An abstract long-distance correlation experiment

[A. Neumaier]

Any outcome predicted by quantum mechanics gets W=0, and any outcome that contradicts a quantum mechanics prediction gets W=1.

If you do this your concept loses contact with those mentioned in the last sentence of your post. But this thread arose through discussions with proponents of the widespread view that quantum mechanics has to be intrinsically weird; so it is appropriate to use their notion of weirdness.

 

[Ken G]

That's kind of my point-- how weird we view the situation depends more on our personal relationship with the concept of weird than on the experimental outcomes. There are experiments that surprise us, are they all weird? Most people were surprised that there was length contraction and time dilation, are they as weird as quantum mechanics? Looking back into history, most people were surprised that the laws of motion can only tell you accelerations, you need to know the current velocity to get the motion going forward, so is that as weird as quantum mechanics? I'd say that all of physics was pretty surprising at one point or another, so if we define "weird" to mean "surprising to anyone", then all of physics is equally weird, and if we define "weird" as "surprising to me", then what is weird is a function of the knowledge of that person, not anything in the physics itself.

 

[A. Neumaier]

if we define "weird" as "surprising to me", then what is weird is a function of the knowledge of that person,

This is what I am investigating in this second stage, trying to give it at least some precision so that one can afterwards make some objective statements about the models of the subjective degree of weirdness.

 

[ddd123]

There are experiments that surprise us, are they all weird? Most people were surprised that there was length contraction and time dilation, are they as weird as quantum mechanics? Looking back into history, most people were surprised that the laws of motion can only tell you accelerations, you need to know the current velocity to get the motion going forward, so is that as weird as quantum mechanics? I'd say that all of physics was pretty surprising at one point or another, so if we define "weird" to mean "surprising to anyone", then all of physics is equally weird, and if we define "weird" as "surprising to me", then what is weird is a function of the knowledge of that person, not anything in the physics itself.

The main difference between QM and all your other examples is that, once, say, SR was established, the debate virtually stopped and all physicists agreed about length contraction from then onwards. QM is the only case in which there is no agreement: foundations is a thorny subject with a lot of, often colorful, disagreements, even some 100 years after QM started being investigated.

 

[jimgraber]

OK, let me suggest two candidates for degree of weirdness.

Candidate number one is the degree of entanglement. I believe there are serious papers in the literature tdefining a degree of entanglement. I think they should be transformable into your form fairly readily.Candidate number two is the degree of nonlocality. Maybe some function from Bell or CHSH can be adapted to your form.A third possible candidate which I do not advance is the degree of nonreality. First of all, every theory of nonreality gets an automatic weirdnesss rating of one, if not infinity from me. Second of all, I am not aware of a serious attempt to measure the degree of nonreality, similar to nonlocality or entanglement.

I am willing to learn more about nonreality, but so far, I think that like Oakland, “there is no there there.”

Best.

Jim Graber

 

[Ken G]

The main difference between QM and all your other examples is that, once, say, SR was established, the debate virtually stopped and all physicists agreed about length contraction from then onwards. QM is the only case in which there is no agreement: foundations is a thorny subject with a lot of, often colorful, disagreements, even some 100 years after QM started being investigated.

There is some value in that approach to "weirdness", it is a rather different one but perhaps the most useful. It sounds like you are saying we should not regard "weird" as equal to "surprising to us", and certainly not "surprising to someone who does not know physics", but instead "still exhibiting a remarkable variation in foundational concepts among practicing physicists." That makes weirdness not a personal issue any more, but rather something that can be observed across a community of scientists, which is what I think is good about it. Still, to play devil's advocate to that version of "weirdness", I would point out that classical mechanics supports many different interpretations as well-- is it forces, is it a principle of least action, or is it just the macroscopic correspondence of quantum mechanics? Notice that if we take that last view, then even something as simple as rolling a die inherits all the same interpretations as quantum mechanics does, and always did-- it's just that few took those other possible interpretations seriously before.

 

[jimgraber]

PS I'm assuming that something similar to these three proposals would be the "mainstream response" of many other physicists.

 

[A. Neumaier]

Candidate number one is the degree of entanglement. I believe there are serious papers in the literature defining a degree of entanglement. I think they should be transformable into your form fairly readily.
Candidate number two is the degree of nonlocality.

Can you please give formulas, or at least definite references, so that it is documented what you refer to?

Note that according to my setting, the degree of weirdness must be a function of the experimental results $R=(A,B,D,E)$ only, not a function of the prepared state, which we do not know if we don't have access to what precisely Norbert prepared. My degree of weirdness is about observable things only, not about the theory behind it.

 

[ddd123]

.

I answered in the other thread to keep things tidy.

 

[edguy99]

... In a continuous run you start with all matrices initialized with zeros, and (assuming for simplicity that all events are valid and paired) after each such event add a 1 to the correct entry in both $A$ and $B$, and a 1 to the correct entry in either $E$ or $F$, depending on the pointer settings...

Animation at: http://www.animatedphysics.com/games/photon_longdistance_nonlocality.htm

Thank you for your assistance, I believe the animation matches the specifications. It stops between each photon to show what each of the people is recording and it allows you to shoot additional photons to verify proper operation and build up a data set (continuous operation will come with the NEXT button, not yet done). Looking forward to use of the data to study "wierdness". At this point, I am still using the classical description of the photon used by Dehlinger and Mitchell in http://arxiv.org/PS_cache/quant-ph/pdf/0205/0205171v1.pdf. When a photon meets a polarizer set to an angle γ, it will always register as Vγ if λ is closer to γ than to γ + π/2, i.e.,

• if |γ − λ| ≤ π/4 then vertical
• if |γ − λ| > 3π/4 then vertical
• horizontal otherwise.

A sample run is shown below after 4 photon shots.

 

[maline]

My "weird result" is the standard one that I assume stevendaryl intends: A and B both have 1/2 in all entries, E=[(0,1);(1,0)], F=[(3/4,1/4);(1/4,3/4)].
This is generated by transmitting pairs of entangled electrons with total spin 0, but as required, the weirdness is in the result itself.
I consider this weird for two reasons:
1. I expect a model of reality to be expressible in terms of variables- of any form-(field strength, wavefunction amplitude, particle momentum, or anything else) that are defined at each point in spacetime and that evolve, at each point, according to incoming information from the past lightcone, plus possibly some completely random changes, where the distribution depends only on such incoming information. Bell proved that no such model can predict the above outcome. This forces me to try to define & adopt some other concept of reality. As of yet I have not done so, so all I can say is "the results are weird".
2. If we assume that measurement of a quantum variable with more than one possible value is fundamentally nondeterministic (as most people seem to conclude), meaning that before the measurement, the universe does not contain the information of what the result will be, then I see the above result as showing a superluminal effect, as follows:
Let us work in a reference frame such that Alice's measurements occur before Bob's. Consider a pair of measurements that occur immediately after a change in pointer direction, such that Alice's choice of setting & Bob's measurement are spacelike separated. An observer who sees Alice's red light flash, and sees her pointer setting, can say with confidence, "if Bob has this setting, then his blue light is about to flash". If Bob does in fact have the same setting, as noted afterward by Yvonne, then his blue light will indeed flash. But before the measurement, the information (that the blue light will flash) does not exist in Bob's region! If information about a result exists in one part of the universe & not in another part, and afterward this "prediction" comes true in the second region, I don't see how to escape the conclusion that the information traveled, in this case superluminally. (this point is also stevendaryl's, from the earlier thread). Superluminal effects are weird because time order is not defined at spacelike separation. The fact that this effect cannot be used to transmit information only makes it weirder: "But I was thinking of a plan to dye my whiskers green/ And always use so large a fan that they could not be seen".

 

[ddd123]

Yes, it's as if nature chose the weirdest mathematical possibility: before we thought either you have action at a distance and thus ftl communication or not. But instead it preserves relativistic signaling while still having nonlocal influences.

My weirdness function is just Bell inequality violation. But as I said if Neumaier puts it in a perspective I haven't thought of, I might change my mind about weirdness, of course I can't predict my reaction.

 

[A. Neumaier]

My weirdness function is just Bell inequality violation.

Can you please formulate it in terms of the matrices $A,B,E,F$ and transform the amount of violation by some function that maps it into [0,1]?

 

[ddd123]

I'm not an expert, I'm afraid of making mistakes. I hope steveandaryl does it.

 

[maline]

I don't think the weirdness should be expressed as a function of particular results. What is weird is the implications of the theory for a conception of reality. As long as these problems can be demonstrated by some set of results, I give W=0.8 for the theory as a whole.

 

[A. Neumaier]

I don't think the weirdness should be expressed as a function of particular results. What is weird is the implications of the theory for a conception of reality.

But the implications are visible only through the results. I deliberately based this thread on stevendaryl's post because he focussed exclusively on results and their weirdness.

 

[zonde]

4. The degree of weirdness should be quasiconvex, i.e., $W(\lambda R+(1-\lambda)R')\le\max(W(R),W(R'))$ for $0<\lambda<1$. This is needed since if Norbert can create signals leading to the valid results $R$ and $R'$, he can always present in another experiment signals prepared in a statistical mixture of the original signals, in this way producing a signal leading to the valid result $\lambda R+(1-\lambda)R'$. Clearly, this should not increase the degree of weirdness.

I don't think my weirdness function would satisfy this. The reason is that A and B matrices are evaluated differently than E, F. Say we have result where A, B show rare statistical fluke and in another result we have the same but when we take average of A, B from two sets they are rather quite what was expected. Then initial sets would have low weirdness (I do not consider results in E, F reliable) but in averaged set E, F can be considered and they show Bell inequality violation so the weirdness is considerably higher.

 

[A. Neumaier]

result where A, B show rare statistical fluke

If you allow rare statistical flukes to count, then nothing can be considered weird since it might be an extremely rare statistical fluke - even classically, like the proverbial brick that flies upward since all random motions of its atoms happen to go upwards for sufficently long time.

Rare statistical flukes are excluded with probability arbitrarily close to 1 if you make your experiment of sufficient long duration. You may specify if you like an average signaling rate for Norbert, and a lower bound on the data collection time $\Delta t$ that reduces such a possibility to the level specified to regard reactions in CERN as conclusive proof of something real.

 

[maline]

But the implications are visible only through the results.

Of course. But once one "weird result" is experimentally detected, what we have is a weirdness about Nature. The result in question was just a demonstration of the fact. Thus I see no point in assigning a weirdness function to the space of possible results. Either we see Nature as weird or we don't.

 

[A. Neumaier]

once one "weird result" is experimentally detected, what we have is a weirdness about Nature.

Of course. But there is a continuum of results ranging from not at all weird to completely weird, and this thread is going to explore this.

The reason is the same as why brain specialists who want to understand consciousness are not content with stating that there is a fully working brain and describing its features. They learn much more from analyzing a whole spectrum of less well functioning brains since this tells them much more about the possible mechanisms bringing the function (and malfunction) about.

 

[jimgraber]

Can you please give formulas, or at least definite references, so that it is documented what you refer to?

Note that according to my setting, the degree of weirdness must be a function of the experimental results $R=(A,B,D,E)$ only, not a function of the prepared state, which we do not know if we don't have access to what precisely Norbert prepared. My degree of weirdness is about observable things only, not about the theory behind it.

You asked for references:

https://quantiki.org/wiki/entanglement-measure
Here (above) is a link to a discussion and list of entanglement measures. Any one could be taken as a basis for further work. Below is a reference for Quantum nonlocality which indicates that it is different from entanglement in at least some cases.
https://en.wikipedia.org/wiki/Quantum_nonlocality

More later.

 

[jimgraber]

Of course. But there is a continuum of results ranging from not at all weird to completely weird, and this thread is going to explore this.

The reason is the same as why brain specialists who want to understand consciousness are not content with stating that there is a fully working brain and describing its features. They learn much more from analyzing a whole spectrum of less well functioning brains since this tells them much more about the possible mechanisms bringing the function (and malfunction) about.

Four steps on the road from not at all weird to completely weird:
1.) Classical communication
2.) Quantum communication.
3.) PRBox Communication.
4.)FTL communication.

 

[maline]

Of course. But there is a continuum of results ranging from not at all weird to completely weird, and this thread is going to explore this.

The reason is the same as why brain specialists who want to understand consciousness are not content with stating that there is a fully working brain and describing its features. They learn much more from analyzing a whole spectrum of less well functioning brains since this tells them much more about the possible mechanisms bringing the function (and malfunction) about.

Weirdness is not a phenomenon. "What is bringing the weirdness about" is not a meaningful question- the weirdness is built into Nature. The relevant question is "How can we learn to think about reality & nature that will allow these results -all physically possible results- to not seem weird?"

 

[A. Neumaier]

the weirdness is built into Nature

No, definitely not. What is weird is a subjective value judgment. Otherwise everyone would agree that Nature is weird. But I am not theonly counterexample.

 

[A. Neumaier]

The relevant question is "How can we learn to think about reality & nature that will allow these results -all physically possible results- to not seem weird?"

Yes, this question is relevant to this thread, and without understanding the nature of weirdness we cannot answer this question about the seeming weirdness of Nature. Part of what I am doing here is prepare for a better understanding of the nature of weirdness in the context of the above class of experiments.

 

[A. Neumaier]

Please restrict discussion to weirdness in the context of this class of experiments, not in general. Otherwise we'll end up nowhere.

 

[Ken G]

Why does it have to be information? And even then, everything would be that way, all experimental results, all physical settings, including slower-than-light interactions. Why would you conclude that FTL should be ignored then?

I was referring to maline's quote: "But before the measurement, the information (that the blue light will flash) does not exist in Bob's region! If information about a result exists in one part of the universe & not in another part, and afterward this "prediction" comes true in the second region, I don't see how to escape the conclusion that the information traveled, in this case superluminally. (this point is also stevendaryl's, from the earlier thread)." Since the view is also attributed to stevendaryl, it is apparently a common way to view the situation. Without this concept of a meaning of information outside of an information processor, there is never any need for any FTL anything. You can put information processors everywhere you like in this experiment, and you'll never need any of them to get any information FTL, in order for them to apply the laws of physics to understand their observations.

 

[georgir]

To the OP - what is your end goal? Are you just doing this to better understand why we think QM is weird, without the goal of convincing us it is not? Or are you going to argue how some of our proposed weirdness functions are "wrong"? Are you planning to show us your own subjective weirdness function, so we understand why you say QM is not weird?

So far, I'm with ddd123 - matrixes that show bell violations would be weird to me, despite the fact they match reality. As to more precise formulation of that, I do not understand your quasiconvex requirement, and my function would not be continuous but binary or at most ternary if I add for 0.5 to mean I don't think we have enough data yet to rule out statistical flukes or experimental errors (although I am not certain where I would place the boundary).

 

[A. Neumaier]

what is your end goal? Are you just doing this to better understand why we think QM is weird, without the goal of convincing us it is not? Or are you going to argue how some of our proposed weirdness functions are "wrong"? Are you planning to show us your own subjective weirdness function, so we understand why you say QM is not weird?

I revealed already my weirdness function, it is identically zero. And I tried to explain in the originating thread why I do not find quantum mechanics weird.

My goal is to understand the true origin of any subjectively perceived weirdness. Since it is subjective, it cannot be wrong. Perhaps I can convince some readers of my point of view; it is more unlikely but not impossible that I can convince some of the participants.

I spend my time on this and the originating thread because I learn through participating in the discussion about the usually neglected language side of the matter, and because I think I can contribute something nontrivial to understanding. After the end of this discussion, I'll write an Insight article for PF in which my insights into the matter are systematically presented.

 

[A. Neumaier]

I do not understand your quasiconvex requirement, and my function would not be continuous but binary or at most ternary

1. Quasiconvexity is a consistency requirement whose origin I had explained when defining it. It is needed since Norbert is free to choose the signals he sends. He can choose in a random order the signals that lead to $R$ in a fraction $\lambda$ of all signals and the signals that lead to $R'$ in the remaining fraction $1-\lambda$ of all signals. This changes the observed statistics as stated.

2. Imagine Norbert sends more than enough signals to rule out statistical flukes with probability $1-10^{-10}$, but changes the input gradually from something that appears weird to you to something that appears ordinary to you. (He doesn't perform yet any of these infinitely many experiments.) If your weirdness function is not continuous but $\{0,1\}$-valued then at some point the weirdness drops suddenly from ''completely weird'' to ''not at all weird''. Close to this threshold there would be two sets of results whose numerical values differs only in the 10th decimal place, one of which you'd regard as weird, the other one to be ordinary. Norbert only needs to perform these two experiments. I want to exclude such bizarre proposals, since it is obvious that in this case the weirdness measure is itself too weird to be taken seriously.

To be able to argue like this I deliberately didn't want to discuss a single setting - where one cannot analyze anything except taking note that some find the result weird but others don't. Having a continuum of possibilities reveals important nuances in what the weirdness is about.

 

[ddd123]

And I tried to explain in the originating thread why I do not find quantum mechanics weird.

Again, you haven't done that in the context of EPR yet. I'm not clear on if you intend to do that later or at all.

 

[A. Neumaier]

you haven't done that in the context of EPR yet. I'm not clear on if you intend to do that later or at all.

I'll do that in the next stage. (I don't want that my interpretation influences your weirdness criteria.)

 

[ddd123]

For the record this is my weirdness function:

matrixes that show bell violations would be weird to me, despite the fact they match reality [...] and my function would not be continuous but binary

I'd say weirdness 1 for Bell violation of any kind, weirdness 0.1 for no violation (because of particle-wave duality, tunneling, HUP, and similar stuff for which however I could potentially make the weirdness function 0 with some further clarification). If it's unclear whether there's violation or not, my function is undefined because I'd wait for a settlement.

 

[maline]

If your weirdness function is not continuous but {0,1}{0,1}\{0,1\}-valued then at some point the weirdness drops suddenly from ''completely weird'' to ''not at all weird''. Close to this threshold there would be two sets of results whose numerical values differs only in the 10th decimal place, one of which you'd regard as weird, the other one to be ordinary. Norbert only needs to perform these two experiments. I want to exclude such bizarre proposals, since it is obvious that in this case the weirdness measure is itself too weird to be taken seriously.

There is nothing bizarre about such a proposal. The idea of a binary weirdness function is that we assign weirdness "1" to`any result that makes our basic conception of reality untenable, and "0" to a result that can be "explained away" in terms that make sense to us. There is a hard boundary between these sets, given by the Bell inequality. A tiny counterexample to our understanding is just as problematic as a "huge" one, unless we allow for statistical flukes, which we are discounting here.

 

[ddd123]

I would add that the weirdness function has memory. If a series of events "undoubtedly" violates Bell inequalities, then they're gradually toned down until they stop doing so, it's not like I can forget what has happened (if anything, weirdness increases).

Also the assessment is already statistical and thus must be done over a number of events.

 

[Nugatory]

A note from the mentors:
A number of off-topic posts have been removed from this thread, and will be moved to another one. Please, please, try to respect the ground rules of this thread.

 

[A. Neumaier]

There is a hard boundary between these sets, given by the Bell inequality.

This is not hard, since it assumes unrealistic perfect experimental conditions. Tiny counterexamples are statistically not significant, not even with experiments of arbitrarily long duration.

 

[maline]

Tiny counterexamples are statistically not significant, not even with experiments of arbitrarily long duration.

As far as I understand the Law of Large Numbers, as long as the deviations are averaged over all trials, any finite deviation can reach any level of significance by increasing the trials sufficiently. But anyhow, let us agree to ignore cases where the reason for non-weirdness is mere lack of statistical significance.

 

[A. Neumaier]

It seems that the task of defining a proper degree of weirdness is too difficult. I am still waiting for stevendaryl's response to this task (post #49), and then will go on to the next stage.

 

[georgir]

Ok, "too difficult" is not quite it. There are actually too many ways, and I'm not sure which will suffice for you. Heres an attempt.
1. Most trivial: Remap the degree of correlations between different-setting measurements from the range of [0.6666, 0.75] to [0, 1].
2. Account for statistical significance of the accumulated data: Produce the weighted average between the above trivial function and a constant 0.5 function, based on the number of measurements made. At close to 0 measurements, return mostly 0.5. At close to say 1000 measurements and after, return function 1.
3. Account for observed "errors" - for example ratio of mismatched results with same-settings should be close to 0, if it is more again return weighted average of above function and constant 0.5. The weight threshold here like above is a bit arbitrary, but let's say I could go with 0.05 and above leading to entirely 0.5 outputs.
This seems to me to be changing slowly enough with each next measurement to make you happy.

 

[maline]

Remap the degree of correlations between different-setting measurements from the range of [0.6666, 0.75] to [0, 1].

You are assuming we actually get perfect correlations for identical settings. If we want to cover all possible results, we need to point out that if those correlations are not perfect then the weirdness is less, or zero, because LHV models (like edguy99's) can theoretically explain such results. Note that in the OP there was no specification about what Norbert sends! We need to consider "ordinary", entanglement-free results as well.
As for me, I still don't see any point of calling some results weirder than others. What is weird is the clash between how Nature operates and how we (at least I) are able to conceive of it. We only need one proof to establish that such a clash exists. Once that is true, we judge results only by how they accord with our theory, namely QM, and of course, they do. The discussion about weirdness/ understanding/ interpretation becomes independent of experiment, which is one reason it is so disliked. But for me something is definitely weird...

 

[georgir]

Maline, you seem to have missed point 3 in my post, allowing for somewhat less than perfect correlations.

 

[maline]

I just mean that Bell's reasoning starts with the point that to get definite results for any pair of equal settings, a local & realist model needs definite hypothetical results for all possible individual settings. A full "weirdness function" based on a Bell violation needs to depend explicitly on both matrices E & F to make sure the logic holds.

 

[wle]

If it's a Bell inequality using only the matrices defined in the beginning of this thread that you want, then a simple one depending only on $E$ and $F$ is $$- \bar{E} + 3 \bar{F} \leq 2 \,, \qquad (*)$$ with $$\begin{eqnarray*}
\bar{E} &=& E_{00} - E_{01} - E_{10} + E_{11} \,, \\
\bar{F} &=& F_{00} - F_{01} - F_{10} + F_{11} \,.
\end{eqnarray*}$$ This is taking results from the different measurement settings to contribute equally to $E$ and $F$, i.e., in terms of the conditional probabilities, $$\begin{eqnarray*}
E_{ab} &=& \frac{1}{3} \sum_{x} P(ab \mid xx) \,, \\
F_{ab} &=& \frac{1}{6} \sum_{x \neq y} P(ab \mid xy)
\end{eqnarray*}$$ for $a, b \in \{0, 1\}$ and $x, y \in \{0, 1, 2\}$. (The simplest way to ensure this is just to make this the definition of the matrices $E$ and $F$.) Otherwise, the inequality holds under the same sort of assumptions as other Bell inequalities, e.g., if you don't do any postselection or you make the fair sampling hypothesis.

I'll skip the details on how (*) was derived unless someone asks. In any case, with (*) given it's not especially difficult to check that it must hold for any LHV. Using a state and measurements similar to those that maximally violate the 1964 Bell inequality it's possible to have $\bar{E} = -1$ and $\bar{F} = 1/2$, so QM can attain at least $- \bar{E} + 3 \bar{F} = 2.5$, which violates (*). Finally, since $\lvert \bar{E} \rvert, \lvert \bar{F} \rvert \leq 1$, the algebraic bound is $- \bar{E} + 3 \bar{F} \leq 4$, and there's a $3 \times 3$-measurement version of the PR box that can attain this while still respecting the no-signalling principle. So if you want a function of value between 0 and 1 when it detects Bell-nonlocal correlations then one possibility is $$W = -1 + \frac{- \bar{E} + 3 \bar{F}}{2} \,.$$

 

[A. Neumaier]

So if you want a function of value between 0 and 1 when it detects Bell-nonlocal correlations then one possibility is $W = -1 + \frac{- \bar{E} + 3 \bar{F}}{2} \,.$

This can become negative (uniformly random output independent of input), hence is not yet good. What about $W=\max(0,\min(6\bar F-2\bar E-4,1))$? This would satisfy my requirements, and makes the specific case you described completely weird while leaving local hidden variable results not weird at all.

 

[A. Neumaier]

I'm not sure which will suffice for you

Any that satisfies the criteria stated in post #49 and reflects your personal view of weirdness. But it must be an explicit formula that I can evaluate for any chocie of E and F.

 

[edguy99]

If it's a Bell inequality using only the matrices defined in the beginning of this thread that you want, then a simple one depending only on $E$ and $F$ is $$- \bar{E} + 3 \bar{F} \leq 2 \,, \qquad (*)$$ with $$\begin{eqnarray*}
\bar{E} &=& E_{00} - E_{01} - E_{10} + E_{11} \,, \\
\bar{F} &=& F_{00} - F_{01} - F_{10} + F_{11} \,.
\end{eqnarray*}$$ This is taking results from the different measurement settings to contribute equally to $E$ and $F$, i.e., in terms of the conditional probabilities, $$\begin{eqnarray*}
E_{ab} &=& \frac{1}{3} \sum_{x} P(ab \mid xx) \,, \\
F_{ab} &=& \frac{1}{6} \sum_{x \neq y} P(ab \mid xy)
\end{eqnarray*}$$ for $a, b \in \{0, 1\}$ and $x, y \in \{0, 1, 2\}$. (The simplest way to ensure this is just to make this the definition of the matrices $E$ and $F$.) Otherwise, the inequality holds under the same sort of assumptions as other Bell inequalities, e.g., if you don't do any postselection or you make the fair sampling hypothesis.

I'll skip the details on how (*) was derived unless someone asks. In any case, with (*) given it's not especially difficult to check that it must hold for any LHV. Using a state and measurements similar to those that maximally violate the 1964 Bell inequality it's possible to have $\bar{E} = -1$ and $\bar{F} = 1/2$, so QM can attain at least $- \bar{E} + 3 \bar{F} = 2.5$, which violates (*). Finally, since $\lvert \bar{E} \rvert, \lvert \bar{F} \rvert \leq 1$, the algebraic bound is $- \bar{E} + 3 \bar{F} \leq 4$, and there's a $3 \times 3$-measurement version of the PR box that can attain this while still respecting the no-signalling principle. So if you want a function of value between 0 and 1 when it detects Bell-nonlocal correlations then one possibility is $$W = -1 + \frac{- \bar{E} + 3 \bar{F}}{2} \,.$$

Thank you for the post. Can you add a sample calculation with 2 matrices to help some of us that may be a little confused by the terminology. Regards.

 

[stevendaryl]

I'm not particularly comfortable with this "degree of weirdness" variable W. To me, what's weird is the lack of an answer to some basic questions about QM, particular to the EPR experiment (a variant of which is being discussed here).

Let's enumerate the critical events:

• e_0 where the twin pair (or whatever it is) is created.
  
• e_{a1} at which Alice picks her setting.
• e_{a2} at which Alice gets her result (one of two possibilities)
• e_{b1} at which Bob picks his setting.
• e_{b2} at which Bob gets his result.

To simplify the discussion, let me first assume that Alice and Bob choose the same setting, and they both know ahead of time which setting that is. For definiteness, let's assume that in Alice's rest frame, e_{a2} takes place slightly before e_{b2}.

Before e_{a2}, Alice doesn't have any idea what result Bob will get at e_{b2}. Then suppose she gets spin-up at event e_2. Afterward, she knows exactly what he will get (because of the perfect anti-correlations): spin-down.

So my question is about Alice's change of knowledge about Bob. It seems to me that there are three possibilities:

1. Bob's (future) result was already determined prior to Alice's measurement, and the only thing that changed by her measurement was her knowledge about that outcome.
2. Bob's result becomes definite as a result of Alice's measurement.
3. Something more exotic, such as Many-Worlds.

Choice number 1 seems to be a hidden-variables theory of the type that is ruled out by Bell's inequality (unless we get into loopholes such as retrocausality or superdeterminism). Choice number 2 seems to require a nonlocal interaction. Choice 3 is weird, for reasons that I won't get into here.

The complications allowing Alice and Bob to choose a setting in-flight doesn't really change the weirdness. It only serves to rule out possibility 1.

So if #1 is ruled out, then it would seem to me that the EPR experiment implies either nonlocality, or various exotically weird possibilities (retrocausality, superdeterminism, many-worlds).

But most people who deny that there is anything weird about QM seem to reject all the possibilities:

1. QM is not retrocausal.
2. QM is not superdeterministic.
3. QM is not nonlocal.
4. QM does not imply Many-Worlds.

--
Daryl McCullough

 

[ddd123]

Choice 2's nonlocal interaction is ill-defined in special relativity. You have to pick a direction arbitrarily or invoke an preferred spacetime foliation. This makes it even weirder for me.

 

[A. Neumaier]

1. Bob's (future) result was already determined prior to Alice's measurement, and the only thing that changed by her measurement was her knowledge about that outcome.

Choice number 1 seems to be a hidden-variables theory of the type that is ruled out by Bell's inequality

... ruled out only under the assumption of a local hidden variable theory with signals moving independently along the rays to Alice and Bob. But this assumption is too strong to have implications when the signal is a field rather than particles.

This is apparent from a (simpler) single-photon nonlocality experiments such as that discussed in my slides here (slides 46-59). The argument there doesn't extend to the setting under discussion here but shows that the assumptions of Bell are tied to an implicit particle assumption.

I'd appreciate if (in a new thread) you'd assess the weirdness of my setting in the slides according to your criteria. For I think the same concerns that you raise above apply to the single-photon nonlocality experiment, although the latter has a fully classical field explanation.