a different view of the reality that applies under EPRB. For we are at an important point of difference. That is, HERE you can proceed via "#1" to Bell's famous inequality and HERE I cannot. So you proceed to something that does not hold under EPRB whereas, via "#1A" I get the result that holds classically and quantum mechanically and under pure mathematics.

Since I am blocked from your continuing mathematics here by my understanding of the underlying EPRB reality (as expressed above), please see my next underline below; in your terms, "the worrisome bit."

2. Since $B_n = \pm 1$, $B_n B_n = 1$. So we can rewrite the right-hand side of equation 1 as
$\frac{1}{N} \sum_n (A_n B_n + A_n B_n B_n C_n)$
$= \frac{1}{N} \sum_n (A_n B_n (1 + B_n C_n))$

3. Taking absolute values, we get:
$\frac{1}{N} |\sum_n (A_n B_n (1 + B_n C_n))| \leq \frac{1}{N} \sum_n |A_n B_n| |1+B_n C_n|$

4. Since $|A_n B_n| = 1$, we get:
$\frac{1}{N} \sum_n |A_n B_n| |1+B_n C_n|$
$= \frac{1}{N} \sum_n |1+B_n C_n|$
$= \frac{1}{N} \sum_n (1+B_n C_n)$
$= 1 + \frac{1}{N} \sum_n B_n C_n$
$= 1 + E(b, c)$

5. So we conclude that:
$|E(a,b) + E(a, c) | \leq 1 + E(b,c)$

But experimentally, we find that:

• $E(a,b) = E(a, c) = -1/2$

Technically, we can prove that $E(a,b) = cos(120) = -1/2$

That violates the inequality:

$|E(a,b) + E(a,c)| = |-1/2 + -1/2| = 1$
$1 + E(b,c) = 1 + -1/2 = 1/2$

There is one technical assumption that may or may not be worrisome. Alice and Bob can't actually measure $E(a,b)$ for the entire run, because some of the runs, they will measure the spins along axes $a$ and $c$. For that run, they have no idea what the value of $B$ is. For other runs, they will have no idea what the value of $A$ or $C$ is.

What's assumed (and I'm not sure if there is a name for this assumption) is that the correlation $E(a,b)$ computed using only those runs where $A$ and $B$ are measured gives the same result as if we had computed $E(a,b)$ using all the runs.

I agree with the above two statements, but I cannot make the next assumption. It looks like what I would call CFI = ContraFactual Inferencing.

That is, we're assuming that the statistics for unmeasured quantities is the same as for the quantities that were actually measured.

But this CFI. It is an Inference (or an Assumption), contrary to the facts (and will likely lead to trouble). For there are NO statistics for $A, B, C$ if they are NOT measured: see my "#1A" above, which is OK in this regard.

And if you proceed properly, with those statistics done properly via measurements under $(a,b), (a,c), (b,c)$: then Bell's inequality is false under valid statistics.

So, in my world, the above prediction "re trouble" comes true?

As is well-known; and (for me) the resolution of this "trouble" requires no need to invoke nonlocality, etc. It is pure stastitics in the face of an inequality that is algebraically false: see post #95 and the remedy that I rely upon in "#1A" above.

And thus the question: What "reality" are you and Bell effectively assuming when you allow $B_nB_m=1$? It seems to me that it is a rudimentary classical assumption of some sort? But one unsuited to EPRB?

So, returning to the OP. Can we say this: the reality in Bell (1964) -- whether assumed or accidental -- is that which satisfies Bell's famous inequality?

And am I right in thinking that only elementary classical realities deliver that satisfaction?

 

[stevendaryl]

But Bell is studying EPRB and seeking a more complete specification, one beyond quantum theory. And A,B,C are measurement results. So, if there is no measurement how can there be a measurement result?

The assumption is that a "measurement" is something that reveals information about the world. If you flip a coin and look at the coin and see heads, the coin was already "heads" before you looked at it. The assumption is that the same is true of quantum measurements. So A, B, C are properties of the particles. They only become measurement results after you perform the measurement. Therefore, they have statistics even if you haven't measured them.

Of course, there can be things like "measurement results" that don't reveal pre-existing properties. The result could be some kind of cooperative effect of the thing being measured and the thing doing the measurement. Classically, you could describe this more complicated situation this way:

$P(A | \lambda, \sigma)$

Instead of saying that the result $A$ is a deterministic function of some property of the particle's state $\lambda$, it might be randomly produced with a certain probability distribution that depends on both facts about the particle, $\lambda$, and facts about the measuring device, $\sigma$.

However, this more general possibility is not compatible with the perfect anti-correlations observed in the EPR experiment. If Bob already got the result "spin-down in the z-direction", then there is no way for Alice to get anything other than spin-up in the z-direction. So detailed facts about her measuring device, other than the fact that it's measuring the z-component of spin, can't come into play.

 

[lodbrok]

The assumption is that a "measurement" is something that reveals information about the world. If you flip a coin and look at the coin and see heads, the coin was already "heads" before you looked at it.

I think measurement in this case is analogous to tossing.

 

[stevendaryl]

I think measurement in this case is analogous to tossing.

But if two particles have anticorrelated spins, then a measurement of one particle's spin (along a specific axis) reveals the value for the measurement of the other particle's spin, even before that measurement is made. So for perfect anti-correlations, the measurement seems more like peeking at the result than tossing the coin.

 

[AgentSmith]

I think the concept of real is like the concept of time - its one of those things that's hard to pin down. Time is what a clock measures - real is the common-sense idea that what we experience comes from something external to us that actually exists. All these can be be challenged by philosophers, and often are circular, but I think in physics pretty much all physicists would accept you have to start somewhere and hold views similar to the above.

For what its worth I think Gell-Mann and Hartel are on the right track:
https://www.sciencenews.org/blog/context/gell-mann-hartle-spin-quantum-narrative-about-reality

The above, while for a lay audience, contains the link to the actual paper.

Thanks
Bill

The only job left for philosophers is to question everything and anything while never reaching answers. Time is change in the spacetime continuum.

 

[lodbrok]

But if two particles have anticorrelated spins, then a measurement of one particle's spin (along a specific axis) reveals the value for the measurement of the other particle's spin,

... if measured at the same time along the same axis. Aren't the terms in Bells inequality experiment measured in different experiments? Thus, the realism assumption seems to involve the idea that heads from tossing/observing one coin at one moment is anticorrelated with tails from tossing/observing a similar coin at a different time.
Seems obvious from equation 2 of your derivation where you factor AnBn.

The assumption is introduced with the term AnBnBnCn. A term which is impossible in any EPRB experiment for the simple reason that AnBn is one toss, BnCn should be a different toss in EPRB. But you used the same subscript. By factoring out AnBn, you are saying the heads-up correlation persists between tosses which is not true in my humble opinion.

 

[DrChinese]

The assumption is introduced with the term AnBnBnCn. A term which is impossible in any EPRB experiment for the simple reason that AnBn is one toss, BnCn should be a different toss in EPRB. ...

You're missing the point entirely. The realist says there are values for measurement outcomes at any A/B/C simultaneously. If so, what are they? Turns out no matter what you make them - and you can select them yourself - they WON'T match the actual measured value (or the quantum value).

The experiment simply demonstrates that the entangled particle results don't match the realistic prediction - no matter what it is. If you don't believe me, take the DrChinese challenge.

 

[N88]

You're missing the point entirely. The realist says there are values for measurement outcomes at any A/B/C simultaneously. If so, what are they? Turns out no matter what you make them - and you can select them yourself - they WON'T match the actual measured value (or the quantum value).

The experiment simply demonstrates that the entangled particle results don't match the realistic prediction - no matter what it is. If you don't believe me, take the DrChinese challenge.

I would call this the NAIVELY REALISTIC position. So I wonder, since this brand of realism needs to be distinguished from other brands; for example Bohrian realism which allows for perturbative measurements:

1: Is NAIVELY REALISTIC a valid name for this belief?

2: And is it taken seriously today?

I ask because c1810, Malus in Paris could transmit photon beams of any linear polarization. And a recipient could put such beams through a linear polarizer and likewise generate beams of almost any linear polarization. But surely no one then thought that the generated beams were the same as the input beams?

Because they all knew Malus Law?

Which then raises a question relevant to the OP: Is this then all that Bell's theorem shows? That naive realism is false in quantum settings?

 

[lodbrok]

You're missing the point entirely. The realist says there are values for measurement outcomes at any A/B/C simultaneously.

No question. But does the realist say heads of one toss is anti-correlated with tails of a different toss? I doubt it.

That is what AnBnBnCn implies. BnBn = 1 only within the same toss not across different tosses even if it's the same coin. I'm simply pointing out assumptions implied in equation 2 irrespective of worldview.

BTW, what is the DrChinese challenge? I'll appreciate a citation so I can read it up.

 

[DrChinese]

1. No question. But does the realist say heads of one toss is anti-correlated with tails of a different toss? I doubt it.

That is what AnBnBnCn implies. BnBn = 1 only within the same toss not across different tosses even if it's the same coin. I'm simply pointing out assumptions implied in equation 2 irrespective of worldview.

2. BTW, what is the DrChinese challenge? I'll appreciate a citation so I can read it up.

1. Agreed that no one has implied that the results of one toss have a correlation to the results of another.

On the other hand, the realist believes that measure of A or B or C by Alice can lead to a certain prediction by Bob measuring at the same A or B or C. That was the EPR result.

2. The DrChinese challenge is where you hand pick the results on both sides (Alice and Bob) for the angle settings A=0, B=120, C=240 degrees. Then I pick which pair of angles Alice and Bob actually measure. The challenge is to produce a dataset that will match the statistical predictions of QM.

 

[DrChinese]

I would call this the NAIVELY REALISTIC position. So I wonder, since this brand of realism needs to be distinguished from other brands; for example Bohrian realism which allows for perturbative measurements:

1: Is NAIVELY REALISTIC a valid name for this belief?

2: And is it taken seriously today?

I ask because c1810, Malus in Paris could transmit photon beams of any linear polarization. And a recipient could put such beams through a linear polarizer and likewise generate beams of almost any linear polarization. But surely no one then thought that the generated beams were the same as the input beams?

Because they all knew Malus Law?

Which then raises a question relevant to the OP: Is this then all that Bell's theorem shows? That naive realism is false in quantum settings?

1. No.
2. Yes. This is EPR realism.

There is no specific connection between Malus and entangled photon stats other than they work out to be the same. In other words, there is no reason to cite Malus as you have. It is not the same example, at least how you describe.

 

[N88]

1. No.
2. Yes. This is EPR realism.

There is no specific connection between Malus and entangled photon stats other than they work out to be the same. In other words, there is no reason to cite Malus as you have. It is not the same example, at least how you describe.

1. I should have been clearer. I am seeking the answer to the OP; to me it is a good and important question. So my Malus example was given in the "classical context" of what Malus discovered when working with light beams. For I think it fair to say that Malus' work was in "classical optics".

So if perturbation was known -- classically -- from c1810, why would EPR (of all people) abandon Bohrian-realism? Or, in your terms, how please does "EPR realism" differ from naive realism and Bell's realism?

2. In my terms, naive realism implies that the outcomes of measurement interactions pre-existed. Isn't this also the basis for your Dr Chinese challenge?

Thus -- in EPRB -- if the analyzer reports "+1", the naive realist believes the spin was "UP" prior to that interaction.

Thus, and I hope I have this correctly: your challenge aims to refute this naivety?

3. But in EPR, their "realism" allows that 'there is an element of physical reality corresponding to the '"UP" outcome'. So it seems to me that they allow that what went into the polarizer in this instance was "an element of physical reality [an unpolarized particle, according to Bell] which, upon interaction, came out as polarized-particle, spin-UP." [Thus, different particles correspond to the "DOWN" outcome.]

So I am thinking that the "EPR element of physical reality" here is the unpolarized particle that went into the polarizer. For it corresponds to the element of physical reality that -- after that interaction -- came out.

So I am still thinking that there is some considerable naivety in Bell's realism, and that this is:

A. relevant to getting a clear answer to the OP.

B: and is shown to be the case by your challenge?

 

[DrChinese]

2. In my terms, naive realism implies that the outcomes of measurement interactions pre-existed. Isn't this also the basis for your Dr Chinese challenge?

Thus -- in EPRB -- if the analyzer reports "+1", the naive realist believes the spin was "UP" prior to that interaction.

Thus, and I hope I have this correctly: your challenge aims to refute this naivety?

3. But in EPR, their "realism" allows that 'there is an element of physical reality corresponding to the '"UP" outcome'. So it seems to me that they allow that what went into the polarizer in this instance was "an element of physical reality [an unpolarized particle, according to Bell] which, upon interaction, came out as polarized-particle, spin-UP." [Thus, different particles correspond to the "DOWN" outcome.]

So I am thinking that the "EPR element of physical reality" here is the unpolarized particle that went into the polarizer. For it corresponds to the element of physical reality that -- after that interaction -- came out.

So I am still thinking that there is some considerable naivety in Bell's realism, and that this is:

A. relevant to getting a clear answer to the OP.

B: and is shown to be the case by your challenge?

Yes, that's part of the challenge. But I have no idea why the word "naive" would be attached to EPR's elements of reality. It was a well made argument, best possible at the time. Bell refuted that (at least showed that it was incompatible with the predictions of QM.

And as I stated earlier in this thread, realism IS an assumption of Bell. And I explained where.

 

[bhobba]

Yes, that's part of the challenge. But I have no idea why the word "naive" would be attached to EPR's elements of reality. It was a well made argument, best possible at the time. Bell refuted that (at least showed that it was incompatible with the predictions of QM.

I just want to mention in order not be bogged down in philosophical baggage which can obscure the physics, I think counterfactual definiteness is the better term. Of course in the original EPR paper they used elements of reality - but that terminology was, IMHO, one of the issues Bell had to sort out when clarifying the whole thing.

Thanks
Bill

 

[N88]

The assumption is that a "measurement" is something that reveals information about the world. If you flip a coin and look at the coin and see heads, the coin was already "heads" before you looked at it. The assumption is that the same is true of quantum measurements. So A, B, C are properties of the particles. They only become measurement results after you perform the measurement. Therefore, they have statistics even if you haven't measured them.

Of course, there can be things like "measurement results" that don't reveal pre-existing properties. The result could be some kind of cooperative effect of the thing being measured and the thing doing the measurement. Classically, you could describe this more complicated situation this way:

$P(A | \lambda, \sigma)$

Instead of saying that the result $A$ is a deterministic function of some property of the particle's state $\lambda$, it might be randomly produced with a certain probability distribution that depends on both facts about the particle, $\lambda$, and facts about the measuring device, $\sigma$.

However, this more general possibility is not compatible with the perfect anti-correlations observed in the EPR experiment. If Bob already got the result "spin-down in the z-direction", then there is no way for Alice to get anything other than spin-up in the z-direction. So detailed facts about her measuring device, other than the fact that it's measuring the z-component of spin, can't come into play.

The "coin-flip" does not work for me; it's difficult to interpret in the EPRB context. The coin had a Head and a Tail before it was flipped -- via a thumb.

Now I take naive-realism -- from ancient days -- to be that primitive realism which supposes that what is observed is what was real before the observation. Thus the coin had a Head and it showed UP when the coin hit the floor

So, in my terms, only ancient naive realism [though it persists in these modern times] allows that the EPRB particles had A, B, C before they were "flipped" -- via the measurement interaction.

So it seems to me that "modern realism" -- interpreting EPR's realism -- allows that "so-called measurements" do NOT ALWAYS reveal pre-existing properties; instead it allows that "measurement" interactions MAY construct and reveal something new: some correspondence.

Remember that Bell's goal was to provide a "more complete specification" of EPRB. So you seem to be saying that he thought that result could be achieved by naive realism? See next.

Yes, that's part of the challenge. But I have no idea why the word "naive" would be attached to EPR's elements of reality. It was a well made argument, best possible at the time. Bell refuted that (at least showed that it was incompatible with the predictions of QM.

And as I stated earlier in this thread, realism IS an assumption of Bell. And I explained where.

But I am not attaching naive to "EPR's realism". I am attaching it to what you say is "Bell's realism." And from your challenge we know that it does not work; me saying that before Bell it was well-known that it could not work. Thus the classical example that I offered from Malus' time.

To be clearer re how I see it: EPR-realism addresses the modern view: ie, EPR-realism is that realism which allows that there was something corresponding to the observed values.

THUS: A pure measurement [in any field] would reveal that the observed value corresponded 100% to that which pre-existed; like charge. Thus naive-realism holds in such limited cases.

BUT: A perturbative "measurement" [in any field] would reveal that the observed value corresponded < 100% to that which pre-existed. Thus the classical example that I offered from Malus' time. Thus naive-realism does NOT hold in such limited cases. So Bell's realism does not hold here either.

So, to possibly clarify many differing views, and eliminate some: What is the name of the realism that Bell assumes in his famous 1964 paper? And where is it introduced in his mathematics?

 

[bhobba]

So, to possibly clarify many differing views, and eliminate some: What is the name of the realism that Bell assumes in his famous 1964 paper? And where is it introduced in his mathematics?

Its called counterfactual definiteness. A counterfactual theory is one whose experiments uncover properties that are pre-existing:
http://www.johnboccio.com/research/quantum/notes/paper.pdf

Thanks
Bill

 

[stevendaryl]

The "coin-flip" does not work for me; it's difficult to interpret in the EPRB context. The coin had a Head and a Tail before it was flipped -- via a thumb.

Now I take naive-realism -- from ancient days -- to be that primitive realism which supposes that what is observed is what was real before the observation. Thus the coin had a Head and it showed UP when the coin hit the floor

So, in my terms, only ancient naive realism [though it persists in these modern times] allows that the EPRB particles had A, B, C before they were "flipped" -- via the measurement interaction.

Well, that's the realism that was falsified by Bell's theorem.

 

[DrChinese]

1. The "coin-flip" does not work for me;...

2. So, to possibly clarify many differing views, and eliminate some: What is the name of the realism that Bell assumes in his famous 1964 paper? And where is it introduced in his mathematics?

1. Really, your argument bears no relation to either EPR or Bell. So there should be no surprise that its conclusion escapes you.

2. It can be called either Bell realism or EPR realism. EPR uses "elements of reality". If you want to draw a distinction, you can, since Bell does not use the word "realism" in his paper. See my post #3 for details.

 

[lodbrok]

1. Agreed that no one has implied that the results of one toss have a correlation to the results of another...

2. The DrChinese challenge is where you hand pick the results on both sides (Alice and Bob) for the angle settings A=0, B=120, C=240 degrees. Then I pick which pair of angles Alice and Bob actually measure. The challenge is to produce a dataset that will match the statistical predictions of QM.

1. Equation 2 of Steven Daryl's derivation definitely makes that assumption so long as it applies to an EPRB experiment.

2. As decribed, I don't yet see the relevance of your challenge to the discussion. To avoid going off topic please provide a citation, so I can read up the details.

 

[stevendaryl]

... if measured at the same time along the same axis. Aren't the terms in Bells inequality experiment measured in different experiments? Thus, the realism assumption seems to involve the idea that heads from tossing/observing one coin at one moment is anticorrelated with tails from tossing/observing a similar coin at a different time.
Seems obvious from equation 2 of your derivation where you factor AnBn.

The assumption is introduced with the term AnBnBnCn. A term which is impossible in any EPRB experiment for the simple reason that AnBn is one toss, BnCn should be a different toss in EPRB. But you used the same subscript. By factoring out AnBn, you are saying the heads-up correlation persists between tosses which is not true in my humble opinion.

I don't really know what you are talking about. $A_n, B_n, C_n$ are three numbers, each one is either +1 or -1. So it's just a fact of arithmetic that:

$A_n B_n + A_n C_n = A_n B_n + A_n (B_n B_n) C_n = A_n B_ n (1 + B_n C_n)$

The assumption is that twin-pair number $n$, has an associated triple of numbers $A_n, B_n, C_n$, where $A_n$ gives the result of a measurement of spin along the first axis, $B_n$ along the second axis, $C_n$ along the third axis. The manipulation above is just arithmetic. There is no additional assumptions involved.

 

[stevendaryl]

BUT: A perturbative "measurement" [in any field] would reveal that the observed value corresponded < 100% to that which pre-existed. Thus the classical example that I offered from Malus' time. Thus naive-realism does NOT hold in such limited cases. So Bell's realism does not hold here either.

Yes, it's certainly possible to have a "perturbative" measurement, where the result is not 100% determined by the pre-existing properties. However, there are no ideas (as far as I know of) how such a perturbative measurement could result in perfect correlation between measurements of distant particles.

That's really Einstein et al's whole point: If measurements DON'T reveal pre-existing properties, then how can two distant measurements produce perfect correlations?

 

[atyy]

So it seems to me that "modern realism" -- interpreting EPR's realism -- allows that "so-called measurements" do NOT ALWAYS reveal pre-existing properties; instead it allows that "measurement" interactions MAY construct and reveal something new: some correspondence.

Yes, it's certainly possible to have a "perturbative" measurement, where the result is not 100% determined by the pre-existing properties. However, there are no ideas (as far as I know of) how such a perturbative measurement could result in perfect correlation between measurements of distant particles.

Isn't the perturbative measurement case dealt with by allowing the measurement apparatus to also have hidden variables? The measurement outcome is the result of interaction between the hidden variables of the apparatus and the hidden variables of the system.

 

[stevendaryl]

Isn't the perturbative measurement case dealt with by allowing the measurement apparatus to also have hidden variables? The measurement outcome is the result of interaction between the hidden variables of the apparatus and the hidden variables of the system.

Yes, I think so. But the fact that in EPR there are perfect correlations between distant measurements implies that in fact, nothing about the measuring apparatus is relevant except the orientation of the spin measurement.

 

[Auto-Didact]

That's really Einstein et al's whole point: If measurements DON'T reveal pre-existing properties, then how can two distant measurements produce perfect correlations?

I quote Bell on this from his well-known paper on Bertlmann's socks:

Could it be that the first observation somehow fixes what was unfixed, or makes real what was unreal, not only for the near particle but also for the remote one ? For EPR that would be an unthinkable 'spooky action at a distance' /8/.

To avoid such action at a distance they have to attribute, to the space-time regions in question, real properties in advance of observation, correlated properties, which predetermine the outcomes of these particular observations. Since these real properties, fixed in advance of observation, are not contained in quantum formalism /9/, that formalism for EPR is incomplete. It may be correct, as far as it goes, but the usual quantum formalism cannot be the whole story.

It is important to note that to the limited degree to which determinism plays a role in the EPR argument, it is not assumed but inferred. What is held sacred is the principle of "local causality" - or "no action at a distance".

Of course, mere correlation between distant events does not by itself imply action at a distance, but only correlation between the signals reaching the two places. These signals, in the idealized example of Bohm, must be sufficient to determine whether the particles go up or down. For any residual undeterminism could only spoil the perfect correlation. It is remarkably difficult to get this point across, that determinism is not a presupposition of the analysis.

There is a widespread and erroneous conviction that for Einstein /10/ determinism was always the sacred principle. The quotability of his famous "God does not play dice" has not helped in this respect.

Among those who had great difficulty in seeing Einstein's position was Born. Pauli tried to help him /11/ in a letter of 1954 :
" ... I was unable to recognize Einstein whenever you talked about him in either your letter or your manuscript. It seemed to me as if you had erected some dummy Einstein for yourself, which you then knocked down with great pomp. In particular Einstein does not consider the concept of "determinism" to be as fundamental as it is frequently held to be (as he told me emphatically many times) ... he disputes that he uses as a criterion for the admissibility of a theory the question : "Is it rigorously deterministic?"..-he was not at all annoyed with you, but only said you were a person who will not listen".

Born had particular difficulty with the Einstein-Podolsky-Rosen argument. Here is his summing up, long afterwards, when he edited the Born-Einstein correspondence /12/ : "The root of the difference between Einstein and me was the axiom that events which happens in different places A and B are independent of one another, in the sense that an observation on the states of affairs at B cannot teach us anything about the state of affairs at A".

Misunderstanding could hardly be more complete. Einstein had no difficulty accepting that affairs in different places could be correlated. What he could not accept was that an intervention at one place could influence, immediately, affairs at the other.

These references to Born are not meant to diminish one of the towering figures of modern physics. They are meant to illustrate the difficulty of putting aside preconceptions and listening to what is actually being said. They are meant to encourage you, dear listener, to listen a little harder.

Here, finally, is a summing-up by Einstein himself /13/ :
'If one asks what, irrespective of quantum mechanics, is characteristic of the world of ideas of physics, one is first of all struck by the following : the concepts of physics relate to a real outside world.. . It is further characteristic of these physical objects that they are thought of as arranged in a space time continuum. An essential aspect of this arrangement of things in physics is that they lay claim, at a certain time, to an existence independent of one another, provided these objects "are situated in different parts of space".

'The following idea characterizes the relative independence of objects far apart in space (A and B) : external influence on A has no direct influence on B...

'There seems to me no doubt that those physicists who regard the descriptive methods of quantum mechanics as definitive in principle would react to this line of thought in the following way : they would drop the requirement ... for the independent existence of the physical reality present in different parts of space ; they would be justified in pointing out that the quantum theory nowhere makes explicit use of this requirement.

'I admit this, but would point out : when I consider the physical phenomena known to me, and especially those which are being so successfully encompassed by quantum mechanics, I still cannot find any fact anywhere which would make it appear likely that (that) requirement will have to be abandoned.

'I am therefore inclined to believe that the description of quantum mechanics ... has to be regarded as an incomplete and indirect description of reality, to be replaced at some later date by a more complete and direct one'.

Bell goes on to say "we will argue that certain particular correlations, realizable according to quantum mechanics, are locally inexplicable. They cannot be explained, that is to say, without action at a distance."

He concludes with a few possible ways to interpret his Theorem:

By way of conclusion I will comment on four possible positions that might be taken on this business - without pretending that they are the only possibilities.

First, and those of us who are inspired by Einstein would like this best, quantum mechanics may be wrong in sufficiently critical situations. Perhaps Nature is not so queer as quantum mechanics. But the experimental situation is not very encouraging from this point of view /19/. It is true that practical experiments fall far short of the ideal, because of counter inefficiencies, or analyzer inefficiencies, or geometrical imperfections, and so on. It is only with added assumptions, or conventional allowance for inefficiencies and extrapolation from the real to the ideal, that one can say the inequality is violated. Although there is an escape route there, it is hard for me to believe that quantum mechanics works so nicely for inefficient practical set-ups and is yet going to fail badly when sufficient refinements are made. Of more importance, in my opinion, is the complete absence of the vital time factor in existing experiments. The analyzers are not rotated during the flight of the particles. Even if one is obliged to admit some long range influence, it need not travel faster than light - and so would be much less indigestible. For me, then, it is of capital importance that Aspect /19, 20/ is engaged in an experiment in which the time factor is introduced.

Secondly, it may be that it is not permissible to regard the experimental settings a and b in the analyzers as independent variables, as we did /21/. We supposed them in particular to be independent of the supplementary variables X, in that a and b could be changed without changing the probability distribution p(X). Now even if we have arranged that a and b are generated by apparently random radioactive devices, housed in separate boxes and thickly shielded, or by Swiss national lottery machines, or by elaborate computer programmes, or by apparently free willed experimental physicists, or by some combination of all of these, we cannot be sure that a and b are not significantly influenced by the same factors X that influence A and B /21/. But this way of arranging quantum mechanical correlations would be even more mind boggling that one in which causal chains go faster than light. Apparently separate parts of the world would be deeply and conspiratorially entangled, and our apparent free will would be entangled with them.

Thirdly, it may be that we have to admit that causal influences do go faster than light. The role of Lorentz invariance in the completed theory would then be very problematic. An "ether" would be the cheapest solution /22/. But the unobservability of this ether would be disturbing. So would the impossibility of "messages' faster than light, which follows from ordinary relativistic quantum mechanics in so far as it is unambiguous and adequate for procedures we can actually perform. The exact elucidation of concepts like 'message' and 'we', would be a
formidable challenge.

Fourthly and finally, it may be that Bohr's intuition was right - in that there is no reality below some 'classical' 'macroscopic' level. Then fundamental physical theory would remain fundamentally vague, until concepts like 'macroscopic ' could be made sharper than they are today.

Abner Shimony (of CHSH fame), in Chapter 5 "John S. Bell: Some Reminiscences and Reflections" of Bertlmann, Zeilinger et al. 2002, Quantum (Un)speakables says the following:

5.3 In What Direction Does Bell's Theorem Point?

The conclusion of Bell's famous paper "Bertlmann's Socks and the Nature of Reality" [13], states four possibilities, with no pretense at exhaustiveness, concerning the interpretation of his theorem and of the experiments inspired by it. He expresses reservations about all of them, but seems guardedly to prefer the third: "it may be that causal influences do go faster than light. The role of Lorentz invariance in the completed theory would then be very problematic. An 'aether' would be the cheapest solution ... But the unobservability of this aether would be disturbing. So would the impossibility of 'messages' faster than light, which follows from ordinary relativistic quantum mechanics in so far as it is unambiguous and adequate for procedures we can actually perform. The exact elucidation of concepts like 'message' and 'we', would be a formidable challenge."

Similar reservations are expressed in "Beables for Quantum Field Theory" [14], where elements of reality, "beables", are used to construct an alternative to ordinary quantum field theory, which centers on "observables". Near the conclusion he says,

"So I am unable to prove, or even formulate clearly, the proposition that a sharp formulation of quantum field theory, such as that set out here, must disrespect serious Lorentz invariance. But it seems to me that this is probably so."

Bell was evidently willing to entertain radical innovations in physics, and to devote some effort to investigating them mathematically, in order to do justice to the apparent nonlocality revealed by correlation experiments and at the same time to hold on to his conception of a physical reality underlying observations.

We should like Bell, be fearless in the face of such questions. As far as I can tell, there are thus a few ways to logically interpret Bell's Theorem; especially the interpretation that Bell himself preferred seems to be a viable one, albeit dangerously non-standard, flying directly in the face of relativistic QFT. Similar words can be said about Bohmian Mechanics, which I do not necessarily advocate as the solution.

Physicists are actually right to give pause to entertaining such an interpretation, and treading very lightly instead; doing so carelessly - especially if they do not explain why to their students - however leads to them missing something crucial, which I will attempt to explain.

Even if experimentally the answer seems clear, given the well-known mathematical problems of QFT itself, the issue itself however remains mathematically unclear, therefore theoretically also no such clarity can be claimed to be evident. In other words, unless one is willing to step out and claim that the full still unknown physical theory of nature will in principle simultaneously leave QM and relativity fully mathematically intact as we know them, this issue can not be legitimately claimed to be settled, expert consensus or no expert consensus.

I believe that the answers to these foundational questions not only deserve but require clear answers; more strongly, I believe, because of the mathematical problems of QFT that they actually require a mathematical reformulation of both QM and SR/GR, one in which both theories arise as appropriate limits of this new mathematical formulation. I am, of course, not the only one who has taken this stance; Shimony himself offers a similar point of view on the resolution of this matter:

I wish now to say why I think Bell's Theorem points to a yet more radical proposal than that of the foregoing quotations. Bell's Theorem shows that there is some tension between quantum mechanics and the space-time structure of special relativity, even though the impossibility of using quantum mechanical entanglement to send superluminal messages prevents outright inconsistency between the two theories. There are, however, two other areas of tension between quantum mechanics and contemporary space-time theory, concerning general rather than special relativity. First, in general relativity the metric field is a dynamical entity rather than a fixed structure, and therefore it has to be quantized if quantum mechanics applies to physical reality in full generality. But the difficulties encountered in attempting to quantize general relativity have been so great that one suspects them to be not be merely technical and mathematical in character but conceptual. Second, at the Planck level - around $10^{-33} \mathrm{cm}$ - the quantum fluctuations of the space-time metric become as large as or larger than the expectation values of lengths under consideration, so that the metric structure may no longer be well defined. Both of these difficulties suggest that the union of quantum mechanics with general relativity may require the modification of one or both. It is not unreasonable to hope that if these difficulties are resolved, the tension that Bell's Theorem exhibits between quantum mechanics and special relativistic locality may find a deep and natural resolution

I will end this post by citing Clauser (Chapter 6 "Early History of Bell's Theorem", in the same book Quantum (Un)speakables.):

Given historical hindsight, I assert that our basic understanding of quantum theory has been significantly improved via Bell's Theorem and via its associated experimental testing, long after it was confidently asserted by many textbooks to be well understood. It is truly amazing that so many "killer" details slipped through cracks that existed between experimentalists and theorists. It is clearly of continuing importance for experimentalists and theorists to scrutinize each other's work with great care to try to eliminate such cracks.

Given such hindsight, I also assert that it is clearly counterproductive to scientific progress for one camp smugly to hold to a belief that all problems are solved in any given area. It is even more counterproductive for this camp then further to rely on this belief to formulate a religious stigma against others who do not share their cherished belief.

Indeed, history also shows that a prohibition against open discussion and experimental testing of the foundations of quantum theory, in turn, led to a significant delay of
the discovery of important new applications of these foundations. Quantum cryptography, distributed entanglement, etc. undoubtedly would never have been envisioned without the intellectual challenges posed by Bell's Theorem.

My own final conclusion is that the only real loser here has been the "stigma", itself. I hope that John Bell would have agreed.

 

[DrChinese]

Bell goes on to say "we will argue that certain particular correlations, realizable according to quantum mechanics, are locally inexplicable. They cannot be explained, that is to say, without action at a distance."

If you are going to quote Bell so extensively, I am going to assume you agree with the above. And I am going to assume that by action at a distance, we mean either a propagation of a direct cause and its related effect in excess of c - or a direct connection between distant points which is absolutely simultaneous (instantaneous action at a distance, a la Bohmian Mechanics).

The bolded statement is obviously incorrect, unless you suitably redefine terms to make it correct. For example, it is generally agreed that Many Worlds does not involve action at a distance. Retrocausal and certain acausal theories do not feature action at a distance (the context slice is a collection of points which are themselves accessible at c or less. Those couldn't be accepted interpretations if you are correct. Basically, we all have our favorite interpretations (or non-interpretations as the case may be), so we naturally believe our baby is prettier.

Sadly, Bell did not live to see experimental quantum entanglement swapping. If he had, he would realize that entangled particle pairs need not have ever co-existed. Co-existing presumably being a requirement for instantaneous action at a distance - i.e. what a non-local theory purports to explain.

I agree that there is something called "quantum non-locality", which would be *whatever* kind of non-locality that is exhibited in quantum experiments. Such is not constrained by distance in space or time, no does it necessarily involve cause and effect. However, that does not map directly to the kind of action at a distance per the bold above. Lines of "action" in quantum non-locality are constrained to c, and can move either forward or backward in time direction. This is very clear when you look at a diagram of entanglement swapping. Note: The entangled particles themselves can superficially appear to demonstrate effects in excess of a large multiple of c, approaching infinity; while the growth of the cone of action does not grow faster than a traditional light cone.

So my point is that experimental evidence would force the very careful Bell to modify the above statement, were he to have lived longer. I don't think he ever came out as a full blooded Bohmian anyway, although I am not certain about that. I definitely don't get the point of your Clauser quote, which doesn't seem to bear any relation to quantum nonlocality.

 

[Auto-Didact]

As I said before I'm not a Bohmian nor am I advocating BM, certainly not as the correct fundamental theory of physics. I believe we do not yet have such a theory.

I'm merely arguing, like Shimony (who I also quoted extensively) as well as others, that quantum non-locality is something actually occurring in nature and that mathematically explaining it will probably require the modification of both QM and relativity.

 

[DrChinese]

I'm merely arguing, like Shimony (who I also quoted extensively) as well as others, that quantum non-locality is something occurring in nature and that mathematically explaining it will probably require the modification of both QM and relativity.

I quite agree, but I didn't read what you wrote previously in this manner.

 

[lodbrok]

I don't really know what you are talking about. $A_n, B_n, C_n$ are three numbers, each one is either +1 or -1.

Yes. The three numbers are assumed to exist together in some context denoted by the subscript $n$ which could represent a particle pair for example. Not unlike when we have three coins $A, B, C$ which we toss in pairs, so that we have three numbers $A_n, B_n, C_n$ representing the results of the toss each either +1, -1 (Heads or Tails) in the context of a single toss $n$. Note also that in principle you could imagine the third coin would have produced a value had you tossed it at the same time so it makes sense to think of $A_n, B_n, C_n$ counter-factually. In fact, you could toss all three but observe only two values and still satisfy your requirement that the third value exists even if not known. All is well and good up to here.

So it's just a fact of arithmetic that:

$A_n B_n + A_n C_n = A_n B_n + A_n (B_n B_n) C_n = A_n B_ n (1 + B_n C_n)$

The assumption is that twin-pair number $n$, has an associated triple of numbers $A_n, B_n, C_n$, where $A_n$ gives the result of a measurement of spin along the first axis, $B_n$ along the second axis, $C_n$ along the third axis. The manipulation above is just arithmetic. There is no additional assumptions involved.

Yes. So long as you are still talking about that specific context, $n$, that is, that specific particle pair, or that specific toss of a pair of coins, your arithmetic is all good, no additional assumptions required. For the coin toss analogy, we can also do simple arithmetic to obtain

$A_n B_n + A_n C_n = A_n B_n + A_n (B_n B_n) C_n = A_n B_ n (1 + B_n C_n)$

However, in an EPRB experiment $A_n B_n + A_n C_n$ is incorrect because $A_n B_n$ is measured on one particle pair, and $A_n C_n$ is measured on an entirely different particle pair. It does not matter if both particle pairs are similarly prepared. This means your notation should really be $A_n B_n + A_m C_m$, to acknowledge the different context $m$. Similarly, using the coin toss analogy, we have really have $A_n B_n + A_m C_m$ since we toss only two coins each time, not three and $A_n B_n$ originates from a different toss than $A_m B_m$. And even if we decided to toss three coins each time, we only read two outcomes from each toss and therefore we necessarily use the results from two different context for the terms.

You introduce the analogy by applying your simple arithmetic to the EPRB experiment or to the analogous coin toss experiment. Because according to your arithmetic

$A_n B_n + A_m C_m = A_n B_ n (1 + B_m C_m) = A_n B_n + A_n B_n B_m C_m$

implying $A_n B_n B_m C_m = A_m C_m$
This is only true if $B_n B_m = 1$ and $A_n = A_m$. This means you expect The outcomes from one context to be perfectly correlated with the outcomes from the other. Or in terms of the coin toss experiment. You assume heads from the first toss of coin $B$ is perfectly correlated with heads from the second toss of coin $B$.

I know it is subtle, but the assumption is introduced in the application of the derived expression to the experiment.Nobody would suggest that coins are non-local. But note the problem you face by applying your logic to such a simple coin toss experiment.

 

[stevendaryl]

However, in an EPRB experiment $A_n B_n + A_n C_n$ is incorrect because $A_n B_n$ is measured on one particle pair, and $A_n C_n$ is measured on an entirely different particle pair.

I think you are misunderstanding the distinction between what is observed, and a model intended to EXPLAIN those observations. Yes, what is observed is that Alice measures the spin on one particle, and Bob measures the spin on another particle. The class of models that Bell is interested in (the class that Einstein, Podolsky and Rose were interested in) are models of the following form:

1. When a twin-pair is produced, there is an associated state variable $\lambda$ describing the pair.
2. When Alice measures her particle, the result depends on (1) facts about her measuring device, and (2) the value of $\lambda$
3. Similarly, when Bob measures his particle, the result depends on facts about his measuring device, and the value of $\lambda$.

So in general, Alice's result $R_A$ can be written as a function $R_A(\lambda, \alpha, \lambda_A)$, where $\alpha$ is the choice of which measurement to perform, and $\lambda_A$ is other facts about Alice's device. Bob's result $R_B$ is similarly a function $R_B(\lambda, \beta, \lambda_B)$ where $\beta$ is his choice of measurement, and $\lambda_B$ is other facts about Bob's device.

So now, we have to take into account a stark fact about these twin-particles: There is PERFECT anti-correlation (or correlation, depending on the exact type of EPR experiment performed). That means that for each $\lambda$, if it happens to be that $\alpha = \beta$ (that is, if Alice and Bob perform the same measurement), they always get opposite result: No matter what $\lambda_A$ and $\lambda_B$ are, we always have:

$R_A(\lambda, \alpha, \lambda_A) = - R_B(\lambda, \alpha, \lambda_B)$

This implies that $R_A$ and $R_B$ don't actually depend on $\lambda_A$ and $\lambda_B$ at all. If Alice's result is NOT determined by $\lambda$ and $\alpha$, then sometimes she would get a result that would not be anti-correlated with what Bob gets.

So we pick three possible measurements for Alice, $\alpha_1, \alpha_2, \alpha_3$. For each $\lambda$, let $A(\lambda)$ be $R_A(\lambda, \alpha_1)$, let $B(\lambda)$ be $R_A(\lambda, \alpha_2)$ and let $C(\lambda)$ be $R_A(\lambda, \alpha_3)$.

Now, if we create a sequence of twin-pairs, each twin pair is associated with some value of $\lambda$. So we let

$A_n = A(\lambda_n)$
$B_n = B(\lambda_n)$
$C_n = C(\lambda_n)$

where $\lambda_n$ is the value of $\lambda$ for the $n^{th}$ twin pair. These are not measurement results, they are just numbers, unknown functions of $\lambda$ evaluated at $\lambda = \lambda_n$. But we're ASSUMING that the significance of these numbers is that

$A_n$ is the result that Alice WOULD get, if she chose to measure her $n^{th}$ particle using device setting $\alpha_1$.
$B_n$ is the result that Alice WOULD get, if she chose to measure her $n^{th}$ particle using device setting $\alpha_2$.
$C_n$ is the result that Alice WOULD get, if she chose to measure her $n^{th}$ particle using device setting $\alpha_3$.

Under the assumption of perfect anti-correlation, Bob would get $-A_n, -B_n, -C_n$ for the corresponding measurements.

So now, the numbers $A_n, B_n, C_n$ are just three numbers, each are assumed to be $\pm 1$. So we can do manipulations as real numbers to come to the conclusion that:

$|\langle A B \rangle + \langle A C \rangle | \leq 1 + \langle B C \rangle$

where $\langle A B \rangle = \frac{1}{N} \sum_n A_n B_n$ and $\langle A C \rangle = \frac{1}{N} \sum_n A_n C_n$ and $\langle B C \rangle = \frac{1}{N} \sum_n B_n C_n$, and where $N$ is the number of twin pairs produced.

This is simply a mathematical fact about ANY sequence of triples of numbers $A_n, B_n, C_n$ where each number is $\pm 1$.

Now, the question is whether it is possible to measurement the quantities $\langle A B \rangle$, $\langle A C \rangle$, $\langle B C \rangle$. We can't, actually, because the definition of (for example) $\langle A B \rangle$ is that it is the average of $A_n B_n$ over all values of $n$. But we don't measure $A_n$ and $B_n$ over all possible values of $n$. We only measure it on for some of the $n$. So to compare theory with experiment, we have to assume that the average of $A_n B_n$ over some of the $n$ is approximately the same as the average over all $n$.

 

[stevendaryl]

Yes. So long as you are still talking about that specific context, $n$, that is, that specific particle pair, or that specific toss of a pair of coins, your arithmetic is all good, no additional assumptions required. For the coin toss analogy, we can also do simple arithmetic to obtain

$A_n B_n + A_n C_n = A_n B_n + A_n (B_n B_n) C_n = A_n B_ n (1 + B_n C_n)$

However, in an EPRB experiment $A_n B_n + A_n C_n$ is incorrect because $A_n B_n$ is measured on one particle pair, and $A_n C_n$ is measured on an entirely different particle pair.

The point of rewriting that expression involving $A_n, B_n, C_n$ is to come up with a relationship about the averages for $A_n B_n$, $A_n C_n$ and $B_n C_n$, averaged over all $n$. Bell's inequality is about the averages, not about particular $n$.

 

[N88]

Yes, it's certainly possible to have a "perturbative" measurement, where the result is not 100% determined by the pre-existing properties. However, there are no ideas (as far as I know of) how such a perturbative measurement could result in perfect correlation between measurements of distant particles.

That's really Einstein et al's whole point: If measurements DON'T reveal pre-existing properties, then how can two distant measurements produce perfect correlations?

"How can two distant measurements produce perfect correlations?"

It is my understanding that the spin-half particles in EPRB are pairwise anti-correlated via the pairwise conservation of total angular momentum.

Support for that such correlation is seen in eqn (14) from Bell (1964):

$A(a,\lambda)_{Alice}=\pm1.\:(1)\:$ $B(b,\lambda)_{Bob}=\pm1=-A(b,\lambda)_{Bob}.\:(2)\:$.

So $A(a,\lambda)_{Alice}=-A(a,\lambda)_{Bob}.\:(3)$ And $A(b,\lambda)_{Alice}=-A(b,\lambda)_{Bob};etc.\:(4).$

And, dropping the subscripts temporarily: such correlation invokes the general product rule:

$P(AB)=P(A)P(B|A).\:(5)$*

* Causation between outcomes would also invoke the same rule, but the outcomes are spacelike separated. So causation between outcomes is not possible.

Thus consistent with Bayesian updating; ie, interpreting (5) epistemically: (5) says that the occurrence of $A(a,\lambda)_{Alice}$ reveals information that is probabilistically relevant to the occurrence of $B(b,\lambda)_{Bob}.$

So we need a deterministic relation that links eqns (3) and (4) and (5), etc., via the angular relation $(a,b)$.

Let $P(A=1,B=1)=P(A=1)P(B=1|A=1)=\frac{1}{2}sin^2\frac{1}{2}(a,b).\:(6)$

Then (6), in agreement with standard QM, supplies the local deterministic particle-detector relations that (in my view) answer your question.

 

[DrChinese]

Yes. The three numbers are assumed to exist together in some context denoted by the subscript $n$ which could represent a particle pair for example. Not unlike when we have three coins $A, B, C$ which we toss in pairs, so that we have three numbers $A_n, B_n, C_n$ representing the results of the toss each either +1, -1 (Heads or Tails) in the context of a single toss $n$. Note also that in principle you could imagine the third coin would have produced a value had you tossed it at the same time so it makes sense to think of $A_n, B_n, C_n$ counter-factually. In fact, you could toss all three but observe only two values and still satisfy your requirement that the third value exists even if not known. All is well and good up to here.

Yes. So long as you are still talking about that specific context, $n$, that is, that specific particle pair, or that specific toss of a pair of coins, your arithmetic is all good, no additional assumptions required. For the coin toss analogy, we can also do simple arithmetic to obtain

$A_n B_n + A_n C_n = A_n B_n + A_n (B_n B_n) C_n = A_n B_ n (1 + B_n C_n)$

However, in an EPRB experiment $A_n B_n + A_n C_n$ is incorrect because $A_n B_n$ is measured on one particle pair, and $A_n C_n$ is measured on an entirely different particle pair. It does not matter if both particle pairs are similarly prepared.

Well of course there are *thousands* of particle pairs in an experiment. If you are a (non-contextual) realist, you believe that Alice's outcome does NOT depend on Bob's setting, and vice versa. So it doesn't matter which pair AB, BC or AC you select on a specific iteration. As stevendaryl says, you are looking for an average.

And the purpose of an experiment is simply to confirm the predictions of QM. The outcome would NOT directly affect Bell's Theorem either way, but would only affect it indirectly. Bell answers the question of whether there is a realistic (and local) theory which can match the predictions of QM, and the answer is NO. The experiment shows then that we do not live in a local realistic world.

 

[stevendaryl]

"How can two distant measurements produce perfect correlations?"

It is my understanding that the spin-half particles in EPRB are pairwise anti-correlated via the pairwise conservation of total angular momentum.

Support for that such correlation is seen in eqn (14) from Bell (1964):

$A(a,\lambda)_{Alice}=\pm1.\:(1)\:$ $B(b,\lambda)_{Bob}=\pm1=-A(b,\lambda)_{Bob}.\:(2)\:$.

So $A(a,\lambda)_{Alice}=-A(a,\lambda)_{Bob}.\:(3)$ And $A(b,\lambda)_{Alice}=-A(b,\lambda)_{Bob};etc.\:(4).$

That's right. If Bob's result is a deterministic function of $\lambda$ and his setting, and Alice's result is a deterministic function of $\lambda$ and her setting, then the requirement that $A(a,\lambda)_{Alice} = - A(a, \lambda)_{Bob}$ explains the perfect correlations. But Bell's inequality shows that there are no such functions $A(a, \lambda)_{Alice}$ and $A(b, \lambda)_{Bob}$.

 

[DrChinese]

"How can two distant measurements produce perfect correlations?"
...

EPR answered that question, and yes, Bell walked through the EPR reasoning on the perfect correlations. But that is not the problem! The problem is when you extend that reasoning - as Bell did - to other angle settings. Then the contradiction occurs, because the results are skewed slightly towards Alice's measurement setting affecting the results of Bob.

If you attempt to write down values for results at A=0, B=120 and C=240 degrees for *both* Alice and Bob (6 per trial) - and remember, you think they are predetermined - you will quickly see that the math does not work out. You can hand pick them even, and they will not work out. Try it.