Does the Bell theorem assume reality?

[DrChinese]

... So Alice's subjective likelihood of Bob getting +1 changed instantaneously from 50% to 100% (or 0%, whichever it is). Einstein reasoned that there were two possible explanations for this sudden change:

1. Somehow Alice's measurement affected Bob's particle, even though it was far away. This would be "spooky action at a distance".
2. Alternatively, maybe Bob's measurement result was pre-determined to be whatever before Alice performed her measurement, and her measurement only informed her of this fact. This would be a hidden variable.

Bell's argument showed that interpretation 2 is not possible. So spooky action at a distance it is.

I call it "quantum nonlocality" rather than "spooky action at a distance" for the simple reason that the "distance" is in spacetime, not space. "Spooky action at a distance" is often thought to be the same as "instantaneous action at a distance", which (IMHO) it is not. There are obvious limits that can be seen in any diagram showing quantum nonlocality (i.e. anywhere there is entanglement).

 

[N88]

Bell is assuming that $A(\overrightarrow{a}, \lambda)$ is a function returning $\pm 1$, and the interpretation is that if a particle has the hypothesized hidden variable $\lambda$, and you measure the spin along direction $\overrightarrow{a}$ (or polarization), then you will get the result $A(\overrightarrow{a}, \lambda)$.

So $A(\overrightarrow{a}, \lambda) A(\overrightarrow{b}, \lambda) - A(\overrightarrow{a}, \lambda) A(\overrightarrow{c}, \lambda)$ can be written:

1. $A(\overrightarrow{a}, \lambda) A(\overrightarrow{b}, \lambda) - A(\overrightarrow{a}, \lambda) A(\overrightarrow{c}, \lambda)$
$= A(\overrightarrow{a}, \lambda) (A(\overrightarrow{b}, \lambda) - A(\overrightarrow{c}, \lambda))$

At this point, we can use that $A(\overrightarrow{b}, \lambda) = \pm 1$, which implies that $A(\overrightarrow{b}, \lambda) A (\overrightarrow{b}, \lambda) = 1$. So we can rewrite

$A(\overrightarrow{c}, \lambda)) = A(\overrightarrow{b}, \lambda) A (\overrightarrow{b}, \lambda) A(\overrightarrow{c}, \lambda)$

Since the first two factors multiplied together yield +1. So we can plug this in for $A(\overrightarrow{c}, \lambda))$ into the right-side of equation 1 to get:
2. $A(\overrightarrow{a}, \lambda) A(\overrightarrow{b}, \lambda) - A(\overrightarrow{a}, \lambda) A(\overrightarrow{c}, \lambda)$
$= A(\overrightarrow{a}, \lambda) (A(\overrightarrow{b}, \lambda) - A(\overrightarrow{b}, \lambda) A (\overrightarrow{b}, \lambda) A(\overrightarrow{c}, \lambda))$
$= A(\overrightarrow{a}, \lambda) A(\overrightarrow{b}, \lambda)(1 - A(\overrightarrow{b}, \lambda) A(\overrightarrow{c}, \lambda))$

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

Bell and you say: $A(\overrightarrow{b}, \lambda) = \pm 1$, which implies that $A(\overrightarrow{b}, \lambda) A (\overrightarrow{b}, \lambda) = 1$

But, to me, this implication only holds if the pair of $A (\overrightarrow{b}, \lambda) = ±1$ come from the same instance. See last line on Bell 1964, p.195.

And we can see from

1. $A(\overrightarrow{a}, \lambda) A(\overrightarrow{b}, \lambda) - A(\overrightarrow{a}, \lambda) A(\overrightarrow{c}, \lambda)$

that there are two sets of instances: one over the

$(\overrightarrow{a},\overrightarrow{b})$ settings and one over the $(\overrightarrow{a},\overrightarrow{c})$ settings.

So it seems to me that Bell is combining two independent non-correlated variables [because they come from different instances]: so each combination will = +1 or -1.

This why I am asking about the realism that Bell is postulating when he uses

$A(\overrightarrow{b}, \lambda) A (\overrightarrow{b}, \lambda) = 1$.

Or, to put my problem another way: Bell seems to combine results from two different sets of instances [a no-no?] and creates a third set of instances: those over

$A(\overrightarrow{b}, \lambda) A(\overrightarrow{c}, \lambda)$.

Could this be the reason that QM disgrees with Bell's inequality? I do not see QM combining results from two differents instances:.

From the QM formula in Bell 1964:(3): σ₁ = -σ₂ . Is it not the case then, in QM, that the sigmas are pairwise-matched from the same instance?

And that they are therefore pairwise antiparallel via the pairwise conservation of total angular momentum in each instance?

 

[DarMM]

On the other hand, if there are multiple outcomes compatible with the observer's state, then I would say worlds corresponding to each outcome exist and SLU applies. In regards to your example, I looked up the Unruh effect but didn't find much relevant to this aspect. However, I don't see how the details could matter. One can imagine 4 worlds, as viewed from outsiders, each containing a stationary observer with the same state and accelerating observers with different states.

In the Unruh effect there is only one accelerating observer, not multiple copies, even when viewed by others.

The whole point is that this is a property of the states alone prior to measurement, so you can't invoke multiple copies of the observer.

Just posit one inertial observer and one accelerating observer, the Bogolyubov transformation on the field's modes induced by the coordinate transformations between their frames alone causes the transition from a pure state to a mixed state, with no entanglement with the observer/measurer.

Of course none of this is meant to be a killing argument, it's just suggestive of an epistemic view (like the no cloning theorem, teleportation, $\psi$ obeying things like diFinetti's theorem, etc) and in the original context of why I mentioned this, why QBists aren't obviously wrong to reject the reality of $\psi$.

Basically there are properties of $\psi$ that appear epistemic. One so far can give them a $\psi$-ontic reading, but that's not an argument against deciding to read them epistemically. You've said that nature could be such that the ontic stuff is structurally the same as epistemic knowledge of it. You might be right, but of course some people are going to look at those epistemic-like structures and read them purely epistemically. Until there is some sort of no-go theorem this is the end point and an objective analysis not favoring any interpretation based on one's own preferences or intutions can't proceed further.

 

[stevendaryl]

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

Any assumptions about reality were made prior to the manipulations here. From this point on, it's just mathematics.

Bell and you say: $A(\overrightarrow{b}, \lambda) = \pm 1$, which implies that $A(\overrightarrow{b}, \lambda) A (\overrightarrow{b}, \lambda) = 1$

But, to me, this implication only holds if the pair of $A (\overrightarrow{b}, \lambda) = ±1$ come from the same instance.

As I said, this is just mathematics. We're deriving a property of a function of two variables. If a number is either +1 or -1, then its square is 1.

It doesn't make sense to ask about whether it "comes from the same instance". It's a function.

See last line on Bell 1964, p.195.

And we can see from

1. $A(\overrightarrow{a}, \lambda) A(\overrightarrow{b}, \lambda) - A(\overrightarrow{a}, \lambda) A(\overrightarrow{c}, \lambda)$

that there are two sets of instances:

As I said, Bell is proving a fact about functions. At this point, "instances" don't come into play. I really don't understand what you're objection is. It might make sense to object to Bell's assumption that the outcome of a particle measurement is described by some unknown function $A(\overrightarrow{a}, \lambda)$, but if you grant that assumption, then everything after that point is just mathematics.

 

[zonde]

Acausal describes physics where you take a 4D chunk of spacetime with some matter in it and declare the events that occur are those that satisfy a specific constraint given conditions on hypersurfaces at opposite ends. This basically occurs in Classical Mechanics in the least action formalism, however due to the resulting least action trajectory obeying the Euler-Lagrange equations, you can convert this to a 3+1D picture of what is going on, i.e. of a particle moving in response to a potential.

Acausal views of QM declare the set of events is a result of a constraint different to the least action principal, however one where the resulting set of events can't be understood in a 3+1D way, i.e. as initial conditions evolving in time under a PDE or something similar.

Reichenbach's common cause principle doesn't hold, because later events don't result from previous ones which are their causes, rather the set of events as a whole is selected by a constraint. However things still have a fairly clear scientific explanation.

What are your considerations for your conclusion that acausal explanations are scientific?
Is your considerations along the lines that in classical mechanics acausal explanation can be converted into causal explanation which we consider scientific?

 

[DarMM]

What are your considerations for your conclusion that acausal explanations are scientific?
Is your considerations along the lines that in classical mechanics acausal explanation can be converted into causal explanation which we consider scientific?

If they can select out the statistics you see in experiments then you can confirm them like any other scientific theory.

 

[zonde]

If they can select out the statistics you see in experiments then you can confirm them like any other scientific theory.

We are speaking about interpretation, so this is not going to work as statistics are already known and confirmed by experiments.
Then say superdeterministic interpretation might give the same statistics, but you won't consider such explanation scientific, right? So there should be other considerations too that are more relevant for interpretations.

 

[akvadrako]

The whole point is that this is a property of the states alone prior to measurement, so you can't invoke multiple copies of the observer.

It's still a bit unclear to me how this mixed state forms, but I assume the way it works is that hidden (or ignored) initial conditions determine which of several outcomes occur. So the copies already exist before the start of the transformation, sitting in different worlds where those critical variables differ.

Until there is some sort of no-go theorem this is the end point and an objective analysis not favoring any interpretation based on one's own preferences or intutions can't proceed further.

We already have Bell's theorem, but QBism seems immune to no-go theorems. 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. The interpretations which deal with a shared objective reality shed light on my experience by analyzing them through the eyes of others. This is something QBists refuse to do; their line of reasoning may eventually lead to some insights, but for now it doesn't provide an alternative.

 

[DrChinese]

Then say superdeterministic interpretation might give the same statistics...

There are no superdeterministic interpretations that give the quantum mechanical stats. People hypothesize that there could be such, and there have been a few toy models. That is far and away different from, say, Relational Blockworld. That acausal theory is explained in an entire book, and numerous peer-reviewed papers.

 

[DarMM]

We are speaking about interpretation, so this is not going to work as statistics are already known and confirmed by experiments.
Then say superdeterministic interpretation might give the same statistics, but you won't consider such explanation scientific, right? So there should be other considerations too that are more relevant for interpretations.

Well for different "interpretations" of the same theory then if they have the exact same predictions as each other there is no way to select one from the other experimentally period, doesn't matter if they are retrocausal, acausal, nonlocal, etc

My response was more for acausal theories in general where you'd confirm them like anything else, via checking their predicted statistics for experiments.

I have interpretations in quotes above, as many of the supposed interpretations of QM either have some scenarios where they make different predictions, or in fact have never been shown to obtain the predictions of QM, so whether they are actually interpretations can be strictly false or unknown currently.

 

[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.