Bell's theorem and local realism

[atto]

QM, and in particular, the EPR experiment, does not provide us with such a set of 4 lists of numbers. That's because in a twin-pair experiment, the experimenters (call them Alice and Bob) must make a choice: For each run i of the experiment, Alice must decide whether to measure A_i, or to measure A'_i. She can't measure both. Similarly, Bob must decide whether to measure B_i or B'_i. He can't measure both.

The experiments must provide these lists, and this is just the raw data, measured during experiment on both sides.

So an EPR experiment, you don't get 4 lists of numbers, each of which is either -1 or +1. You get 4 values, 2 of which are +1 or -1, and 2 of which are ?, meaning unmeasured.

In that case you lose completely the context of these Bell-type inequalities.

If you assume that there really are 4 numbers for each i, and that those 4 numbers are either +1 or -1, but we just don't know what two of them are, that leads to a contradiction. But QM doesn't say that there are 4 numbers associated with each run. It only says there are two numbers, the numbers actually measured by Alice and Bob. To assume that there are 4 numbers goes beyond QM to some "hidden variables theory" that is supposed to explain QM. Bell proved that there is no such hidden variables theory. There is no way to replace the ? by +1 or -1 everywhere so that the statistics for unmeasured values obey the predictions of QM.

To such, let say: a free-version of the problem, applies quite different inequality,
and it has higher limit, because up to 4, thus QM still breaks nothing!

So QM, together with Bell's theorem, shows us that quantum measurements are not simply a matter of measuring a variable that had a pre-existing value whether you measured it or not. What is it, if not that? Well, that's the big question.

QM shows nothing special in this area. We know very well the formal mathematical truths are universal, unbreakable, indestructible.
The experimental tests/verification of the mathematical theorems are completely pointless.

 

[Nugatory]

The experimental tests/verification of the mathematical theorems are completely pointless.

The experimental tests are not verifying the correctness of the mathematical theorem - we know that it's correct (unless there's an error in the proof and no one has found one in the past century, so that's not a serious possibility).

The theorem is stated in the form (I've already posted a link to Bell's original paper, and for this discussion we probably need to focus on that) "If A then B", and therefore "If not B then not A". That's the theorem, and no one is arguing about it.

The purposes of the experiments is to discover whether B is false; if it is then the mathematical logic of the theorem "if not B then not A" tells us that A is false.

 

[atto]

The theorem is stated in the form (I've already posted a link to Bell's original paper, and for this discussion we probably need to focus on that) "If A then B", and therefore "If not B then not A". That's the theorem, and no one is arguing about it.

The purposes of the experiments is to discover whether B is false; if it is then the mathematical logic of the theorem "if not B then not A" tells us that A is false.

I don't know what represent the A, B.

The EPR-tests were designed to verify some inequalities, never the whole reality, nor the basics of math.

 

[DrChinese]

The point I am trying to make is about the failure of the freedom assumption, not about locality. ...

The "failure of freedom assumption" means that Alice and Bob's choice of measurement settings are not free. In that view, those too is a function of the parameters you claim are somehow tied up in the other parameters you are mentioning. But that cannot be! There is no known influence of those parameters on the human brain! (Except of course in superdeterminism.) And if you care to postulate some connection, it can be ("easily") falsified.

Just to remind everyone what is at stake here, let's use my usual example of Type II entangled photons with possible angle settings 0, 120 and 240 degrees. The QM prediction for Alice and Bob to match is 25% when their settings are different. The local realistic prediction is not less than 33%. So for an example where Alice is checking at 0 degrees and Bob is checking at 120 degrees for a run, we might expect something like this (and in this case Alice and Bob are told to make their setting choices according to DrChinese):

0 120 240 Alice&Bob Match / Total Matches / Total Permutations
+ - - 0 1 3
- + - 0 1 3
- - + 1 1 3
+ - + 0 1 3
+ - + 0 1 3
+ - - 0 1 3
+ + - 0 1 3
- + + 1 1 3
Total 2 8 24
(sorry these don't line up quite right)

Note that it is certainly possible to have Alice and Bob see 25% match rate (2 of 8 runs). But regardless of how you pick ‘em, the total match rate cannot be less than 33% (8 of 24 permutations, and note the 16 of the permutations are counterfactual). So whenever we say a local realistic theory is occurring, we have something like the above. And that means that there is something privileged about Alice and Bob’s choice of settings. That is because the 0/120 degree combination of settings has a 25% match rate (matching QM), while the 0&240 combo has a 37.5% match rate (3 of 8) and the 120&240 combo also has a 37.5% match rate (3 of 8). In any local hidden variable theory purporting to mimic QM via loopholes or failed implied assumptions, the true universe (including counterfactuals) cannot match the observed sample.

Now suppose Alice and Bob left their settings alone for long enough to have 1,000,000 runs instead of just 8. The Alice&Bob pair has 250,000 matches (same 25%) and this is the local realistic summary (give or take a few) when we extrapolate:

0&120: 250,000 of 1,000,000, or 25% (this is the Alice&Bob setting)
0&240: 375,000 of 1,000,000, or 37.5% (this is a counterfactual setting)
120&240: 375,000 of 1,000,000, or 37.5% (this is a counterfactual setting)

Clearly, there is something “preferred” about the Alice & Bob setting pair, else the results would be consistent! If you are getting 25% there, you are getting something much different on the counterfactual ones. Note that we have agreed that we get the same result when Alice and Bob make independent decisions (ignoring the instructions from DrChinese) and they make their decisions outside each others’ light cones. Weihs et al (1998).

So I appreciate that you are saying it is possible to have a violation of the freedom assumption if classical dependencies exist. But perhaps you can explain how, out of the 1000000 runs, the results match the QM prediction AND yet are wildly different from the local realistic average, using any classical idea at all. Because you are essentially saying that the results are observer dependent (Alice&Bob results are different from the 0&240 and 120&240 combos) while simulanteoulsy saying that the choice of Alice&Bob’s settings is correlated to DrChinese’s instructions above, transmitted through PhysicsForums.com via this post.

Wait, perhaps I have special powers! That would explain a lot. Or perhaps you can acknowledge that superdeterminism, that mystical theory which is yet to be unveiled, is nothing at all like determinism. And if we follow the requirements of superdeterminism to their logical conclusion, it will be seen that a local realistic rendering must bear elements that are unscientific by almost any standard.

 

[atto]

You just try to analyze the consequences of the impossible lists of outcomes/measurements, which break the inequality.
But these lists don't exist, fortunately, there is nothing to analyze.

Although, on the other hand, you can analyze this scenario.
We assume that Alice has a knowledge of the Bob's chooses;
for example she has a magic crystal ball, through which she sees images instantly from a distance, and so on.

That fantastic 'possibility' has been even implemented in many movies, for example: Star Trek, Stargate, etc. :)

 

[Nugatory]

I don't know what represent the A, B.

The EPR-tests were designed to verify some inequalities, never the whole reality, nor the basics of math.

Have you read the paper yet? (I'll repeat the link so people who new to this thread won't have to dig back through it to find it: http://www.drchinese.com/David/Bell_Compact.pdf)

When I say that the theorem is of the form If A then B, A represents the "vital assumption" stated after equation 1 in the paper, and formalized in the integral in equation 2; and B represents the inequality.

 

[atto]

Have you read the paper yet? (I'll repeat the link so people who new to this thread won't have to dig back through it to find it: http://www.drchinese.com/David/Bell_Compact.pdf)

When I say that the theorem is of the form If A then B, A represents the "vital assumption" stated after equation 1 in the paper, and formalized in the integral in equation 2; and B represents the inequality.

OK. This is a formal proof of the inequality.
So, it's true - it can't be violated in any way.

 

[rubi]

OK. This is a formal proof of the inequality.
So, it's true - it can't be violated in any way.

It can be violated by a theory in which $P(\vec a,\vec b)$ isn't of the form as given by equation (2). This is the case for QM.

 

[Nugatory]

OK. This is a formal proof of the inequality.
So, it's true - it can't be violated in any way.

No, it is a formal proof that if a certain precondition holds, then the inequality cannot be violated in any way.

It's like the Pythagorean Theorem, which says that if a triangle is a right triangle then the sum of the squares of the lengths of the two sides will equal the square of the length of the hypotenuse - it can be and is violated by any triangle that is not a right triangle.

The experiments that measure whether Bell's inequality is violated are analogous to measuring the sides of a given triangle to see if the sum of the squares of the lengths of the two shorter sides is equal the square of the length of the long side. If it's not, then the Pythagorean theorem tells us that that triangle is not a right triangle.

 

[stevendaryl]

The experiments must provide these lists, and this is just the raw data, measured during experiment on both sides.
...
QM shows nothing special in this area. We know very well the formal mathematical truths are universal, unbreakable, indestructible.
The experimental tests/verification of the mathematical theorems are completely pointless.

I think you are confused about this topic. You seem to be expressing a view that is at odds with what everyone else has said about Bell's inequality. As I said recently in a different thread, the fact that something is an establishment view doesn't make it right, but it makes Physics Forums the wrong place for you to be arguing about it.

 

[atto]

It can be violated by a theory in which $P(\vec a,\vec b)$ isn't of the form as given by equation (2). This is the case for QM.

No. This means only the condition can be violated, not the inequality itself.

Triangle has 3 sides - yes?
This means: triangle with 4 sides is imposible.

And the QM reasoning - logics work in this way:
the square is a triangle with 4 sides; so, this fact breaks the reality;
and we are very naive beings, because we always believe it's impossible. :)

 

[atto]

I think you are confused about this topic. You seem to be expressing a view that is at odds with what everyone else has said about Bell's inequality. As I said recently in a different thread, the fact that something is an establishment view doesn't make it right, but it makes Physics Forums the wrong place for you to be arguing about it.

I understand, but this is a fact:
there are no series, which can break the inequalities, the same the inequality has never been broken, despite the many sensational reports of experimenters.

 

[stevendaryl]

OK. This is a formal proof of the inequality.
So, it's true - it can't be violated in any way.

Let me try one more time. QM gives us a function

P(\vec{a}, \vec{b}) = the probability of Alice measuring spin-up in direction \vec{a} and Bob measuring spin-up in direction \vec{b}. This function is given by (in the spin-1/2 EPR case):

P(\vec{a}, \vec{b}) = \frac{1}{2} cos^2(\frac{\theta}{2})

where \theta = the angle between \vec{a} and \vec{b}. What Bell proved is that it is impossible to write this function in the form:

P(\vec{a}, \vec{b}) = \sum P_1(\lambda) P_A(\lambda, \vec{a}) P_B(\lambda, \vec{b})

where the sum ranges over possible values of the hidden variable \lambda, and P_1 is the probability for each value.

So Bell showed that the joint probability distribution did not "factor" into local probability distributions. He did not prove that the original probability distribution is impossible. Of course, it's possible, and experiments bear out that it correctly describes the EPR results.

 

[rubi]

No. This means only the condition can be violated, not the inequality itself.

Bell's inequality is $1+P(\vec b,\vec c) \geq \left|P(\vec a,\vec b)-P(\vec a,\vec c)\right|$. This can be violated by a function $P(\vec a,\vec b)$ that is not of the form $P(\vec a,\vec b) = -\int\mathrm d\lambda\rho(\lambda)A(\vec a,\lambda) A(\vec b,\lambda)$. For example if $P(\vec a,\vec b) = -2$, then the inequality says $-1 \geq 0$.

 

[stevendaryl]

I understand, but this is a fact:
there are no series, which can break the inequalities, the same the inequality has never been broken, despite the many sensational reports of experimenters.

Yes, Bell's theorem is a theorem. There is no way to produce 4 lists of numbers that violate his inequality. Everybody agrees with that. Quantum mechanics is not in violation of Bell's theorem, because it's a theorem, and you can't come up with a counter-example to a theorem. Bell's theorem, together with the predictions of QM, can be used to prove that there is no "local realistic" implementation of the predictions of QM.

QM does not contradict Bell's theorem. QM plus Bell's theorem contradicts local realism.

 

[atto]

Bell's inequality is $1+P(\vec b,\vec c) \geq \left|P(\vec a,\vec b)-P(\vec a,\vec c)\right|$. This can be violated by a function $P(\vec a,\vec b)$ that is not of the form $P(\vec a,\vec b) = -\int\mathrm d\lambda\rho(\lambda)A(\vec a,\lambda) A(\vec b,\lambda)$. For example if $P(\vec a,\vec b) = -2$, then the inequality says $-1 \geq 0$.

Of course. For example the inequality is easily breakable:
1 + x >= |y-z|; where: x,y,z - free, independent - arbitrary parameters.

In the oryginal inequality the x,y,z are inter correlated - dependend.

 

[atto]

Let me try one more time. QM gives us a function

P(\vec{a}, \vec{b}) = the probability of Alice measuring spin-up in direction \vec{a} and Bob measuring spin-up in direction \vec{b}. This function is given by (in the spin-1/2 EPR case):

P(\vec{a}, \vec{b}) = \frac{1}{2} cos^2(\frac{\theta}{2})

This is just the fantastic scenario, i mentioned earlier, ie. we assume the knowledge about the outcome on other arm... or maybe the setting angle alone will be sufficient.

So Bell showed that the joint probability distribution did not "factor" into local probability distributions. He did not prove that the original probability distribution is impossible. Of course, it's possible, and experiments bear out that it correctly describes the EPR results.

It's impossible - the measured series of {1,-1} don't break the inequality - with probability 1 exactly, and certainly.

 

[rubi]

Of course. For example the inequality is easily breakable:
1 + x >= |y-z|; where: x,y,z - free, independent - arbitrary parameters.

In the oryginal inequality the x,y,z are inter correlated - dependend.

The correlation of the x, y, z is exactly defined by the form of $P(\vec a,\vec b)$ that is given by the integral that I quoted earlier. So if we experimentally find that the inequality is broken, we have automatically ruled out all theories that require $P(\vec a,\vec b)$ to be of that form. However, we haven't ruled out QM, since QM doesn't require $P(\vec a,\vec b)$ to be of that form.

 

[atto]

QM does not contradict Bell's theorem. QM plus Bell's theorem contradicts local realism.

No. The realism is just the math.

The results of experiments rather show there must be an error in the realisation of the experiments... maybe in the further data processing.

 

[stevendaryl]

No. The realism is just the math.

The results of experiments rather show there must be an error in the realisation of the experiments... maybe in the further data processing.

Really, I'm going to have to ask you to stop posting on this topic. If you believe that the standard results are all wrong, Physics Forums is not the place to argue about them.

Personally, I don't think that you know what you're talking about, but this forum is not the place to argue about it.

I am notifying a moderator.

 

[rubi]

The results of experiments rather show there must be an error in the realisation of the experiments.

This claim is equivalent to the claim that there can be no consistent theory that doesn't predict $P(\vec a,\vec b) = -\int\mathrm d\lambda \rho(\lambda) A(\vec a,\lambda) A(\vec b,\lambda)$. Do you have any evidence for this bold claim?

 

[stevendaryl]

This claim is equivalent to the claim that there can be no consistent theory that doesn't predict $P(\vec a,\vec b) = -\int\mathrm d\lambda \rho(\lambda) A(\vec a,\lambda) A(\vec b,\lambda)$. Do you have any evidence for this bold claim?

I really don't think that Physics Forums is the correct avenue for breaking new results. I don't think it's appropriate to discuss this here. Atto's opinion is contrary to just about all published papers about Bell's theorem. So if there is anything to it, it's new research, and this is not a forum for new research.

Like Jeopardy, he should have put it in the form of a question: "I don't understand...how is Bell's theorem compatible with the predictions of QM?" instead of declaring that it's not.

 

[rubi]

I really don't think that Physics Forums is the correct avenue for breaking new results. I don't think it's appropriate to discuss this here. Atto's opinion is contrary to just about all published papers about Bell's theorem. So if there is anything to it, it's new research, and this is not a forum for new research.

I agree. When I started writing my post, yours wasn't there yet.

 

[atto]

Really, I'm going to have to ask you to stop posting on this topic. If you believe that the standard results are all wrong, Physics Forums is not the place to argue about them.

Personally, I don't think that you know what you're talking about, but this forum is not the place to argue about it.

I am notifying a moderator.

OK.
By the way: do not forget to ask the moderator for these fantastic binary series, which breaks the Bell's-type tautology or eventually the whole mathematical world, at least.

 

[Doc Al]

Closed pending moderation.