Mathematical expression of Bell's local realism

[DrChinese]

I still think you need to read La Nouvelle Cuisine. Is there something preventing you from doing this? Or maybe you read it and didn't find it clarifying?

I would purchase & enjoy it if it were at Barnes & Noble, which it's not. Or if someone gave me a copy for Christmas, which probably won't happen. So I guess I will have to get from Amazon...

 

[DrChinese]

The point is, there are a ton of subtly different things that might plausibly be thought of as a kind of parameter independence. Being clear about which is which and which is justified/relevant in a given context, however, is crucial to this discussion.

I agree. And I have tried to discuss this point. I think the context is: what does it take to test Bell's Inequality.

 

[Sherlock]

But the point is that Bell Locality is a stronger condition that signal locality. Bell Locality can be violated, even by a theory that still respects signal locality.
...
... details like the speed of light don't appear in Bell's locality condition. It is just a condition saying that one thing is independent of another. Then, from that condition, the inequality follows -- so it is an indirect test of whether or not the one thing depends on the other. If it doesn't so depend, the inequality should be respected by the experimental results. If there is some dependence on the distant setting/outcome, the inequality will be violated.
...
If entanglement simply meant that the initial spins of the particles were correlated (such that there was, later, no influence of Alice's measurement on Bob's outcome or vice versa) Bell's inequalities would *not* be violated. That's the whole point here. The violation of Bell inequalities proves that testing one affects the other.
The correlations can *not* be accounted for in terms of pre-correlated properties which locally determine the outcomes.

So, what does it mean for the correlations to violate Bell locality but not signal locality? It means that the measurements at A and B are related to each other, but not causally. This is what quantum non-locality means, an acausal relationship between A and B, and because the relationship is acausal there is no conflict with special relativity.

Bell locality is a more encompassing condition than signal locality. Bell locality encompasses correlations as well as causal relationships. So, a violation of Bell locality does not necessarily discern the presence of a causal relationship between A and B.

As DrChinese has mentioned, it's been shown experimentally that the results at one end don't depend on the settings at the other end.

We know that the statistics at A and B aren't independent of each other, because the observations at A and B aren't independent of each other. What is sampled at one end depends on what is sampled at the other end. "Testing one affects the other" in that a detection at one end gates open the coincidence circuitry, and therefore determines what is selected for sampling at the other end.

The sampling method is of primary importance, because it's assumed that the spins of the photons are correlated upon creation of the entangled state, which occurs during emission, and exists prior to filtration by the polarizers. And it's not just the spins, and therefore the polarizations, of the incident disturbances associated with suitably paired detection attributes that are related. The phase relations and relative amplitudes of the emitted disturbances also determine the strength of the correlations.

Now, I don't see in any of this where it is necessary to postulate the existence of superluminal propagations to account for the correlations. QM doesn't do that, and it accounts (approximately, quantitatively) for the correlations. I have to conclude that it hasn't been shown that non-locality is a fact of nature.

Via EPR we can think of local hidden variables as, in some sense, existing. However, because of limitations on what can be experimentally determined, which limit the content of any, explicitly local, hidden variable formulation, the development of an experimentally viable local hidden variable theory which matches or exceeds the accuracy of qm is disallowed. Maybe it's worth noting again that it is local hidden variable theories, not locality, and not non-local hidden variable theories which are ruled out. But since locality isn't ruled out, and therefore the assumption of locality is retained, then developing an explicitly non-local hidden variable theory wouldn't make much sense (unless you just wanted to show that some, any, hidden variable formulation was possible).

 

[Sherlock]

Yes, in the paper you have in mind, Bell says what you say he says -- that the outcomes on each side can't depend on the distant setting. But this is *not* what is technically known in the literature as PI (parameter independence). You have to be extremely careful about what all these things mean precisely, mathematically -- in particular, you have to be careful about what the probabilities involved are conditionalized on. To say something like "the outcome here doesn't depend on the parameter there" is indeed a kind of "parameter independence". But it simply is *not* the same as the "PI" satisfying "PI + OI = Bell Locality".

The "PI satisfying PI + OI = Bell Locality" is,
P(A|a) = P(A|a,b) and P(B|b) = P(B|b,a), isn't it?
If so, then this is just a shorthand way of saying that the "outcomes on each side can't depend on the distant setting", isn't it?
If this isn't "the same as the PI satisfying PI + OI = Bell Locality", then what is the PI that satisfies PI + OI = Bell Locality?

These things are very subtle and you can't just plow through them at light speed. For example, you say: "If Alice's measurement apparatus is not affecting Bob's results..." Well what do you mean exactly that Bob's results aren't affected? That in a given run of the experiment (i.e., for a given particle pair) the outcome is the same as it would otherwise have been? That the probabilities for different outcomes are the same as they would have been? Or that the long-time-average of Bob's results over many runs are the same as they would have been? Or that the correlation coefficient for Alice's and Bob's outcomes are independent of Alice's settings? etc. The point is, there are a ton of subtly different things that might plausibly be thought of as a kind of parameter independence. Being clear about which is which and which is justified/relevant in a given context, however, is crucial to this discussion.

I think the definition I wrote above is ok. But if not, let me know.
Parameter independence is the part of the Bell Locality condition that isn't violated. Parameter independence means that the rate of detection, A, does not vary with the setting, b, of the distant polarizer, and that the rate of detection, B, does not vary with the setting, a, of the distant polarizer.
What is violated is outcome independence, which is equivalent to statistical independence. A and B are not statistically independent. So, the Bell Locality condition (which is not, strictly speaking, a locality condition) is violated.
The idea that there is a common cause explanation for the correlations, rather than a non-local causal link between space-like separated filtration and detection processes and events, is further supported by the joint detection schemes and experimental protocols of the tests of Bell inequalities.

 

[ttn]

The "PI satisfying PI + OI = Bell Locality" is,
P(A|a) = P(A|a,b) and P(B|b) = P(B|b,a), isn't it?
If so, then this is just a shorthand way of saying that the "outcomes on each side can't depend on the distant setting", isn't it?
If this isn't "the same as the PI satisfying PI + OI = Bell Locality", then what is the PI that satisfies PI + OI = Bell Locality?

You *have* to also conditionalize on a complete specification of the state of the system prior to measurements. Otherwise, it just isn't Bell Locality you're talking about. P(A|a,b,B,L) = P(A|a,L). That's Bell Locality. (L here is a complete specification of the state.)

Note that this is *not* something you can just go into a lab and measure. How would you ever know that you had conditionalized on a *complete* specification of the state? That's why I've said that this is a condition that only (directly) applies to *theories* -- because a given theory gives an account of what a complete specification consists of.

Parameter independence is the part of the Bell Locality condition that isn't violated.

I don't know how you could possibly know that. The inequality is derived from Bell Locality (both OI and PI) and is violated. Which is to blame? There's no way to answer. All one can say is that Bell Locality is violated.

So, the Bell Locality condition (which is not, strictly speaking, a locality condition) is violated.

Yes, it is. (Bell Locality is a locality condition, I mean. If you leave out the idea of conditionalizing the probabilities on L, then I can see why you'd think it's merely a statistical correlation condition. But once you have specified that complete state description, the *additional* dependence of a probability on some spacelike separated event implies a nonlocal causation.)

The idea that there is a common cause explanation for the correlations, rather than a non-local causal link between space-like separated filtration and detection processes and events, is further supported by the joint detection schemes and experimental protocols of the tests of Bell inequalities.

No. You've been repeating this mantra for years, but it's just wrong. You need to rethink it.

 

[Sherlock]

You *have* to also conditionalize on a complete specification of the state of the system prior to measurements. Otherwise, it just isn't Bell Locality you're talking about. P(A|a,b,B,L) = P(A|a,L). That's Bell Locality. (L here is a complete specification of the state.)

Afaik, parameter independence doesn't mean something different as a component of Bell's Locality condition, P(A|a,L) = P(A|a,b,B,L), than that the outcome, A, doesn't depend on the distant setting, b.

But if it does, then I would like to learn what it means.

Note that this is *not* something you can just go into a lab and measure. How would you ever know that you had conditionalized on a *complete* specification of the state?

Exactly. How would you ever know? So, we ask the physically meaningful question: does P(A) -- ie., the detection rate, A --vary with the distant setting, b, or P(B) with a? The QM answer is no, and the QM answer has been experimentally corroborated.

Parameter independence is the part of the Bell Locality condition that isn't violated.

I don't know how you could possibly know that. The inequality is derived from Bell Locality (both OI and PI) and is violated. Which is to blame? There's no way to answer. All one can say is that Bell Locality is violated.

I believe that PI isn't violated because of the physical evidence that it isn't, and also because the theory that does make correct predictions wrt Bell tests, QM, doesn't violate PI --- and, afaik, there isn't any physical evidence to suggest that PI is violated.

On the other hand, we know that OI is violated (A and B are not statistically independent) because the sampling method imposes an observational dependency. (The sampling method is based on the assumption, which is part of the experimental design of all Bell tests that I'm familiar with, that measurable properties of the incident disturbances are related due to common cause or common interaction prior to filtration and/or detection.)

So, I think more can be said than just that the Bell Locality condition is violated. There are pretty good indications of which part of it is violated.

If you leave out the idea of conditionalizing the probabilities on L, then I can see why you'd think it's merely a statistical correlation condition. But once you have specified that complete state description, the *additional* dependence of a probability on some spacelike separated event implies a nonlocal causation.

The more complete state description promised by L only works if nature is non-local. But, there's no reason to believe that nature is non-local.

There are two classes of spacelike separated events involved in Bell tests, polarizer/analyzer settings and detector registrations. The setups and results have been analysed enough to show, pretty convincingly I think, that parameter independence is not violated and outcome independence is violated --- and from this we can't conclude that non-locality is a fact of nature.

The idea that there is a common cause explanation for the correlations, rather than a non-local causal link between space-like separated filtration and detection processes and events, is further supported by the joint detection schemes and experimental protocols of the tests of Bell inequalities.

No. You've been repeating this mantra for years, but it's just wrong. You need to rethink it.

The experimental protocols reveal a lot. They're what tell you that OI is violated, that PI isn't violated, and that Bell tests are not a test of locality vs. non-locality, but rather are only testing the viability of realistic or hidden variable theories of quantum phenomena. The conclusions (so far) are that nature is local, and that hidden variable theories of nature's quantum processes are ruled out --- and while non-local hidden variable theories can be constructed which are quantitatively viable, they represent a departure from the conceptual direction that the extant experimental evidence indicates should be taken.

 

[ttn]

Afaik, parameter independence doesn't mean something different as a component of Bell's Locality condition, P(A|a,L) = P(A|a,b,B,L), than that the outcome, A, doesn't depend on the distant setting, b.
But if it does, then I would like to learn what it means.

Yes, it means the (probability of the) outcome on one side doesn't depend on which measurement is performed on the other side -- *given* the pre-measurement complete state description *and* the outcome on the far side.

Exactly. How would you ever know?

If you were a mindless drone in a lab doing the experiment, you *wouldn't* know. That's why I keep saying that violation of Bell Locality (or PI) isn't something that you can directly test in a lab. You can only test it subject to an *assumption* about the completeness of one's state description.

This is all just a different perspective on the importance of recognizing that Bell's proof of nonlocality has two parts. The first part is essentially the EPR argument: Bell Locality *requires* the existence of a certain type of local hidden variables which determine the outcomes of the measurements on each side. The second part of the argument is then Bell's Theorem: these HV's entail something which can be directly tested empirically, and is found to be false.

So, we ask the physically meaningful question: does P(A) -- ie., the detection rate, A --vary with the distant setting, b, or P(B) with a?

You can certainly ask this question, but it isn't the same as the original question (namely: is Parameter Independence true). Just because the original question is unanswerable in the way you hoped, doesn't mean that the question you ask instead is the same question you started with.

I believe that PI isn't violated because of the physical evidence that it isn't,

What evidence is that exactly?

and also because the theory that does make correct predictions wrt Bell tests, QM, doesn't violate PI

But other theories which *also* make correct predictions *do* violate PI! So you definitely cannot say that the experimental results show that PI isn't violated!

On the other hand, we know that OI is violated (A and B are not statistically independent) because the sampling method imposes an observational dependency.

Either what you say is false, or you've switched to your alternate definition of OI. I really don't know, and don't care, which. In either case, it simply is not true that "we know that OI is violated."

What we know for sure is violated is Bell Locality, because of the combination of the two parts of Bell's argument (EPR and Bell's thm). Whether OI or PI is to blame is not known -- and, as Maudlin has very convincingly argued, not even a meaningful question.

So, I think more can be said than just that the Bell Locality condition is violated. There are pretty good indications of which part of it is violated.

As I've said, I don't agree. But even if this were right, to say that one "part" is violated is to confess that Bell Locality *is violated* -- which seems to be something you deny -- e.g., here:

But, there's no reason to believe that nature is non-local.

 

[Sherlock]

Yes, it [parameter independence wrt the Bell Locality condition] means the (probability of the) outcome on one side doesn't depend on which measurement is performed on the other side -- *given* the pre-measurement complete state description *and* the outcome on the far side.

This isn't quite clear to me. Are you saying that how, or whether, the rate of detection, A, varies with the distant polarizer setting, b_hat, depends on whether a pre-measurement complete state description accompanies the test *and* on the rate of detection, B --- or what?

If you were a mindless drone in a lab doing the experiment, you *wouldn't* know [that you had conditionalized on a *complete* specification of the state].

And if you do, or don't, "conditionalize on a *complete* specification of the state" --- how does that affect the detection rate? Why should it, for that matter?

That's why I keep saying that violation of Bell Locality (or PI) isn't something that you can directly test in a lab. You can only test it subject to an *assumption* about the completeness of one's state description.

Why should an assumption about the completeness of one's state description affect the results? In fact, the results, A, don't vary with b_hat, or with a_hat either. Same with B. The results, A or B, don't vary at all, ever, in Bell tests. The rate of detection at A is the same as B in every run, and doesn't vary from run to run.

This is all just a different perspective on the importance of recognizing that Bell's proof of nonlocality has two parts. The first part is essentially the EPR argument: Bell Locality *requires* the existence of a certain type of local hidden variables which determine the outcomes of the measurements on each side. The second part of the argument is then Bell's Theorem: these HV's entail something which can be directly tested empirically, and is found to be false.

What is being directly tested are the predictions of an LHV theory. There seems to be some disagreement as to whether it is the L part or the HV part, or both, that is responsible for the discrepancy between the LHV predictions and the experimental results.


The L part is usually identified as the Bell Locality condition which is further analysed into PI and OI. Neither PI nor OI are violated because the detection rates, A and B, at either end of the experimental setup remain constant.


Which leaves the HV part as the most likely, if not the usual, suspect in the non-viability of LHV theories.

But other theories which *also* make correct predictions *do* violate PI! So you definitely cannot say that the experimental results show that PI isn't violated!

Whether or not PI is violated depends on whether the observed detection rate varies with the polarizer setting, doesn't it? So if the detection rate doesn't vary with the polarizer setting, then how can a theory that says the detection rate does vary with the polarizer setting be making correct predictions?

... it simply is not true that "we know that OI is violated."

I agree. After thinking about this a bit, it became clear to me that all we can say is that, as far as can be ascertained, OI isn't violated.

What we know for sure is violated is Bell Locality, ...

What we know for sure is that, as far as can be ascertained, neither PI nor OI is violated experimentally. So, Bell Locality is not violated.


Wrt individual results and settings, there are no interesting correlations.

But if we impose a certain structure on the apparently causally unrelated individual events, then a correlational pattern is revealed. The rate of coincidental detection, AB, varies with the angular difference between the polarizer settings, Theta, as cos^2 Theta. How can this be, if the two sides of the setup are causally isolated from each other?


One answer is that the components (AB, Theta, and the disturbances that are jointly analyzed by the polarizers during a particular coincidence interval) of the structure that has been imposed, the combined context, all have some common cause. Individual A and B results are paired via their occurance during the same coincidence interval, and the combined AB result is then associated with the Theta in effect during that interval. Coincidence intervals are determined wrt the presumed elapsed time between the creation of an entangled pair of incident disturbances and a detection event at one end or the other which initiates the coincidence circuitry. In the case of eg. the Aspect et al. experiments, the presumed common cause of paired photons, and of their entanglement, is that they were emitted from the same atom -- in which case the observation of predictable correlation patterns between their joint detection and a common measurement operator doesn't seem too surprising --- even if somewhat resistant to a detailed explanation of exactly how it happens.

 

[ttn]

This isn't quite clear to me. Are you saying that how, or whether, the rate of detection, A, varies with the distant polarizer setting, b_hat, depends on whether a pre-measurement complete state description accompanies the test *and* on the rate of detection, B --- or what?

Look, the condition is what it is. It's just what I said before. No, this doesn't necessarily imply anything about the rates of detection. Those rates are some kind of averages over the probabilities involved in Bell Locality. The whole point here is that you can't just test Bell Locality by looking at emprical rates -- not without some subsidiary assumptions provided by a theory (which will tell you what a complete specification of the state consists of, and will tell you something about what initial state(s) are being produced by the setup procedure).

And if you do, or don't, "conditionalize on a *complete* specification of the state" --- how does that affect the detection rate? Why should it, for that matter?

The conditionalization isn't something that affects the detection rates. The probabilities we're talking about are not detection rates, not directly. They are the probabilities for some events predicted by a theory. Those can be related to detection rates with subsidiary assumptions.

Why should an assumption about the completeness of one's state description affect the results?

Of course it doesn't affect the results. But it affects whether the probabilistic dependence in question implies a causal influence. No causal influence is implied just because the probability of some event changes when you conditionalize on some other bit of space-like separated info. But if that probability changes even though one has already specified the contents of the past light cone of the event in question, there is clearly some sort of causal influence involved. That's why Bell thought Bell Locality was the proper mathematical test for "relativistic causality." And it's a test that can be applied directly to *theories* -- not to empirical correlation rates.

What is being directly tested are the predictions of an LHV theory. There seems to be some disagreement as to whether it is the L part or the HV part, or both, that is responsible for the discrepancy between the LHV predictions and the experimental results.
The L part is usually identified as the Bell Locality condition which is further analysed into PI and OI. Neither PI nor OI are violated because the detection rates, A and B, at either end of the experimental setup remain constant.

Wrong.

Which leaves the HV part as the most likely, if not the usual, suspect in the non-viability of LHV theories.

Except that L alone entails HV. So you can't "save" L by "blaming" HV for the violation of Bell's inequality. That's logic 101.

Whether or not PI is violated depends on whether the observed detection rate varies with the polarizer setting, doesn't it?

NO. PI is not simply the claim that the rate on one side doesn't depend on the distant setting.

What we know for sure is that, as far as can be ascertained, neither PI nor OI is violated experimentally. So, Bell Locality is not violated.

But if Bell Locality is true, then there must exist local hidden variables. But there can't exist local hidden variables, because there existence entails Bell's inequality, which is violated.

 

[Sherlock]

Look, the condition is what it is. It's just what I said before. No, this doesn't necessarily imply anything about the rates of detection. Those rates are some kind of averages over the probabilities involved in Bell Locality. The whole point here is that you can't just test Bell Locality by looking at emprical rates -- not without some subsidiary assumptions provided by a theory (which will tell you what a complete specification of the state consists of, and will tell you something about what initial state(s) are being produced by the setup procedure).
The conditionalization isn't something that affects the detection rates. The probabilities we're talking about are not detection rates, not directly. They are the probabilities for some events predicted by a theory. Those can be related to detection rates with subsidiary assumptions.

Ok, ttn ... I suppose I should reread your paper on this stuff, as well as the papers you reference (including Bell's La Nouvelle Cuisine). The comments being traded recently in this thread are not giving me a better understanding of the issues involved -- all of which are related to the larger question of what can be said about theoretical formulations and, especially, nature from experimental tests of Bell inequalities ... but I don't want to divert this thread from its specific program any longer.

Anyway I have some questions for DrChinese, which I want to post later ... and will also start a couple of threads on what I'm not clear about wrt the analysis (including yours) and meaning of Bell's formulation. But first, some reading and ... thinking. Thanks for your comments and Happy Holidays.