Entanglement and FTL signaling in professional scientific literature

[vanhees71]

Once more: The definition of "nonlocality" must be consistent with "microcausality" in any interpretation of standard relativistic quantum field theory. Otherwise you propose a new theory. Then one must find an experiment which can decide between the two theories.

The "minimal statistical interpretation" is a well-defined and consistent interpretation of standard Q(F)T and not just something I mumble about. It just uses the probabilistic interpretation of the quantum state, as it is defined in the mathematical formulation of Q(F)T and nothing else. It's "minimal" in the sense that it just refers to the empirical meaning of the formalism and nothing else.

A very concise comprehensive formulation of it is given in Ballentine, Quantum Mechanics. Unfortunately that's a book on non-relativistic quantum mechanics, i.e., you don't find much about microcausality (locality in the well-defined mathematical sense of relativistic QFT) vs. "non-locality" (in the sense of the possibility of long-ranged correlations between parts of entangled quantum systems that are measured at far distant places with measurement events space-like separated) and why there is no contradiction between the two.

 

[gentzen]

Once more: The definition of "nonlocality" must be consistent with "microcausality" in any interpretation of standard relativistic quantum field theory. Otherwise you propose a new theory. Then one must find an experiment which can decide between the two theories.

Most interpretations have their origin in non-relativistic quantum mechanics, and the "minimal statistical interpretation" is no exception here. Some interpretations can be "upgraded" to interpretations of relativistic quantum field theory. But if nobody invests the work to do such an "upgrade," this doesn't imply that some existent interpretation had suddenly turned into a proposal for a new theory.

Even so I my guess is that the "minimal statistical interpretation" can in principle be "updated," it would be your task to explain to us how to do this, or to point to a reference where it has been done.

But I find the position to accept the nonlocal randomness more convenient for various reasons. One of those reasons is that explaining interpretations is already very hard, even without also having to take into account relativistic quantum field theory. A day only has so many hours, and other things are simply more important to me.

 

[vanhees71]

Most interpretations have their origin in non-relativistic quantum mechanics, and the "minimal statistical interpretation" is no exception here. Some interpretations can be "upgraded" to interpretations of relativistic quantum field theory. But if nobody invests the work to do such an "upgrade," this doesn't imply that some existent interpretation had suddenly turned into a proposal for a new theory.

That's indeed one of the problems. If you discuss the tension between relativistic causality (no faster-than-light causal effects) and "quantum non-locality" you can't argue within non-relativistic QT, where "actions at a distance" are just the normal description of the dynamics as in classical Newtonian mechanics. It's no surprise that there you get faster-than-light signal propagation.

Then the question is, where's the need for an "upgrade" of all kinds of interpretations, which don't solve any "real physical problem"? Why should anybody invent, e.g., a Bohmian theory for relativistic QFT? What could one learn from it?

Even so I my guess is that the "minimal statistical interpretation" can in principle be "updated," it would be your task to explain to us how to do this, or to point to a reference where it has been done.

What do you want to "update"? It's just applied all the time when comparing the results of experiments testing Q(F)T with the predictions of that theory.

But I find the position to accept the nonlocal randomness more convenient for various reasons. One of those reasons is that explaining interpretations is already very hard, even without also having to take into account relativistic quantum field theory. A day only has so many hours, and other things are simply more important to me.

As I said, if you want to understand the question of and relation between "locality" in the sense of relativstic spacetime descriptions versus "quantum nonlocality", you must employ a relativistic consistent description, and the only one we have is microcausal relativistic QFT. So that't the theory you have to apply.

 

[gentzen]

What do you want to "update"? It's just applied all the time when comparing the results of experiments testing Q(F)T with the predictions of that theory.

The instant sharing of random numbers for space-like separated events is a nonlocal effect. Therefore, if an interpretation wants to be truly local, it has to explain why this doesn't count, or what we did wrong to accidentally introduce that nonlocality.

 

[vanhees71]

There is no "instant sharing of random numbers for space-like separated events". If you want to reveal the correlations, you have to exchange this information in a "classical way" by comparing measurement protocols.

 

[PeterDonis]

within a "minimal interpretation", i.e., the assumption that the "kinematics and dynamics" of the theory is complete, including the "probabilistic and only probabilistic" meaning of quantum states.

This is not a "minimal interpretation". It's a claim that the minimal interpretation is the "correct" interpretation. Other interpretations do not agree with this. Not only that, other interpretations do not even agree that the two statements you give here are consistent: the MWI says that the kinematics and dynamics are complete but says quantum states are not probabilistic; they are an exact and complete description of the physical state of the system and they always evolve by unitary dynamics. "Physical collapse" theories, OTOH, say that quantum states are indeed "probabilistic and only probabilistic", but deny that the "kinematics and dynamics" of the theory (meaning standard QM) are complete.

 

[gentzen]

you have to exchange this information in a "classical way" by comparing measurement protocols.

This certainly goes into the direction I had in mind. However, there are many ways to exchange information in a classical way. One way would be to agree on an exact sequence of measurement settings in advance. Another way would be to try use completely random measurement settings, and share those settings later using a classical communication channel.

But let me stop here. It is fine for me now. I am not overly keen on performing such interpretational dances. Because I fear that in the end they will just confuse everybody, without solving any actual substantive problem. At least not until somebody comes along (in a Frauchiger-Renner way) and makes a mistake of this sort. But even then, he probably won't understand the message of such a dance.

 

[PeterDonis]

you have to exchange this information in a "classical way" by comparing measurement protocols.

No, it's not enough to share the protocols. You have to share enough information for Alice and Bob to be able to compare their exact meaurement directions as they were actually realized. In any real experiment, measurement directions cannot be picked with infinite accuracy, so there will be some "error" or "noise" in the actual data; that "error" or "noise" is part of the information produced by the measurements and it can only be known by comparing the results after the fact using classical channels.

 

[PeterDonis]

there are many ways to exchange information in a classical way.

There is only one, though, that ensures that all of the information produced by the measurements is shared by both parties. See my post #218 just now.

 

[vanhees71]

No, it's not enough to share the protocols. You have to share enough information for Alice and Bob to be able to compare their exact meaurement directions as they were actually realized. In any real experiment, measurement directions cannot be picked with infinite accuracy, so there will be some "error" or "noise" in the actual data; that "error" or "noise" is part of the information produced by the measurements and it can only be known by comparing the results after the fact using classical channels.

Of course, all this must be contained in the measurement protocols, and of course there are systematic on top of the statistical uncertainties in any measurement.

 

[PeterDonis]

all this must be contained in the measurement protocols

Ah, ok, by "protocols" you meant to include the things that have to occur after the measurements are complete. Got it.

 

[DrChinese]

a. Say Alice measures the spin of her photon at some angle and observes "up". Is the following statement true:

"If bob performs the same measurement on his photon, there is a 100% chance he will observe 'up', but if Alice had not measured her particle, there would be less than 100% chance that Bob will observe 'up'"

b. But, as has already been pointed out, neither of these can be right, because they are both asymmetric, but the situation is symmetric between Alice and Bob. So whatever is going on "behind the scenes" would need to be symmetric between them as well. ... But part of what we do understand is that the situation is symmetric between Alice and Bob: we know the measurements commute. So any resolution, it seems to me, would have to be consistent with that.

c. Sure there is: the entangled state that was prepared. The issue some people have is that that isn't the kind of "variable" they are looking for as a common cause.

d. I agree. [Yay!]

I disagree. Bell only excludes the kind of common cause that follows his ansatz; which is a hidden correlation of "ignorance type". Why would this be exhaustive??

@Morbert:
a. I agree, Bob would then see a 50% "up" rate rather than 100% assuming Alice's results are unknown. If measured at different angles (but Alice IS known), the results follow the usual expectation value. @PeterDonis:
b. My statements ("Either Alice's measurement casts Bob's particle into a state synchronized with Alice, or Bob's measurement casts Alice's particle into a state synchronized with Bob.") ARE symmetric (or commute), and precisely fit the facts. There are no other facts you can state about the results OTHER than what I say.

c. The entangled state absolutely does NOT predetermine the outcomes of (all possible) measurement choices by Alice and Bob. Bell ruled that out for all possible type of hidden variables (see d. below for quotes). The best you could say is that it is responsible for the random element to the outcomes, which are otherwise unaccounted for in the quantum expectation value for matches.@Fra:
d. You are incorrect about Bell's hidden variable types - there was no limitation or exclusion on them at all. He said (1964): "Let this more complete specification be effected by means of parameters λ. It is a matter of indifference in the following whether A denotes a single variable or a set, or even a set of functions, and whether the variables are discrete or continuous." He does provide one assumption however, but it is not on type or form: "The vital assumption is that the result B for particle 2 does not depend on the setting a, of the magnet for particle 1, nor [vice versa] A on b."

 

[Morbert]

@Morbert:a. I agree, Bob would then see a 50% "up" rate rather than 100% assuming Alice's results are unknown. If measured at different angles (but Alice IS known), the results follow the usual expectation value.

I think there's where the disagreement is: Not all interpretations would agree that Bob would see a 50% up rate under these counterfactual scenarios, and under these interpretations, we cannot conclude a nonlocal influence.

 

[PeterDonis]

My statements ("Either Alice's measurement casts Bob's particle into a state synchronized with Alice, or Bob's measurement casts Alice's particle into a state synchronized with Bob.") ARE symmetric (or commute)

I think we disagree on what "symmetric" and "commute" mean.

The entangled state absolutely does NOT predetermine the outcomes of (all possible) measurement choices by Alice and Bob.

I didn't say it did. I only said it was a candidate for a "common cause". Nothing "predetermines" the outcomes--even the entangled state plus the measurements choices doesn't predetermine the outcomes since there is still a random element involved. But the entangled state is certainly part of what "contributes" (to use as neutral a word as possible) to the outcomes. So are the measurement choices of Alice and Bob. The entangled state is the "common" part of that, and the choices are the "non-common" part (since they can differ between Alice and Bob).

The best you could say is that it is responsible for the random element to the outcomes

That would depend on what interpretation of the quantum state is adopted.

 

[PeterDonis]

Not all interpretations would agree that Bob would see a 50% up rate under these counterfactual scenarios

Which interpretations would not?

 

[Morbert]

Which interpretations would not?

Consistent histories as presented Robert Griffiths (one of its founders). From Chapter 19 of "Consistent Quantum Theory"

"While relativistic quantum theory is outside the scope of this book, an analysis of non-relativistic versions of some of the paradoxes which are supposed to show the presence of superluminal influences indicates that the real source of such ghostly effects is the need to correct logical errors arising from the assumption that the quantum world is behaving in some respects in a classical way. When the situation is studied using consistent quantum principles, the ghosts disappear
[...]
Certain quantum paradoxes are stated in terms of counterfactuals: what would have happened if some state of affairs had been different from what it actually was. Other paradoxes have both a counterfactual as well as in an “ordinary” form. In order to discuss counterfactual quantum paradoxes, one needs a quantum version of counterfactual reasoning
[...]
The result is X+ with probability one. That this is reasonable can be seen in the following way. The actual measurement outcome X + shows that the particle had Sx = +1/2 at time t1 before the measurement took place, since quantum measurements reveal pre-existing values if one employs a suitable framework. And by choosing [x+] at t1 as the pivot, one is assuming that Sx had the same value at this time in both the actual and the counterfactual world. Therefore a later measurement of Sx in the counterfactual world would necessarily result in X+"

That last paragraph is not explicitly referencing the EPR experiment, but a similar scenario characterised by equation 19.12. I have constructed the analogous support for the EPR experiment below: Say Bob measures the spin-x of his particle and Alice either measures spin-x or doesn't measure spin-x of her particle. The support is
$$
\left\{\begin{array}{lll}
\left[x_B^+\right]&\otimes&\left\{\begin{array}{lll}\left[X_A^+,X_B^+\right]&&\\
&&\\
\left[X_A^0,X_B^+\right]&&\end{array}\right.\\
&&\\
\left[x_B^-\right]&\otimes&\left\{\begin{array}{lll}\left[X_A^-,X_B^-\right]&&\\
&&\\
\left[X_A^0,X_B^-\right]&&\end{array}\right.
\end{array}\right.
$$
where $x_B^\pm$ is the x-spin up/down of Bob's particle, $X_A,X_B$ are Alice's and Bob's apparatus for measuring spin-x, $X^\pm$ means up/down was measured and $X^0$ means the measurement was not carried out. If Alice measures spin-x and observes up, we know we are on the top branch of this support, and so we can use counterfactual reasoning to work out what would happen if Alice had not measured spin-x (the 2nd branch).

[edit] - Simplified support and scenario. Some of the support is implicit, but I think it is easier to read.

 

[PeterDonis]

a similar scenario characterised by equation 19.12

Huh? That scenario, like all of the scenarios discussed in that chapter, is about a measurement on a single particle. None of them are about measurements on entangled pairs of particles.

Even if we leave that aside, the only scenario that even discusses counterfactuals where a measurement that is made in the "actual" world is not made in a counterfactual world is the one shown in equation 19.14.

 

[DrChinese]

I think there's where the disagreement is: Not all interpretations would agree that Bob would see a 50% up rate under these counterfactual scenarios, and under these interpretations, we cannot conclude a nonlocal influence.

As you asked the question, there is no counterfactual to consider. This can be experimentally realized, so there's nothing to consider in the way of interpretations. It's a fact.

 

[DrChinese]

I didn't say it did. I only said it was a candidate for a "common cause". Nothing "predetermines" the outcomes--even the entangled state plus the measurements choices doesn't predetermine the outcomes since there is still a random element involved. But the entangled state is certainly part of what "contributes" (to use as neutral a word as possible) to the outcomes. So are the measurement choices of Alice and Bob. The entangled state is the "common" part of that, and the choices are the "non-common" part (since they can differ between Alice and Bob).

So... the entangled state is NOT the common cause (nor a candidate for same) of the quantum outcomes/expectation precisely because it does NOT factor into the equation - only Alice and Bob's choices do. We agree that it is POSSIBLE that the entangled state holds the key to the random component we see in individual trials. But mathematically, it cannot contribute because the choices of Alice and Bob account for everything we can predict accurately. Which is everything except that random element. Of course, that random element could also be contributed by the environment.

Ultimately, it is meaningless to assert the "entangled state" (how the system is prepared) accounts for the results, when clearly the one critical prediction of QM for Bell tests does not incorporate it at all. There is not one iota of a hint that the superposition itself contains any information that contributes to the outcomes.

--------

And again, to address the idea that QFT is needed to understand the phenomena of entanglement vis a vis Bell tests: it doesn't! There is no difference between QM and QFT as regards to Bell test calculations (that I am aware of). As far as anyone knows, the theoretical constraints some think are contained in QFT ("microcausality") are not mentioned in any work I am aware of by teams investigating entanglement. Nor is this mentioned in any comparisons of common quantum interpretations. I say that's because it's completely irrelevant, why else?

------------

I have quoted top teams until I am blue in the face. So if anyone else can provide a suitable counter to those quotes in a generally accepted recent paper by a top team, I'm all ears. All I ever see is self quotes... where's the beef?

Quantum nonlocality is generally accepted physics. It's high time this forum explicitly accepted same, that's our stated standard. I hope that is not too radical to be said here. Saying this does not mean we know anything about the underlying mechanism. It does not mean there is instantaneous or spooky action at a distance. It does not mean that there are signals propagating FTL. It only means that there are elements of quantum systems that when measured, do not respect classical limits for any of a number of reasons (i.e. because local realism is refuted by Bell tests).

 

[PeterDonis]

the entangled state is NOT the common cause (nor a candidate for same) of the quantum outcomes/expectation precisely because it does NOT factor into the equation - only Alice and Bob's choices do.

What equation are you talking about? The entangled state most certainly does "factor into" the QM equations that are used to predict probabilities.

the choices of Alice and Bob account for everything we can predict accurately

I don't understand how you can possibly say this given that we have to know the entangled state that was prepared in order to make accurate predictions.

it is meaningless to assert the "entangled state" (how the system is prepared) accounts for the results, when clearly the one critical prediction of QM for Bell tests does not incorporate it at all

I have no idea how you can say this given that, as above, we have to know the entangled state in order to make correct predictions.

 

[PeterDonis]

Quantum nonlocality is generally accepted physics. It's high time this forum explicitly accepted same, that's our stated standard

The issue is that you are using a term, "quantum nonlocality", whose meaning is not generally accepted: different sources use it to mean different things.

The actual experimental facts are that when we make Bell-type measurements on entangled pairs of particles, the observed correlations violate the Bell inequalities and match the predictions of standard QM. If that is what you mean by "quantum nonlocality", then yes, "quantum nonlocality" is generally accepted physics because it's an experimental fact. But other people, using "quantum nonlocality" to mean different things, might not agree that "quantum nonlocality" is generally accepted physics. That doesn't mean they disagree with you about the experimental facts or about what standard QM predicts. It just means they are using words differently.

The best procedure I know of for dealing with such situations is to taboo the problematic term, in this case "quantum nonlocality". If you mean "Bell inequality violations", say so. If someone else means something different, they can say what they mean. IIRC I've proposed a solution like this before but nobody adopted it.

 

[PeterDonis]

the theoretical constraints some think are contained in QFT ("microcausality") are not mentioned in any work I am aware of by teams investigating entanglement.

That's irrelevant unless those teams are arguing that those constraints don't apply, which AFAIK nobody is arguing.

We don't argue that phenomena within the domain of Newtonian physics don't have to obey relativistic constraints because relativity is "not mentioned" in Newtonian physics. We understand that Newtonian physics is just an approximation to relativity valid in a limited domain, and we understand that, if we're talking about "theoretical constraints", the constraints of relativity apply in the Newtonian domain just as they do everywhere else, even if those constraints aren't "mentioned" in Newtonian treatments.

The position with respect to non-relativistic QM is the same: it's an approximation to QFT valid within a limited domain, and the theoretical constraints of QFT apply even if they aren't "mentioned" in non-relativistic QM treatments.

 

[Fra]

@Fra:
d. You are incorrect about Bell's hidden variable types - there was no limitation or exclusion on them at all. He said (1964): "Let this more complete specification be effected by means of parameters λ. It is a matter of indifference in the following whether A denotes a single variable or a set, or even a set of functions, and whether the variables are discrete or continuous."

The assumption I was thinking of is the ansatz where you partition/index the sample space as per the hidden variable, and assume that the partitioning into hidden variable makes sense. This is exactly what I think constitutes an "ignorance type" mechanism. From my interpretation (abstract interacting agent view, this assumption generally makes no sense, but then I am not looking for the sort of realism bell did.)
I think of it so that the hidden variables can explain the correlation, but not the interaction. The way this is possible is that unlike ignorance HV, which in which the environment is potentially informed (due to decoherence), an isolated entangled pair is strictly forbidden to interact. It's also not possible to infere an interaction rule from such an interaction, as it would break the entanglement.

Instead the sort of HV I imagine, is from the "inside view" of one of the entangled systems, there maybe a uniqe "reality" there, but due to interactions works, this is not inferrable from the outside and thus really HIDDEN unlike the ignorance where the information just lost averaged out or lost by the observer, but it exists in the evironment.

He does provide one assumption however, but it is not on type or form: "The vital assumption is that the result B for particle 2 does not depend on the setting a, of the magnet for particle 1, nor [vice versa] A on b."

I have no problem with THIS assumption.

/Fredrik

 

[PeterDonis]

I have no problem with THIS assumption.

You realize that that assumption is a key one that leads to deriving the Bell inequalities--which are violated by actual experiments and by the predictions of QM--right?

 

[Fra]

You realize that that assumption is a key one that leads to deriving the Bell inequalities--which are violated by actual experiments and by the predictions of QM--right?

Yes.

 

[DrChinese]

The issue is that you are using a term, "quantum nonlocality", whose meaning is not generally accepted: different sources use it to mean different things.

The actual experimental facts are that when we make Bell-type measurements on entangled pairs of particles, the observed correlations violate the Bell inequalities and match the predictions of standard QM. If that is what you mean by "quantum nonlocality", then yes, "quantum nonlocality" is generally accepted physics because it's an experimental fact. But other people, using "quantum nonlocality" to mean different things, might not agree that "quantum nonlocality" is generally accepted physics. That doesn't mean they disagree with you about the experimental facts or about what standard QM predicts. It just means they are using words differently.

The best procedure I know of for dealing with such situations is to taboo the problematic term, in this case "quantum nonlocality". If you mean "Bell inequality violations", say so. If someone else means something different, they can say what they mean. IIRC I've proposed a solution like this before but nobody adopted it.

"Nonlocality" in abstract, past 12 months: 1167. I guess many sources do know how to use the word, and they expect advanced readers to know it as well (without tearing the syntax of every sentence apart as we tend to do here). Paul Kwiat et al (2022): "...the original purpose of Bell tests, providing a measurable criteria for separating local and nonlocal theories, has been largely fulfilled..."

It doesn't make sense to me to make taboo the very term that is in fact generally accepted. We don't live in a local realistic world, we live in one in which there is evidence of nonlocality. And in fact, the title of this thread references the professional literature - which question we have been trying to answer. And the answer is that the professional literature accepts the existence of quantum nonlocality (demonstrated by entanglement, as an example, and there are other examples too), and denies the existence of FTL signaling possibilities (since quantum outcomes individually exhibit randomness).

-DrC

 

[DrChinese]

Bell: "The vital assumption is that the result B for particle 2 does not depend on the setting a, of the magnet for particle 1, nor [vice versa] A on b."

I have no problem with THIS assumption.

And from the results of Bell tests, we know this assumption is false. In reality, they ARE mutually dependent. That is what I have been saying over and over. The ONLY inputs to the quantum expectation value for Bell tests are the a (Alice) and b (Bob) settings. And Bell rules out all other variables, sets, functions, interactions, preconditions, etc as contributing in any way to the statistical results. You cannot even hand create conditions where anything happens prior to Alice and Bob's measurements and get the right answer.

 

[Fra]

And from the results of Bell tests, we know this assumption is false. In reality, they ARE mutually dependent. That is what I have been saying over and over. The ONLY inputs to the quantum expectation value for Bell tests are the a (Alice) and b (Bob) settings. And Bell rules out all other variables, sets, functions, interactions, preconditions, etc as contributing in any way to the statistical results. You cannot even hand create conditions where anything happens prior to Alice and Bob's measurements and get the right answer.

You seem to be ignoring here my most important objection, ie, the assumption that it makes sense to sum over the hidden variable? You need this to prove Bell inequality. I think such a partition is unphysical in the situation. It seems the argument here is which assumptions you object to. the objection I make I rarely see, for some reason. Perhaps because its not an argument that someone looking for old typ realism is likely to come up with I think.

I am not suggesting that there is a HV the predetermines the outcome independently of hte detector settings.

/Fredrik

 

[Fra]

You cannot even hand create conditions where anything happens prior to Alice and Bob's measurements and get the right answer.

We have QM, that describes this. It's the preparatation of the entangled pair that is the condition. And my logic is that the detector is "informed" about the preparation only, NOT about the hidden variable. Also no other part of the environment can be, as that would break the isolation and entangelement. This is why the statistics at both detectors must be independent of hidden variable; yet it explains the "correlation". I find this plausible as information not at hand, should not influence the interaction; as my conjecture beeing part of my interpretation is that interactions are a clash between "expectations".

I still think and agree something is missing here (ie. I am not happy with our understanding of QM), but for me its something different, than what I think Bell had in mind.

/Fredrik

 

[PeterDonis]

The ONLY inputs to the quantum expectation value for Bell tests are the a (Alice) and b (Bob) settings.

And the entangled state that was prepared. I don't understand why you keep leaving this out.

 

[vanhees71]

@Morbert:
a. I agree, Bob would then see a 50% "up" rate rather than 100% assuming Alice's results are unknown. If measured at different angles (but Alice IS known), the results follow the usual expectation value. @PeterDonis:
b. My statements ("Either Alice's measurement casts Bob's particle into a state synchronized with Alice, or Bob's measurement casts Alice's particle into a state synchronized with Bob.") ARE symmetric (or commute), and precisely fit the facts. There are no other facts you can state about the results OTHER than what I say.

But it contradicts the very foundation of relativistic QFT, i.e., the microcausality constraint on local observables, i.e., there cannot be a mutual influence of A's and B's measurements if the "measurement events" (photon-detection events) are space-like separated.

c. The entangled state absolutely does NOT predetermine the outcomes of (all possible) measurement choices by Alice and Bob. Bell ruled that out for all possible type of hidden variables (see d. below for quotes). The best you could say is that it is responsible for the random element to the outcomes, which are otherwise unaccounted for in the quantum expectation value for matches.

Of course the entangled state doesn't predetermine any of the outcomes of the possible single-photon measurements, because the single-particle states are maximum-entropy mixed states, i.e., the single photons in the entangled two-photon states are ideally unpolarized. Nevertheless the preparation of the two-photon state as an entangled state implies the correlations as measured in all possible experiments. It's 100% (anti-)correlated if both A and B measure (or rather test for) linear polarization in the same direction.

@Fra:
d. You are incorrect about Bell's hidden variable types - there was no limitation or exclusion on them at all. He said (1964): "Let this more complete specification be effected by means of parameters λ. It is a matter of indifference in the following whether A denotes a single variable or a set, or even a set of functions, and whether the variables are discrete or continuous." He does provide one assumption however, but it is not on type or form: "The vital assumption is that the result B for particle 2 does not depend on the setting a, of the magnet for particle 1, nor [vice versa] A on b."

Separability means that the probabilities for the outcome of a joined measurement, given the value(s) of the hidden variable(s), $\lambda$, commute, and that's indeed assumed in Bell's original paper (Eq. 2):

J. S. Bell, On the Einstein-Podolsky-Rosen paradox, Physics
1, 195 (1964),
https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195

Of course the reduced probabilities, i.e., the integral/sum over $\lambda$ is not "separable".

 

[Morbert]

As you asked the question, there is no counterfactual to consider. This can be experimentally realized, so there's nothing to consider in the way of interpretations. It's a fact.

Say Alice measures the spin of her electron at some angle and observes "up". Is the following statement true:

"If bob performs the same measurement on his electron, there is a 100% chance he will observe 'up', but if Alice had not measured her particle, there would be less than 100% chance that Bob will observe 'up'"

The bit in bold is the counterfactual. I can also rephrase the question in terms of ensembles:

Given an ensemble of identically prepared electron pairs in the appropriate Bell state, Bob measures spin-x = up 50% of the time and spin-x = down 50% of the time. However, for the subensemble of experimental runs where Alice measures spin-x = up, Bob measures spin-x = up 100% of the time. The counterfactual: If, for this subensemble of experimental runs, Alice had chosen not to perform a measurement, Bob would have instead observed spin-x = up 50% of the time, not 100%

The conclusion of superluminal influence rests on counterfactual reasoning like this, but if this reasoning is not correct, then we cannot yet conclude superluminal influence.

@PeterDonis I was applying the counterfactual reasoning outlined in chapter 19 for supports like 19.12 (or 19.14) to the scenario I posed to @DrChinese to show that, under the consistent histories interpretation, no superluminal influence is implied. I will go through steps in more detail/more explicitly when I get the chance.

[edit] - Changed photons to electrons to keep my scenario physical

 

[vanhees71]

Suppose the two photons are prepared in the polarization-singlet state, then the single-photon states are given by the corresponding partial traces. The outcome is $\hat{\rho}_A=\frac{1}{2} \hat{1}_A$, $\hat{\rho}_B=\frac{1}{2} \hat{1}_B$, i.e., the photons are perfectly unpolarized. All that A and B find when measuring the linear-polarization state both in the same direction are unpolarized photons, i.e., they find with 50% chance H and with 50% chance V. Only when their measurement protocols are storing the results of this measurement (together with the accurate time stamps to be able to know, which photon pair's where prepared as an entangled state when doing their measurements) you are able to "post-select" the subenemble, where A found H, and then, comparing the measurement protocols, it will come out that, given A found H, B always found V, i.e., a 100% anti-correlation, and that's precisely what the preparation in the entangled state implies. This measurement protocol establishes a measured fact. You cannot conclude anything about something, which could have been measured, but hasn't been measured.

 

[PeterDonis]

the microcausality constraint on local observables, i.e., there cannot be a mutual influence of A's and B's measurements if the "measurement events" (photon-detection events) are space-like separated

The "microcausality constraint" is that spacelike separated measurements must commute. That doesn't rule out a "mutual influence" altogether; it just means that any such "influence" cannot depend on the order in which the measurements occur. Some might say that rules out any "influence" at all, but others might disagree.

 

[DrChinese]

And the entangled state that was prepared. I don't understand why you keep leaving this out.

A: There is no variable from the entangled state (other than I guess a selection of type of conservation rule) that is a part of the quantum expectation value. Further, Bell ruled out predetermined elements of the superposition/entanglement as being an explanation of the outcome statistics. That only leaves things that happen from the first measurement to the second measurement (and regardless of order or reference frame) as being part of the mechanism we wish to understand. That's the time during which we go from an entangled 2 particle system to 2 systems of 1 particle. We don't know what happens during this period, when both particles end up in sharply defined spin states (if that is what we are measuring).

Only Alice and Bob's settings matter, that's the math. For polarization entangled photon pairs, the relevant term is: cos^2(theta) where theta is the difference between Alice's setting and Bob's setting.