# Mathematical expression of Bell's local realism

1. Dec 2, 2005

### DrChinese

Mathematical expression of Bell's "local realism"

I have started this thread to continue a discussion with NateTG that was starting to get a bit off-topic there. I will repeat the base comment and then reply to NateTG's last comment. I would invite anyone interested to please join in!

Bell's Theorem rules out local realistic theories, as is well known. Nailing down *exactly* how Bell defines "local" and "realistic" - especially what is necessary for a proof of Bell's Theorem - is a bit more complicated. That is the discussion topic.

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The definition of a "realistic" theory is that particles have observable attributes independent of the act of observation. This is all EPR says; and that is why EPR says QM is incomplete. It is not an assertion of EPR that there are hidden variables that predetermine outcomes. It is that the outcome values themselves exist independent of a measurement.

And Bell follows that thinking completely. So if there are 2 simultaneous values for a single particle (corresponding to 2 different measurement settings), then there are 3 as well.

a, b and c are different settings to measure observables on a single particle. But such a simultaneous measurement is not possible without disturbing the system under view. So if you measure the particle and a "clone", then you might be able to get 2 values simultaneously.

In this example, we are testing the hypothesis - of Bell - that a single particle has 3 simultaneous values. I think your characterization is OK but let me repeat the experimental questions.

I. Experimental test of Bell Locality so it does NOT need to be assumed a priori:

P(Alice+ (at polarizer setting=a), Bob (any setting b)) =
P(Alice+ (at polarizer setting=a), Bob (any setting c))

and similarly (I left this out in the earlier post I think)

P(Alice- (at polarizer setting=a), Bob (any setting b)) =
P(Alice- (at polarizer setting=a), Bob (any setting c))

We are looking at the variations of the setting for Bob and how it affects things over at Alice, but are not concerned with Bob's outcome in this statement. Because this scenario exactly - word for word - maps to Bell's statement as to his locality assumption. That being that the result at Alice is independent of the setting at Bob.

The interesting thing: It just doesn't matter whether there is signal locality or not; if the particles are space-like separated or not; or if there are slower than light influences. None of these can matter in our experiment II IF the experimental result above is first proven. Therefore, there is no need to assume Bell Locality or locality of any kind. In fact, you are free to assume the opposite: that there are such effects because they just won't matter.

II. Experimental test of Bell's Inequality

This would test correlations between Alice and Bob once we have ruled out - by experiment - that the outcome at Alice is affected by the setting at Bob. So now we can see that the correlations are too strong to obey Bell's Inequality - because there is NO SIMULTANEOUS a, b and c to begin with.

Last edited: Dec 2, 2005
2. Dec 2, 2005

### DrChinese

Well, this is pretty much how I see it as well.

If a hidden variable is a hypothetical result/outcome/measurement - i.e. the output side of some black box - then I say that is an "element of reality" in EPR's terms. Since there is a black box, we are silent on the point of whether there is an internal state that exactly maps to the outcome and whether we can see inside the black box itself to see its inner workings - and then learn if it is deterministic or not. So I don't believe the elements of reality needs to extend that far, just to the point that we agree that there were "answers" (the results) to the "questions" (the set of possible measurement settings, i.e. polarizer angle settings to be specific.) You might assume it is stochastic, but it doesn't need to be to make Bell's Theorem important.

EPR says that these elements of reality (P and Q in their example, polarization in ours) exist independently of a measurement. EPR also says insists that they not be dependent on a measurement setting on distant particle: "This makes the reality of P and Q depend on the process of measurement carried out on the first system which does not disturb the second system in any way. No reasonable definition of reality could be expected to permit this." This is the locality assumption - and Bell essentially defines locality in an identical manner.

What I don't agree with is those who insist that the hidden variables are the inputs to a deterministic equation which generates the resultset. That is an extra and unnecessary step that can't be taken until we know if the elements of reality - which are observables - exist simultaneously and independent of observation.

3. Dec 2, 2005

### NateTG

As a note, please quote what you're replying to, or at least link to it, unless the two are in direct sequence. Context can be quite handy.
This is a digression from posts 15+ of the thread:

We've got a pretty clear notion of realistic:
I'm going to assume that it's actually 'all observable attributes exist independant of observation'. So let's say that, in a realistic world, for every particle, $p$ there is a hidden state $\vec{\lamda}_p$ that is sufficient to provide the result any concievable measurement on that particle. One way to describe this hidden state would be to simply list all of these observable quantities.

Note: For the remainder of this post I refer to a particular measurement e.g. $m_1$ I mean not just the act of measurement, but also the past of that measurement, including any 'hidden' state.

Now, we want to look at Signal Locality, and Bell Locality.
Signal Locality is also a fairily clear notion in some sense:

A physical model is signal local if it is impossible to have faster than light communication in it.

That means that if we have two measurements $m_1$ and $m_2$, neither in the past of the other, with possible result events $R_1$ and $R_2$ respectively we have:
$$\forall r_2 \in R_2 p(r_2)=\int_{R_1} p(r_1) p(r_2 | r_1) dr_1[/itex] and [tex]\forall r_1 \in R_1 p(r_1)=\int_{R_2} p(r_2) p(r_1 | r_2) dr_2[/itex] And Bell Locality: A physical model is Bell local if the probability of a particular measurement result can only be affected by things in that measurement's past, so, if the system is Bell Local, we also have: [tex]\forall r_1 \in R_1 \forall r_2 \in R_2 p(r_2)=p(r_2 | r_1)$$
and
$$\forall r_1 \in R_1 \forall r_2 \in R_2 p(r_1)=p(r_1 | r_2)$$

It should be clear that Bell Locality (at least as I have defined it) is a stronger condition than signal locality since it's possible to simply integrate over $R_1$ or $R_2$.

4. Dec 2, 2005

### NateTG

Let's say that we have some hidden state $\lambda \in \Lambda$ and some stochastic, but not necessarily deterministic measurement $m$ which produces a result in the range $R$.
Then we can construct a probabilty distribution on $\Lambda'=\Lambda \times R$ (where $\times$ is the cartesian product) so that $\lambda'$ will deterministically describe the result of measurement $m$.
So, provided the process is stochastic, there will be some way to describe the hidden state so that the measurement results are deterministic based on it.

5. Dec 2, 2005

### ttn

This isn't Bell Locality. You've forgotten one absolutely crucial thing: all the probabilities here need to be conditionalized on a *complete* description of the state of the particles in the past. If you leave that out, then you have to say things like this violate Bell Locality: break an arrow in half and randomly put the head and tail ends into two identical boxes, then carry the boxes to distant locations, then open one of them and see whether it has the tail or the head. Before you look, by assumption, the probability is 50/50 either way. But the conditional probability that you'll find the tail is definitely not 50% given the outcome of the similar experiment on the distant other box -- it's either 100% or 0%, depending on whether the distant box contains the head or tail. So, according to your definition above, this situation would involve a violation of Bell Locality! But Bell wasn't nearly so stupid as to think this kind of thing was problematic from the point of view of relativity. That's why he stressed that the Locality condition involved probabilities that were conditional on a complete specification of the state of the system prior to measurement. Then, in the arrow example, the probability is either 100% or 0% (for finding the tail, say) *regardless* of whether or not you *also* conditionalize on the outcome of the other distant experiment. That's the whole point. Since you've already calculated the probability conditional on all the relevant info in the past light cone of the measurement in question, adding *extra* information pertaining to space-like separated measurements doesn't change things -- that extra info is either *irrelevant* or *redundant*, and in neither case does the probability in question change when you conditionalize on it.

Of course, if the probability *does* change (when you add conditionalization on the distant outcome), even though you've already conditionalized on the complete state in the past of your measurement, that seems to imply some kind of superluminal causation. The probability for what happens *here* doesn't just depend on stuff in the past light cone of "here", but depends also on space-like separated stuff. And normally in science when we find such stochastic dependencies we think they signal a causal dependence. And such causal dependence from outside the past light cone is precisely what special relativity is supposed to have prohibited. That's why people think that there is an inconsistency between relativity and non-Bell-Local theories. And that's why this whole Bell's theorem thing is so damn interesting -- because there is *no* Bell Local theory which can explain what is observed in the experiments.

6. Dec 2, 2005

### DrChinese

1. This is good.

2. This may be a good summary of traditional Bell Locality, but I submit it is absolutely NOT what either EPR defined it as. And importantly, it is NOT necessary and sufficient for Bell's Theorem to work (although it does work). The reason this is relevant is discussed below. I would say that the mathematical formulation should exactly correspond to Bell's words:

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

Note that the location of the particles and the speed of any hypothetical influence is not even mentioned! That is because it is simply NOT needed in any way to get to our conclusion - to prove Bell's Theorem!!

3. This is where I disagree slightly. I say that we want the least assumptions possible, and therefore we want the weakest assumptions possible. So we need signal locality to be stronger than (or equal to) Bell Locality. That makes Bell's Theorem as strong as possible. So by weakening Bell Locality as I have in 2. that result is accomplished. Note that we don't need to test in a wide range of scenarios because we only need to cover enough ground with Locality to allow us to continue on with the rest of our proof. In other words, don't think of Bell as relating to locality at all. Think of it as relating to the realism requirement, which is entirely what EPR was about, and really what Bell is mostly about (except in much subsequent discussion).

After all, what happens exactly if the locality argument was not included? You couldn't rely on his conclusion because in an actual experimental situation, the sample is compromised. I.e. the results at Alice are affected when we measure at Bob. And we need them to be independent as to the statistics. So we only need enough to make the statistics valid, and for that we only need Bell's "vital assumption" and no more. We don't care if superluminal signaling can or cannot occur, or any kind of signalling or influence for that matter, as long as it does not affect our particular test.

Last edited: Dec 2, 2005
7. Dec 2, 2005

### DrChinese

That is wrong. You don't understand WHY Bell Locality exists. It ONLY exists to make the Theorem work. No more, no less.

Using an analogy:

1. I have a coin I want Alice and Bob to analyze, and I can break the coin in half and give it to each. Each will examine a different attribute of the coin. Unfortunately, their test is a destructive test and so they can each do only one test and they need at least half a coin to do a test.

When done, we will know 2 things about the coin and our conclusion should be reliable - as long as we make sure that Alice's test does not distort Bob's - and vice versa. Of course no one is saying that it should distort anything because the 2 halves of the coin are separate in this classical example.

2. But we have a very special case with a Bell test: it must use ENTANGLED particles and so we are at great risk that Alice's test will skew Bob's. But there is good news: we can rule it out by assumption (which weakens our conclusion of course). So the locality assumption is that our separate tests will yield independently valid results and will not skew the outcomes.

Of course, if I could PROVE this type of locality by experiment then I wouldn't need to assume it, would I?

8. Dec 2, 2005

### ttn

What you say initially is correct: 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.

Any information about the speed of light comes exclusively from experiment, not from the theorem. If you do the experiment such that Alice and Bob are 3*10^8 meters apart, and they randomly pick their polarizer settings one second before their particles arrive, then the "signal" (or causal effect or whatever you want to call it) that "carries" the dependence (which we know is present if the inequality is violated) must be propagating at the speed of light or faster. If you change the distance or the timing, you will set some different lower limit on the speed involved in the "nonlocal dependence."

This is all of course just why Aspect's experiment was so important. Before that, there were experiments showing that Bell's inequalities were violated, but this in no way implicated relativistic causality, since the kind of causal dependence one needed to postulate to account for the experimental results moved with a speed that was well less than c. Aspect's experiment finally got a long enough distance between Alice and Bob and a brief enough switch timing, that the "nonlocal action at a distance" has to be happening faster than signals propagating subluminally could possibly explain.

I can't really follow this, but it sounds like you're saying you can derive a Bell type inequality from a weaker locality assumption (weaker than Bell Locality, i.e., something like signal locality). That is of course false, since we already know of at least two signal local theories which correctly predict violations of Bell's inequalities.

Yeah, neither Einstein nor Bell were concerned with locality at all. EPR and Bell's theorem have nothing whatever to do with locality. :rofl:

9. Dec 2, 2005

### ttn

I don't understand the analogy. What are they measuring about the coin?

Um, hello, that is the whole freaking point. If Alice's test skews Bob's -- i.e., the fact that Alice performs an experiment on her particle *changes* the state of Bob's particle and/or the outcome of Bob's experiment -- that is a nonlocal action at a distance. It's exactly the kind of thing that, if really happening, would be a complete and total shock from the point of view of relativity, which is usually taken to prohibit such action at a distance.

Huh? The whole point of Bell's theorem is that it gives you an empirical test for whether Alice's experiment skews Bob's. Why would you want to just "rule it out by assumption"? Do the freaking experiment and find out whether or not the one skews the other. What you find is: it does!

I have no idea what you are talking about.

I'm starting to think you have no interest in really understanding this topic. Nothing you say makes any sense. You go on and on and on about EPR and Bell and Locality and Realism and all sorts of things, but you never get any closer to actually understanding anything. It's starting to look like you have a vested interest in staying confused -- and hence starting to make me think this discussion is a complete and total waste of my time.

10. Dec 2, 2005

### NateTG

I got into trouble because I was (foolishly) trying to shoehorn signal to make it a weaker condition than Bell Locality.
Here's a second attempt:
Let's say we have some environment that can be accurately modeled as a stochastic process.
Let's also say that we have a disturbance principle so we won't deal with anything that has any observations in its past. (There is, of course, the issue that a changed measurement setting qualifies as an event for just about any reasonable notion of observed event, but I'll ignore that for now.)
Then the environment is Bell Local if, for any two events $e_1$ and $e_2$, neither of which is in the other's past, and, which have the common past $h_{1,2}$ (note that this past includes any unobserved events or other hidden state) we have
that
$$p(e_1 | h_{1,2}) = p(e_1 | e_2 \rm{and } h_{1,2})$$
That is to say, the probability of $e_1$ and $e_2$ occuring for any particular common history is equal to the probability of just $e_1$ occuring for that common history.
And, if the environment is signal local if, for any two events $e_1$ and $e_2$, neither of which is in the other's past, and, which have the common past observations $o_{1,2}$ we have
$$p(e_1 | o_{1,2}) = p(e_1 | e_2 \rm{and } o_{1,2})$$
Since, we're only considering events with empty observation histories, that reduces to
$$p(e_1)=p(e_1 | e_2)$$
And, now we can integrate over the possible $h_{1,2}$ to show that signal locality is indeed a weaker condition than Bell Locality, provided the measurement settings are not 'observable events'.
Now, it's unclear whether Bohmian Mechanics should be considered Bell Local because the notion of past is rendered ambiguous by the manifestly non-local aspects of the theory. However, signal locality is (more or less by definition) restricted to considering the observations in the past light-cone so that is still well-defined in Bohmian Mechanics.

11. Dec 2, 2005

### NateTG

Slighlty off-topic, but since we're talking about Bell's theorem, I have the following question:

What is the justification for the assumption that the detector settings are assigned independantly of the spin-state of the particles?

12. Dec 2, 2005

### DrChinese

1. I said they were entangled. But apparently you do not understand that the entanglement is used for Bell tests because the entangled particles essentially have identical STATES (yes, I know, they are usually orthogonal) - and NOT because testing one affects the other. We must rule that out or our results will be skewed.

2. Apparently you keeping missing Bell's words in his paper, which specify it is an assumption:

The vital assumption is that the result B for particle 2 does not depend on the setting a, of the magnet of particle 1, nor A on b.

My point is that this assumption is no longer necessary because since Bell wrote his paper, this particular element has been demonstrated experimentally. This is NOT because the CHSH inequality was violated, because that is a test of a Bell Inequality. I am talking about the assumption above, which is always tested during calibration of a Bell test. The setting at Alice is held constant, and the setting at Bob is varied. (This is done to locate the maximum correlation setting, but it also proves our assumption experimentally. The same calibration is also used to prove that there is entanglement - which is an experimental requirement and not a theoretical one.) There is no change in the outcomes at Alice. Therefore Bell Locality is proven by experiment.

And the result is that there IS NO SKEWING; and if there were any skewing, you could send a message that way.

As everyone knows, the only thing that "changes" is the correlation percentage. A simple reference to the definition above will convince you that correlation is not part of it. The correlation ONLY relates to the Inequalities which is not part of the assumption above.

13. Dec 2, 2005

### DrChinese

Ouch, that stings. :rofl:

I am nearing completion of a revised proof of the Theorem which should be short and sweet. Hindsight is great, especially when you know the answer beforehand. It is a slightly more formal proof than here but uses the same basic logic. So see for yourself - I derived this particular formulation myself and didn't copy anyone else's work.

(So I guess it might be hard to say I am parroting someone else's words... In fact, if you google "Bell's Theorem Negative Probabilities" you won't find another actual derivation of it other than mine - for the angle settings I present the expectation value (for the A=B<>C cases directly) is -.1036.)

No, I don't really think I need to justify my understanding of Bell's Theorem to you. I have provided quotes, formulae, specifics, references, etc. I am here to learn and I hope you are too. I enjoy my participation here and I hope others do as well.

I prefer to stay focused on the substance of the discussion. I think you have plenty to offer to this discussion; perhaps my points are not clear to you and I can do a better job of expressing myself. So even if you are a bit acidic, I value your contributions.

Last edited: Dec 2, 2005
14. Dec 2, 2005

### DrChinese

Do you mean: Alice and Bob's choices of detector settings?

15. Dec 2, 2005

### NateTG

Yes, that's what I mean.

16. Dec 2, 2005

### ttn

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.

You are again simply confusing Bell Locality with signal locality.

I don't know how to say it any clearer than I've said it before, so I'm giving up trying to explain this to you. But let me just say this: you are not understanding Bell yet, and I would urge you to get ahold of "Speakable and Unspeakable" and read through all of his papers. I think you are being thrown by focusing exclusively on his original paper. But he makes things much, much clearer in his later writings. See in particular "La Nouvelle Cuisine", an absolutely brilliant piece of writing that is the last chapter in the 2nd edition of "Speakable..." It will overturn all your confusions if anything will.

Now you're just defining "skewing" as signal locality. Yes, if there were that kind of skewing, you could send a message. Duh. 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. OQM and BM are two theories with just these properties -- they're Bell NonLocal but still signal local. There's non-local action at a distance going on "behind the scenes" according to both theories, but, alas, it cannot be used to transmit messages (in the one case because of ineliminable uncertainty in the initial conditions, and in the other because of the inherent randomness associated with the collapse dynamics).

17. Dec 2, 2005

### DrChinese

I can't say for sure, but I don't think it really can be justified completely. So it would need to be assumed. Maybe that's just a Many World branch...

18. Dec 3, 2005

### DrChinese

You are focused on the phenomenon of entanglement and trying to explain that. Entanglement is a tool to gain knowledge, and we use it to prove Bell's Theorem as follows: You do a direct test on Alice. You can only do one such test and then Alice is disturbed. But then you do a different test on Bob. That give you indirect knowledge about Alice. It's very simple.

(Of course, we know that quantum mechanics limits the validity of this information - but that is the whole point. Bell's Theorem does not assume that Quantum Mechanics is true! Quantum mechanics is not a realistic theory so of course this concept does not apply.)

19. Dec 4, 2005

### DrChinese

Bell's Inequality without Locality

What is Bell Realism?
===============

The definition of realism used in Bell’s paper is tied back to the ideas originally put forth in EPR [2] – after all, the paper was titled “On the Einstein Podolsky Rosen Paradox”. The definition is always tied to observable quantities as these are objectively verifiable and therefore within the realm of science. Of course, one might presume that there is a deeper level too, but we do not need this for our definition.

a. Realism according to EPR: EPR asserts that there must be simultaneous reality – and therefore definite values - for non-commuting operators and by extension all physical operators.[2]
b. Realism according to Einstein: “I think that a particle must have a separate reality independent of the measurements. That is: an electron has spin, location and so forth even when it is not being measured. I like to think that the moon is there even if I am not looking at it."
c. Realism according to Bell: “It follows that c is another unit vector [in addition to observables a and b]…”. [1]
d. Realism of a particle as used here: A particle has simultaneous definite values for any and all observables.

Here, we define “realism” in specific mathematical form for photons:

For any three angle settings a, b and c for an individual photon, measured singly or as a statistical ensemble via a polarizer which yields an observable result labeled as + or – arbitrarily, where p() represents the observable probability of the stated outcomes:

Rule 1 Assumption:
1 >= p(a+, b+, c+) >= 0
(..and similar for all permutations of the above.)

Rule 2 Assumption:
p(a+) = p(a+, b+) + p(a+, b-)
= p(a+, b+, c+) + p(a+, b-, c+) + p(a+, b+, c-) + p(a+, b-, c-)
(…and similar for all permutations of the above.)

So our two assumptions are: the likelihood of any outcome is between 0 and 1 inclusive; and any outcome is the sum of the + and – outcomes permutations with an additional setting variable included – even if they were not observed. These innocent assumptions form our definition of realism.

We will also define 2 helper functions, corr() and noncorr(), which stand for correlation and non-correlation, which will simplify the arithmetic, such that:

Rule 3 Definition:
Where we assume there is a sample of suitable size:

corr(a, b) = p(a+, b+) + p(a-, b-)
(…and similar for all permutations of the above.)

noncorr(a, b) = p(a+, b-) + p(a-, b+)
(…and similar for all permutations of the above.)

Derivation of Bell’s Inequality
=====================
Using these rules, we derive as follows, beginning with the value of 2 specific outcome permutations (out of 8 possible when there are 3 settings):

(1) p(a+, b+, c-) + p(a-, b-, c+)

Double and divide by 2:
= ( (p(a+, b+, c-) + p(a-, b-, c+)) + (p(a+, b+, c-) + p(a-, b-, c+)) ) / 2

Add and subtract the same value to the numerator:
= ( (p(a+, b+, c-) + p(a-, b-, c+)) + (p(a+, b+, c-) + p(a-, b-, c+)) +
(p(a+, b-, c-) + p(a-, b+, c+)) + (p(a+, b+, c+) + p(a-, b-, c-)) –
(p(a+, b-, c-) + p(a-, b+, c+)) + (p(a+, b+, c+) + p(a-, b-, c-)) ) / 2

Rearrange terms:
= ( (p(a+, b+, c+) + p(a+, b+, c-) + p(a-, b-, c+) + p(a-, b-, c-)) +
(p(a+, b+, c-) + p(a+, b-, c-) + p(a-, b+, c+) + p(a-, b-, c+)) –
(p(a+, b+, c+) + p(a-, b+, c+) + p(a+, b-, c-) + p(a-, b-, c-)) ) / 2

Simplify by using Rule 2 assumption substitutions:
= ( (p(a+, b+) + p(a-, b-)) +
((p(a+, c-) + p(a-, c+)) –
((p(b+, c+) + p(b-, c-)) ) / 2

Simplify by using Rule 3 substitutions:
(2) = (corr(a,b) + noncorr(a,c) – corr(b,c) ) / 2

Recalling that by our Rule 1 assumption:
p(a+, b+, c-) >= 0 and
p(a-, b-, c+) >= 0 and therefore also
p(a+, b+, c-) + p(a-, b-, c+) >= 0

By equivalence of (1) and (2), the last becomes:
corr(a,b) + noncorrc(a,c) – corr(b,c) ) / 2 >= 0

To be explicit that we are talking about one photon, which we will call Alice:
(3) corr(Alice.a, Alice.b) + noncorr(Alice.a, Alice.c) – corr(Alice.b, Alice.c) ) / 2 >= 0

Result (3) is a deduction from assuming simultaneous reality to Alice’s a, b and c polarization observables. This is a Bell Inequality, and makes reference only to the internal relationship of 3 hypothetical simultaneous observables. This is an uncontroversial requirement of any “realistic” theory.

References:
[1] J.S. Bell: "On the Einstein Podolsky Rosen paradox" Physics 1 #3, 195 (1964).
[2] A. Einstein, B. Podolsky, N. Rosen: "Can quantum-mechanical description of physical reality be considered complete?" Physical Review 41, 777 (15 May 1935).

Last edited: Dec 4, 2005
20. Dec 5, 2005

### Tez

I'd rather not read through this whole argument - I just have a question for Dr. Chinese. Would you call the so-called "PR boxes" nonlocal? (See e.g. http://arxiv.org/abs/quant-ph/0506180 ) for an intro to them. They are basically a pair of hypothetical magic boxes (used in quantum information to help quantify nonlocal resources) which are "maximally nonlocal" but still do not allow for signalling. Normally they imagined as a pair of boxes into which Alice can feed into her box a 0 or a 1, Bob can feed into his box a 0 or a 1 and the outputs satisfy that when both parties input a 1 their outputs are different (i.e. 01 or 10, though which of these two cases is chosen randomly), but in the other three cases their outputs are the same (00, 11 - again each chosen with probability 1/2). (See Eq. 2 in the above paper). Its unclear to me whether you would call such boxes local or nonlocal, but they certainly don't allow signalling - since locally each party sees a 0 or 1 output with probability 1/2 regardless of what (they or) the other person does.

I think its useful to have a completely operational definition of what one calls local/nonlocal - i.e. theory and philosophy independent, just a definition in terms of the statistics of measurement outcomes. Thus my question - If we had such a pair of boxes would characterise them operationally as local or nonlocal? I suspect you might call them local, which is ok, but then its clear your definition is incompatible with the more common usage.