Bell's Theorem and Nonlocality

My I suggest that you go to the on line version of The Quantum Challenge, by George Greenstein and Arthur G. Zajonc, and see if you can cut and paste. I have a later version in front of me and so the page numbers are different, but if you could use that, I would be comfortable with the source. Suit your self, of course.

 

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I promise you nothing short of spectacular.

Before each measurement deflection angles a1 and a2 are randomly chosen from three constants A, B, C?

Single measurement consist of measuring two binary values v0 and v1, from two different particles p0 and p1?

After, say 100 measurements, I'm supposed to get some percentage of 25%, where it would normally be 35%?

What is this percentage we are calculating, what's the equation?

What are other details, check-boxes it must satisfy to pass?

 

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I have a question I toss out to the community while we wait. Is a hidden variables interpretation of QM the same as a hidden variables theory that can be tested using the postulates of quantum mechanics?

My answer is no, a hidden variables interpretation of QM says QM might be incomplete, and we don't know the particulars of how it is incomplete, while a hidden variables theory that makes a prediction as to percentages of outcomes over a large number of test is a hidden variables theory that can be tested. There is a clear distinction between the meanings of the words "hidden variables" in the two usages of the words.

The meaning of the words "hidden variables" when referring to a hidden variables interpretation means a theory of local reality that cannot be tested, and the meaning of the words "hidden variables" when referring to a hidden variables theory that can be tested using the postulates of QM refers to a prediction, like a percentage of outcomes over a large number of tests.

And finally, in a hidden variables theory, the words "hidden variable" also refer to the predicted percentage which the experiments are going to test for.

For an example of a hidden variable in that last sense, suppose we send six marbles to a friend, who randomly splits them up and places them in two boxes. He sends one box to LA, and one to NY. When the boxes are opened, the box in LA has four marbles. We don't have to open the box in NY because we know there were six to start with, we found four, and so we know there are two in the box in NY.

We repeat the experiment numerous times, and we always start with six, so if we look in one box, we always know how many will be found in the other box.

The percentage of marbles can be different for each experiment as long as the total equals six marbles. That percentage is referred to as a hidden variable of the particular test, and is one event in the testing of the theory which will include a sufficient number of events to allow the percentage in the theory to accumulate to a probability in which we have a high degree of confidence.

The reference to hidden variables when we are talking about a hidden variables interpretation of QM, or a hidden variables theory, or the hidden variable percentage is not same definition, in fact it is a different definition in all three cases.

For future reference, we will call a hidden variables interpretation an HVI, we will call a hidden variables theory that can be tested by the postulates of QM an HVT, and we will call the hidden variable percentage like in the marbles experiment simply a hidden variable or HV.

I want CptBork to acknowledge and agree with these different uses of the words hidden variable, and when asked, I want him to say which usage he is invoking when he uses the words "hidden variable", and I will agree to do the same.

 

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The constants A, B and C must have a \(120^\circ\) separations if you're looking to obtain the 25% value predicted by quantum mechanics, otherwise with different angular separations between them you get a different value in the actual experiments. The local hidden variables prediction is that at least 1/3 of the pairs (possibly slightly less, given small statistical fluctuations) must have a correlation when measured, so you have just a tiny bit more flexibility than 35%. Also the hidden variables of the two particles must be set so that \(v_0=v_1\) is always true without exception whenever \(a_0=a_1\). You can pick whatever scheme you want to select the hidden variables of the particles, but the values for \(a_0\) and \(a_1\) must be chosen by a legitimately unbiased random scheme, and I think you should run at least 1000 trials if not more to get a fair statistical sample.

 

They're most definitely not the same thing, because it's impossible to test anything using the postulates of QM, they're just postulates.

The whole point of Bell's Theorem is that any local hidden variable theory can be tested without knowing its specific details, because the very postulates of locality and determinism automatically lead to a deduced prediction which can be tested. So there's no distinction between a local hidden variable theory in general and one that can be tested and falsified by Bell and related experiments, and you don't need to assume anything about quantum mechanics in order to perform these experiments.

 

Good point. Are you in agreement that a hidden variables interpretation (HVI) says that QM might be incomplete, and we don't know the particulars of how it is incomplete?

Are you in agreement that if an experiment to test a hidden variables theory (HVT) uses pairs of photons created simultaneously with opposite polarization, and each sent down one of two paths, then those photons are considered entangled with their polarization superimposed?

Can you to confirm that the hidden variables theory (HVT) that can be tested is not the same as a hidden variables interpretation of QM (HVI) that cannot be tested?

 

Shouldn't v0 = -v1? I don't think it maters though, it just defines whether the two balls should be sent in opposite or parallel directions. So anyway, do we agree those values can be represented by billiard ball's spin around vertical axis?

Ok. I think we understand each other. I expect to make this at some point during the next three days. I don't want to program it myself, I want to use some free software so you can replicate the same thing yourself and have insight of and access to all the variables and equations used. Before I start it would be useful if someone can suggest or recommend some software for this, preferably where this kind of setup could be done with simple drag and drop. I've seen such software exist, but have no idea where it was or how it was called. In the meantime I'll see if maybe Wolfram can do it online, which would be ideally convenient for everyone else to check it out too. -- How many measurements they usually do in those experiments?

 

Why two billiard balls don't behave like quantum particles:

A pair of quantum particles interact and become entangled. Now you want to measure some property of both particles, like spin.
If the particles are spin-entangled then the measurements will be correlated (resp. anticorrelated).

If you prepare a pair of billiard balls so they spin in opposite or the same directions, there is no correlation, nor is there any interaction; summing their angular momenta gives you a classical result which is not zero whereas total angular momentum is always conserved in quantum interactions: if a spin-0 particle decays, emitting a particle plus antiparticle (an electron and a positron, say), their momenta sum to zero.

As CptBork points out, measurements of the classical angular momentum can always be made on either billiard ball, from any angle. So if you have them spinning initially in the same direction, you can measure them so they appear to be spinning in opposite directions by rotating the frame for the measurement of one of them by 180[sup]o[/sup]. This is not something you see in quantum measurements; correlation does not depend on the spatial orientation of the measurer/equipment.

You might as well just have one black and one white ball, then when you "choose" one of them, you know the color of the other. With otherwise identical spinning billiard balls with arbitrary spin axes, all you can tell is that both are spinning, there is no correlation.

But, hey, knock yourself out.

And p.s. the question "is entanglement a physical law" is moot: particles interact physically; entanglement is therefore a "law of interaction". If this is not the case, do particles interact? Does anything interact? What does "interact" mean, then?

 

In science it is what is called a theory, and it is very unlikely that with all of the unknowns involved in entanglement and superposition that it is moot. It is not a fact, it is not a law, it is a theory, and it is one we do not know how to explain.

 

Are you saying particle interactions is a theory, not a fact? Are you saying we don't know how to explain interactions between particles, it isn't something we can observe or measure? If so, that is obviously incorrect. It's so incorrect it denies the existence of electronic circuits, and light emitting diodes, in fact any kind of manmade device that emits visible light, in fact any device that emits radiation of any kind.

How could we have built the LHC if we can't observe particle interactions?

We don't know how to explain gravity or electric charge in terms of "interactions"? What, btw, are the unknowns involved in entanglement and superposition?

Note carefully, I will say this only once: particle interactions = entanglement.

 

Let's go right to the question of how can FTL communication occur? It is unknown. Quantum theory does not describe how particles can interact instantaneously, or how a particle can go through two slits, or how we could possibly alter the past as in delayed choice experiments. QM does not provide explanations of events, but only represents them in terms of probablilities.

 

So then, what does it mean if they still manage to produce the same result as quantum particles?

There are three constant preset polarization angles: A, B, C

If 'random number generator' happens to choose same polarization for the two particles, they will ALWAYS be correlated?

If 'random number generator' happens to choose different polarization for the two particles, they will NEVER be correlated?

It's a binary value like 1 and 0, plus and minus, or up and down. There is no "summing". It's "comparing", which too can only yield either true or false.

It seems balls or photons are completely irrelevant actually. The only variable I see in the whole setup is how many times will random number generator pick the same polarization angle for both particles. That is, random generator is the only thing influencing results. I made a program to test this, in plain C:

Code:

#include <time.h>

void main()
{
START:;

int a0, a1;
int i0, i1= 10000; //<-- CHANGE NUMBER of MEASUREMENTS
int hits= 0;

//initialize random seed
    srand (time(NULL));

    for(i0=0; i0 < i1; i0++)
    {
        a0 = rand() % 3;
        a1 = rand() % 3;

        printf("%d [%d:%d] ", i0, a0, a1);
        if(a0 == a1)
        {
            hits++;
            printf("SAME\n");
        }
        else
            printf("-xx- \n");
    }

printf("\nRESULT: %d", hits/100);
printf("\n\nPress a key to repeat.");
getch();
goto START;
}

Results:
100 measurements: 25% - 42%
1000 measurements: 30% - 35%
10000 measurements: 32% - 33%


With 100 measurements I got 25% several times in 5 minutes I tested it. Spooky, eh? Naturally it should converge to 33,333... as that's how 100/3 is. In other words there is 1 in 3 chance random number generator will choose the same polarization angle where there is only 3 possible to choose from. There are different types of random number generators, it is not uncommon they will produce different kinds of number distributions. I'd expect different random number generators could easily slide the result to either side of 33.33%. -- What kind of random number generators are used in these experiments? Do they ever publish that kind of information?

 

Not true. What is known is that communication cannot occur at FTL.
The appearance of communication is the "problem", but no actual communication occurs; it's our "need" to see a causal relation that is the real problem.

It doesn't have to explain any of that, but we do. One explanation is that all those effects are due to misunderstanding--particles don't interact instantaneously unless they're in the same location; a particle doesn't go through two slits because it isn't really a particle but a wave; we don't alter the past in a delayed choice experiment . . .

 

It may be settled for you and for me that way, but in the scientific community it is an either/or; either there is FTL communication, or there is no local reality.

I don't think it is true that the entire issue resolves to an "our need" by everyone who is not yet decided on the either/or. That is a little dismissive, but you will probably not agree.

That is true, and there are "weird" things about the quantum realm that we try to explain and can't.

I agree, except that I still an comfortable with wave/particle duality.

 

If two cars are each traveling at a speed of 60 MPH towards each other, the distance between them is closing at a rate of 120 miles per hour. Neither car is going faster than 60 MPH, but the distance between the cars is decreasing greater than 60 MPH. Of course, the distance between the cars is not an object traveling in space, so there is nothing traveling at 120 MPH.

 

Where have you got this from? Which part of the scientific community holds that local reality requires FTL communication?

I maintain that entanglement has nothing to do with what people have "decided" about it. Entanglement appears to be an aspect of the superposition of states which is independent of space and time. What we think are two separated particles is actually a superposition of quantum states; the particles seem to be localised and their wavefunction is delocalised, or spread out over some arbitrary distance. If there is any communication, it's the wavefunction communicating with itself.

 

I was thinking of polarization measurements where the two photons are created with the same polarity, but you can go for a spin-0 system of two spin-1/2 fermions or classical billiard balls with opposite spins if you want.

I would personally either do it in C++ or Matlab, looks like you've chosen the C++ route.

If you're using the same spin-axes for both particles with \(120^\circ\) separations, then you should get approximately 25% anti-correlations for the spin signs (+/-) in a spin-0 system when the two particles are measured on different axes, and 100% anti-correlations when the measurements are done on the same axis.

Correct, if you're measuring both particles on the same axis. Another caveat that follows as a corollary: the particle pair spins have to be decided independently of the measurement axis choices, since one particle in a local hidden variable theory won't have time to communicate the axis of its measurement to the other one and yet they must nonetheless always have opposite spins when measured on the same axis.

Correct, we only care whether a fermion is spinning parallel to or anti-parallel to a given axis, or whether a photon is or isn't polarized along a given plane.

You should look up Bell's Theorem on Wikipedia, it links to an article discussing tests of the theorem in which much of this info is mentioned. The random number generators have ranged from computers running pseudo-random algorithms, to quantum oscillators generating sequences as random as anything we know of.

 

It means you probably made a mistake.

Angular momentum is not a binary value. What you mean is the direction of spin can be like a binary value. You can ask a question like: "is the spin pointing up or down?" and get yes or no by measuring it.
As already pointed out, measuring the spin of a quantum particle is very different from measuring the spin of a billiard ball, or a top.


If you want to model quantum spin measurements (using a random number generator or whatever), wouldn't it be a good idea to first understand what quantum spin measurements are?

P.S. my earlier comment about angular momentum was more about how you can't have a billiard ball plus an anti-billiard ball whose momenta sum to zero. You can't place billiard balls in any kind of superposition.

 

Do you mean "should" as that is what actually happens in quantum experiments?

What about if the axis is different:
- If 'random number generator' happens to choose different polarizations for the two particles, they will NEVER be correlated? Or sometimes they will?

Spins have to be decided by what, how? Don't they always start with either always opposite, or always equal spins, depending on type of experiment?

I don't see any percentages like "25%" are mentioned in the whole of Wikipedia regarding Bell's theorem. Anyway, do you know specific number of measurements for some of those experiments?

Just by measuring how many times RNG produces matching angles, with 1,000 measurements I get 30% very often. Why is 25% spooky and 30% is not? It's at least a little a bit spooky, isn't it?

 

It is an either/or, as I call the position of those who hold to the Copenhagen interpretations of QM. I am under the impression that Copenhagen is the growing consensus in quantum physics. Do you know otherwise?

Interesting.