I Is quantum weirdness really weird?

[zonde]

But then it is speculation, not physics.

It's physics. We just have to call this speculation a hypothesis.
Look here: https://en.wikipedia.org/wiki/Scientific_method#Elements_of_the_scientific_method
Don't know if you like Feynman but anyways. Look at the very start of this video about first step in the process of getting new law:


The problem with interpretations of QM is rather about turning them into something more than interpretation and getting unique predictions ... and maybe getting rid of unacceptable interpretations (interpretations that will newer be anything more than an interpretation).

 

[A. Neumaier]

It's physics. We just have to call this speculation a hypothesis.

This view is certainly fringe, as it would turn everything into physics: flat earth, hollow earth, cold fusion, perpetuum mobiles...

 

[zonde]

Quantum theory tells you that nature is indeterministic and thus you can only have probabilities for the outcome of measurements given the state of the system, even if this knowledge is complete in the sense that you know the exact pure state of the system.

This is not Quantum theory by itself, it's Copenhagen interpretation (wave function is complete description of every quantum system in ensemble).

 

[vanhees71]

The predictions of QT are very well defined within the minimal interpretation, and there's no contradiction so far with any experiment/observation. Everything going beyond the minimal interpretation, leading to the same (probabilistic) predictions of QT are by definition just speculation although not in contradiction to QT, and you can follow these interpretations, but it's not of anything beyond standard QT.

 

[zonde]

This view is certainly fringe, as it would turn everything into physics: flat earth, hollow earth, cold fusion, perpetuum mobiles...

Ah, but therefore you have other steps in scientific method so that you can drop false and unscientific hypotheses.

 

[vanhees71]

This is not Quantum theory by itself, it's Copenhagen interpretation (wave function is complete description of every quantum system in ensemble).

Well, as long as there's no disprove of QT, I don't see any reason in believing otherwise. That's indeed part of standard quantum theory in the minimal interpretation. I'm not sure whether you can call the minimal interpretation a flavor of Copenhagen interpretation. Note that the Copenhagen interpretation consists of at least as many physicists having worked together with Bohr in Copenhagen ;-)). Even Heisenberg's and Bohr's point of view are slightly different although I think one could say that Heisenberg was most closely working together with Bohr on the philosophy of QT, called "interpretation" ;-).

 

[rubi]

Violation of counterfactual definiteness isn't a matter of interpretations of quantum mechanics. It's an experimentally tested fact that must be included in every physical theory that describes every aspect of nature. We can't blame quantum mechanics for this.

 

[zonde]

That's indeed part of standard quantum theory in the minimal interpretation. I'm not sure whether you can call the minimal interpretation a flavor of Copenhagen interpretation.

Some would say that minimal interpretation is Ballentine's statistical interpretation. And Ballentine makes clear distinction between his interpretation and Copenhagen.

 

[stevendaryl]

Quantum theory tells you that nature is indeterministic and thus you can only have probabilities for the outcome of measurements given the state of the system, even if this knowledge is complete in the sense that you know the exact pure state of the system.

I think that the indeterminism is NOT the weird part of quantum mechanics, to me. The weird part is that the Rules of Quantum Mechanics are in terms of results of measurements, when measurements are just complicated interactions. It's weird to describe what goes on microscopically, at the level of individual particles, in terms of something that is macroscopic and very human-centered--measurement. If there is nothing weird going on in the measurement process--it's just the working out of ordinary physics (including QM) on the macroscopic scale, it seems to me that there should be a formulation of QM that doesn't mention measurements, or observations. If measurements are ordinary interactions (just macroscopic), then what happens when you perform a measurement should be derivable from the laws of physics, it shouldn't be postulated.

 

[A. Neumaier]

The weird part is that the Rules of Quantum Mechanics are in terms of results of measurements

Only the textbook rules, not the rules that are actually used, which are much more flexible and more along my thermal interpretation.

 

[A. Neumaier]

what happens when you perform a measurement should be derivable from the laws of physics, it shouldn't be postulated.

And it is derivable. When creating a high performance measurement one uses a lot of QM to ensure that the measurement results are good! There is a whole industry of quantum measurement, with only 0.1% of it being covered by the textbook account.

 

[stevendaryl]

And it is derivable.

In that case, there should be a formulation of quantum mechanics that doesn't mention measurement (in particular, the rule that the result of a measurement is an eigenvalue). Deriving facts about measurements from postulates that already assume facts about measurements seems a little circular. (Although a circular derivation is comforting, to a certain extent, because it shows that at least the whole story hangs together, and doesn't contradict itself.)

 

[rubi]

In that case, there should be a formulation of quantum mechanics that doesn't mention measurement

There is already such a formulation (consistent histories) and it doesn't even require any shift in perspective, just a little more precise use of language.

(in particular, the rule that the result of a measurement is an eigenvalue).

I don't know whether you wanted to put the emphasis on the word "measurement" or on "eigenvalue", but if the latter concerns you, then I don't understand the problem. It's just specifying the set of physically allowed values of an observable, similar to classical mechanics, where the physically allowed values of observables are also restricted (for example, the energy $H=p^2+x^2$ of a harmonic oscillator can only be positive).

 

[stevendaryl]

There is already such a formulation (consistent histories) and it doesn't even require any shift in perspective, just a little more precise use of language.

Whether this is accurate or not, I think of consistent histories as a variant of the Everett Many-Worlds. I actually think that something along those lines might be the right way to think about it. I was mostly complaining about the so-called "minimal interpretation". If "measurement" is mentioned in the formulation, then it's not really very minimal, since measurement is a complicated thing.

 

[rubi]

Whether this is accurate or not, I think of consistent histories as a variant of the Everett Many-Worlds.

Yes, a fair statement would be to say that consistent histories is Everett Many-Worlds with only one world, although that is also not completely accurate.

I was mostly complaining about the so-called "minimal interpretation". If "measurement" is mentioned in the formulation, then it's not really very minimal, since measurement is a complicated thing.

I agree that measurement should not have the status of a fundational concept.

 

[vanhees71]

Well, but physics is about what we can objectively observe (and measurements are just accurate quantitative observations) nature and about regular patterns ("natural laws") of such observtions. So any physics more or less explicitly includes ideas on more or less complicated measurements of quantitatively observable phenomena. Already measuring the width of my table uses a lot of assumptions and physics (sufficient accuracy of the validity of Euclidean geometry for example) although you wouldn't think about this very much as it seems very natural to you, but if you think about it, it's already not so simple. We are just used to it from elementary school on. That QT compared to classical physics seems so "weird" to us is simply because we are not very used to quantum phenomena like interference of probabilities, entanglement etc.

 

[A. Neumaier]

In that case, there should be a formulation of quantum mechanics that doesn't mention measurement (in particular, the rule that the result of a measurement is an eigenvalue). Deriving facts about measurements from postulates that already assume facts about measurements seems a little circular. (Although a circular derivation is comforting, to a certain extent, because it shows that at least the whole story hangs together, and doesn't contradict itself.)

Relativistic quantum field theory (the true foundation of quantum mechanics) is usually developed from scratch without the slightest reference to measurement. The latter comes much later in when interpreting the S-matrix, but that is done mainly to establish contact to tradition rather than to justify the foundations. Moreover, the S-matrix also appears in the derivation of the Boltzmann equation, where it gets a measurement-free interpretation that shows that nothing in the foundations depends on a notion of measurement.

 

[stevendaryl]

Well, but physics is about what we can objectively observe (and measurements are just accurate quantitative observations) nature and about regular patterns ("natural laws") of such observtions. So any physics more or less explicitly includes ideas on more or less complicated measurements of quantitatively observable phenomena.

I would disagree completely. Newtonian physics is about the motions of particles under the influence of forces. It is not about measurements. Of course, you have to do measurements to test Newtonian mechanics, but it isn't about measurements.

 

[madness]

If one interprets "weird" as meaning "defies the common sense expectations we've acquired from a lifetime lived in a classical world", as is the rest of your post does, the quantum mechanics is indisputably weird.

By that definition, classical physics is also "weird". For example, the fact that if I move my hand it will cause the moon to move. Or even more basically, the fact that things will continue at a constant speed unless acted upon by something - that went completely against common sense and took a very long time to be realized or accepted.

 

[Nugatory]

By that definition, classical physics is also "weird".

A point that serves to reinforce the general futility of trying to characterize a discipline as "weird" or "not weird"...

 

[OmCheeto]

A point that serves to reinforce the general futility of trying to characterize a discipline as "weird" or "not weird"...

Actually, I think that's why people "do" science. As everything that is "weird", has to be figured out, so as to make it "not weird".
I suspect this might be why the profs at sixty symbols keep using the term, as their audience is laypeople.

--------
Grady doesn't let them use maths.
And it's my general sense, from listening to them, is that you can't understand QM unless you understand the maths, and their only response, in that case, is "it's weird".

 

[madness]

Actually, I think that's why people "do" science. As everything that is "weird", has to be figured out, so as to make it "not weird".

I think you have things back to front. It's the explanation that makes things become "weird". For example, classical physics telling us that that if we move our hand it causes to the moon to move. That increases rather than decreases the amount of weirdness, because it is a departure from common sense "folk physics".

 

[stevendaryl]

I think you have things back to front. It's the explanation that makes things become "weird". For example, classical physics telling us that that if we move our hand it causes to the moon to move. That increases rather than decreases the amount of weirdness, because it is a departure from common sense "folk physics".

I think this thread is confusing different things. Counter-intuitiveness is something that is a matter of, well, intuitions. Intuitions come from experience. When you are first introduced to something that is different from what you've known before, it is often counter-intuitive.

The weirdness of quantum mechanics, though, is not really the fact that it is counter-intuitive. It might be counter-intuitive to people who are first introduced to it, but after studying it, you develop an intuition for it, and it ceases to be counter-intuitive. But the weirdness is what's left over after it becomes intuitive. Some of those that have complained about quantum mechanics being weird include Einstein, Feynman, Penrose, Weinberg...people who were plenty familiar with quantum mechanics.

 

[secur]

"Weird" is not a scientific term, but subjective. The statement "QM is weird" says absolutely nothing about QM or science, rather it's a statement about the speaker and his/her emotional state. Therefore to figure out why "QM is weird" don't analyze QM, but the speaker. The real question is "Why does person X think (or, better, feel) that QM is weird?"

To make a long story short, Person X is making a "category error". There are some rules, or intuitions, applying to category A being applied incorrectly to category B. In this case category A is the macro world of physics. In that category things behave deterministically, observations don't affect them. Probabilities are due to subjective ignorance, not an inherent quality of objects. Different possible outcomes don't interfere with each other. Category B OTOH is the micro world of physics - QM. The only connection between the two is (basically) the correspondence principle. Loosely speaking, as quantum systems get large enough they must behave classically. IOW in the large limit they become category A and Person X should expect them to behave that way. However apart from that, category B, the micro world, is a brand new situation about which we knew nothing, prior to the advent of advanced technology and instrumentation more than 100 years ago. So there is absolutely no reason to apply category A intuition to category B. They're two different things. Only experimental observations can tell you how micro objects behave.

So if you think QM is weird, you're simply making an elementary philosophical error. The problem is not with QM, but with you.

In contrast, I've never found QM weird. A long time ago I learned a few elementary facts about it, recently I've learned a lot more. None of it is weird, or strange, or contrary to expectation - simply because I didn't expect anything. I have absolutely no basis to judge QM (micro physics) except the relevant experiments. They say what they say, and I accept it - intellectually and emotionally. Recommend you do the same.

 

[vanhees71]

I would disagree completely. Newtonian physics is about the motions of particles under the influence of forces. It is not about measurements. Of course, you have to do measurements to test Newtonian mechanics, but it isn't about measurements.

And I completely disagree in turn with this statement. You used, e.g., the word "force". To be able to do so, you must define how to observe/measure it, and that is what Newton does in the very beginning of his Principia.

 

[stevendaryl]

So if you think QM is weird, you're simply making an elementary philosophical error. The problem is not with QM, but with you.

I think you're completely off base. I think it's inappropriate to lecture people about what they should care about. Scientific progress is guided by the scientist's sense that there is something left to be explained. I think the advice that you seem to be giving would be death to good science.

When people say that QM is weird, they mean that they feel that there is something that is still not understood about it. As I said, in my case, it is the fact that measurement seems to have a special role in the formalism. How can that be, if (as it surely must be) a measurement is just built up out of the same sort of interactions that govern particles? But your complaint that "QM is weird" is a philosophical error is itself an erroneous understanding of what people are saying. Obviously, they aren't making a scientific statement when they claim that it is weird, they're making a statement about the state of our understanding of QM.

 

[stevendaryl]

And I completely disagree in turn with this statement. You used, e.g., the word "force". To be able to do so, you must define how to observe/measure it, and that is what Newton does in the very beginning of his Principia.

As I already said, the fact that you need to be able to measure things in order to experimentally test a theory does not mean that the theory is ABOUT measurements. Newton's laws are about things like planets. They would move around the sun in the same way even if there were no people to measure their motions.

 

[secur]

I think you're completely off base.

BTW although my post happened, accidentally, to appear after yours, it wasn't a response to you specifically, but to anyone in the "QM is weird" camp.

 

[PeterDonis]

It's physics. We just have to call this speculation a hypothesis.

Only if we can test it experimentally. Different speculations that all make exactly the same predictions for all experimental results (which is the case for all interpretations of QM) are not physics, because there is no way to experimentally test which one is right.

The problem with interpretations of QM is rather about turning them into something more than interpretation and getting unique predictions ...

Which has not yet been done--hence, as above, they are not (yet) physics.

 

[PeterDonis]

Violation of counterfactual definiteness isn't a matter of interpretations of quantum mechanics.

Yes, it is, because there are interpretations that don't even include it. In fact, I would argue that even Bell's derivation of his inequalities didn't include it. But of course that depends on how one interprets the term "counterfactual definiteness". Which is my point.

 

[rootone]

...measurement should not have the status of a fundational concept.

Any better ideas about how science is to be done? ( btw, I expect you meant 'fundamental', but could be wrong)

 

[OCR]

Just a quick question, and it's not related to weirdness... well, I don't think it is... anyway...

Is the member that posts with the user name "A. Neumaier", the same person that gave the presentation shown in the picture below?

 

[vanhees71]

As I already said, the fact that you need to be able to measure things in order to experimentally test a theory does not mean that the theory is ABOUT measurements. Newton's laws are about things like planets. They would move around the sun in the same way even if there were no people to measure their motions.

Newton's theory is about the description of this motion. To describe it you have to define all kinds of quantities like the position of the planets and the sun, their velocities, accelerations, forces. This already needs both theory how to define these quantities and an operational description of how to observe them. The very fact that Newton's mathematical construct has to do with the motion of the planets and the sun and is not just a nice mathematical puzzle without any meaning to an aspect of nature forces you to do so. Of course, the sun and the planets couldn't care less whether we describe their motion with mathematical tools and observe them or not.

 

[rubi]

Yes, it is, because there are interpretations that don't even include it.

That's not true. Every theory that reproduces the predictions of standard QM must violate counterfactual definiteness. That applies also to Bohmian mechanics and Bohmian mechanics in fact violates it as well. This isn't a consequence of Bell's theorem. It's a consequence of the Kochen-Specker theorem or the GHZ theorem.

In fact, I would argue that even Bell's derivation of his inequalities didn't include it. But of course that depends on how one interprets the term "counterfactual definiteness". Which is my point.

Well, in a mathematical theorem, the assumptions are just mathematical statements. It doesn't matter how you interpret these statements physically. It's enough for them to hold mathematically. We can assign names all the mathematical statements that appear in the assumptions of Bell's theorem and one of them is often called counterfactual definiteness. It was shown by Stapp and Eberhard that counterfactual definiteness appears as an assumption in all known proofs of Bell type inequalities that also assume locality.

Any better ideas about how science is to be done? ( btw, I expect you meant 'fundamental', but could be wrong)

I was talking about the formulation of physical theories. Of course you will test these theories by measurements, but the concept of measurement should not be required for the formulation of a theory. Measurement devices ought to be governed by the laws of physics as well and thus they should not play a distinguished role in a fundamental theory. Instead, the measurement process should be describable within the theory. (And in fact, this an be accompished in QM.)

 

[Simon Phoenix]

Well I find QM indisputably mysterious. Whether one goes as far as 'weird' or 'strange' or just settles for the more innocuous sounding 'counter intuitive' is a matter of taste I guess.

The lesson of Bell's work, discounting non-local effects, is that nature is not describable by assuming objects have a list of properties (known or unknown) independent of measurement. So what are we saying here - we're saying that if the objects in nature really had some set of properties independent of measurement we could certainly write those down in principle and use them as inputs to a model that attempts to predict observations. We know that any such model is doomed to fail.

In other words, objects in nature do not have properties independent of measurement, it's a much stronger statement than saying we simply don't know those properties. We can say things like "if we make a measurement of $A$ then we'll get the result $a$ with some probability $p$", but it is incorrect to infer from that that the object really had some property $a$ prior to measurement.

Am I the only person on these forums who finds that weird? Not knowing properties is unremarkable, things not even having such properties independent of measurement is remarkable, in my opinion. We're all used to describing QM as a theory that predicts measurement results - that's fine and dandy. We even get used to saying that QM doesn't say anything about properties in between measurements (except in very special circumstances) - but to go the extra step, implied by Bell's work, that these properties themselves don't exist in any meaningful way independent of measurement, that the existence of the very properties we try to measure is inextricably linked to measurement as if the act of measurement itself 'creates' those properties - that I find deeply mysterious and rather wonderful

 

[A. Neumaier]

objects in nature do not have properties independent of measurement

The sources have properties independent of measurement, and the beams have properties independent of measurement. These are the real players and the real objects.

Only the particles in the beams don't. This is because the particle concept is a derived, approximate concept that makes intuitive sense only under very special situations - namely in those where they actually behave like particles. It is a historical accident that one continues to use the name particle in the many microscopic situations where it is grossly inappropriate if one thinks of it with the classical meaning of a tiny bullet moving through space.

Restrict the use of the particle concept to where it is appropriate, or don't think of it as an ''object'' - in both cases all mystery is gone.

 

[A. Neumaier]

Is the member that posts with the user name "A. Neumaier", the same person that gave the presentation shown in the picture below?

Yes, that was me in an earlier life (it is more than two years old). Why does it matter?

By the way, since you write that you sometimes edit Wikipedia: The articles on virtual particles and related stuff need a lot of cleaning up. See https://www.physicsforums.com/insights/misconceptions-virtual-particles/

 

[Simon Phoenix]

and the beams have properties independent of measurement

I'm not sure what you are suggesting here - do you mean that there is a (local) non-contextual variable description for, say, the output of a pulsed laser that is equivalent to the 'quantum state' version and will, in all circumstances, reproduce the experimental predictions (for example if we use the beam in a parametric downconversion process)?

 

[A. Neumaier]

I'm not sure what you are suggesting here - do you mean that there is a (local) non-contextual variable description for, say, the output of a pulsed laser that is equivalent to the 'quantum state' version and will, in all circumstances, reproduce the experimental predictions (for example if we use the beam in a parametric downconversion process)?

Both sources and beams are extended macroscopic objects describable by quantum field theory and statistical mechanics, and hence have associated nearly classical observables (densities, intensities, correlation functions).

The output of a laser (before or after parametric down conversion or any other optical processing) is a coherent laser beam or beam collection, in a well-defined state that can be probed and is always found to have the properties ascribed to it by the preparation procedure. Thus the properties exist independent of any measurement - just as the moon when nobody is looking at it!

 

[Simon Phoenix]

and is always found to have the properties ascribed to it by the preparation procedure

Of course, but does it have those properties before measurement? All you're saying is that if we prepare a pure state then it has the 'properties' consistent with any subsequent measurement of that state - but that's kind of stating the obvious. What we can't always do is to describe this state properly with (local) contextual variables.

And the output from a parametric downconverter with coherent state input isn't a coherent state but a squeezed (and entangled) state.

 

[A. Neumaier]

the output from a parametric downconverter with coherent state input isn't a coherent state but a squeezed (and entangled) state.

Coherent has multiple meanings. A squeezed state is still very coherent, just with a different group defining the coherent state.

Of course, but does it have those properties before measurement?

Of course. Neither the state of the laser nor of the beam is changed by a measurement at the end of the beam. And its properties are reproducible. So why should anyone (except those who want to maintain a weird view of Nature for other reasons) think that these properties should depend on measurement?

 

[Nugatory]

Neither the state of the laser nor of the beam is changed by a measurement at the end of the beam. And its properties are reproducible.

How does this compare with the EPR criterion: "A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system"?

 

[A. Neumaier]

The EPR criterion, "A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system", is satisfied by stationary (or sufficiently slowly varying) optical sources and arrangements of beams.

But quantum particles do not. That's why an inappropriate focus on the particle aspect of quantum mechanics creates the appearance of weirdness.

In this sense, sources and beams are much more real than particles.

 

[PeterDonis]

Every theory that reproduces the predictions of standard QM must violate counterfactual definiteness.

How do you define "counterfactual definiteness"?

We can assign names all the mathematical statements that appear in the assumptions of Bell's theorem and one of them is often called counterfactual definiteness.

Which one?

It was shown by Stapp and Eberhard that counterfactual definiteness appears as an assumption in all known proofs of Bell type inequalities that also assume locality.

What about the ones that don't?

 

[ljagerman]

I like a simplistic approach to "weirdness" in quantum mechanics, particularly when teaching amateur scientists. The big three weirdnesses are (1) the uncertainty principle, (2) wave-particle duality, and (3) entanglement.
1: Everything in the universe, notably subatomic particles, and always in at least some random motion. So if we try to pin down location, momentum is uncertain, and vice versa.
2: Particles are particles, but their locations in space-time may be wave-like if graphed or plotted. I.e., the waves in this duality are waves of probability in the behavior of particles.
3: Two entangled particles may show interdependent behavior, but that behavior is always, to at least some extent, uncertain. So only probabilities are entangled, not firm unequivocal information. Also, there may be more than four dimensions, and entangled particles may be immediately adjacent in one of those additional dimensions.

 

[secur]

I like a simplistic approach to "weirdness" in quantum mechanics, particularly when teaching amateur scientists. The big three weirdnesses are (1) the uncertainty principle, (2) wave-particle duality, and (3) entanglement.

1: Everything in the universe, notably subatomic particles, and always in at least some random motion. So if we try to pin down location, momentum is uncertain, and vice versa.
2: Particles are particles, but their locations in space-time may be wave-like if graphed or plotted. I.e., the waves in this duality are waves of probability in the behavior of particles.
3: Two entangled particles may show interdependent behavior, but that behavior is always, to at least some extent, uncertain. So only probabilities are entangled, not firm unequivocal information. Also, there may be more than four dimensions, and entangled particles may be immediately adjacent in one of those additional dimensions.

That's too simplistic for this discussion, I'd say, and perhaps not entirely correct.

Point 1 implies the particle is real and has a definite position. Its random motion makes it impossible to "pin down" location and momentum simultaneously. But that's (more or less) true only in one interpretation, pilot wave. Most people wouldn't agree. This is called the "realism" assumption (in EPR). Its apparent violation is key to so-called quantum weirdness and can't be ignored even in elementary discussion. BTW, I don't feel QM is at all weird.

Point 2 emphasizes that "particles are particles". Personally, I have no problem with that, but again it's probably not mainstream. QFT represents particles as "excitations of the field".

Point 3 seems misleading. The correlation between the two entangled particles is "certain" - theoretically, at least. The value they have when measured is, as you indicate, uncertain. Finally, I'd say extra dimensions are out of scope for a simplistic explanation.

Apart from that it's on the right track!

 

[rubi]

How do you define "counterfactual definiteness"?

It means that you can assign values to unperformed measurements. Mathematically, it's just the requirement that you have functions $O_\xi : \Lambda\rightarrow \mathbb R$ from the space $\Lambda$ of states to the real numbers for all possible measurements $\xi$. If you want to prove Bell's theorem, it's enough to have these functions for all spin measurements $\xi=(\text{Alice},\alpha)$ of Alice and $\xi=(\text{Bob},\beta)$ for Bob. (Concretely, this means that the functions $A(\alpha,\lambda)$ and $B(\beta,\lambda)$ exist.)

What about the ones that don't?

I don't know a proof of the inequality that doesn't assume counterfactual definiteness.

 

[Nugatory]

What about the ones that don't?

Is there such a thing? Asking, not arguing.

 

[PeterDonis]

Is there such a thing?

It depends on what "assume locality" means. AFAIK every proof has some form of "factorizability" assumption for the probability distribution, but I don't know that all sources agree on whether that assumption captures "locality".

 

[rubi]

Is there such a thing? Asking, not arguing.

Yes, you can prove Bell-type inequalities for general random variables in a probability space. For example let $A,B,C,D:\Lambda\rightarrow\{-1,1\}$. Then it is easy to show that $\left|A(\lambda)B(\lambda)+A(\lambda)C(\lambda)+B(\lambda)D(\lambda)-C(\lambda)D(\lambda)\right|\leq 2$. Thus $\left|\left<AB\right>+\left<AC\right>+\left<BD\right>-\left<CD\right>\right|\leq 2$. It doesn't matter whether $A$, $B$, $C$ and $D$ represent locally separated observables of a physical theory or not.