I A computational model of Bell correlations

[entropy1]

You need to actually try to rigorously formulate your requirement in a way that can be mathematically analyzed and used to make predictions. (That's what Bell was trying to do with his definition of "locality".)

If I am not mistaking, Bell's theorem deals with models of local hidden variables. But does it deal with non-locality?

 

[PeterDonis]

Bell's theorem deals with models of local hidden variables

Bell's Theorem defines "locality" as follows: the function that describes the correlation between the measurement results factorizes, i.e., it is the product of two functions, each of which depends on the measurement settings at only one of the two measurements. It includes hidden variables in the mathematical expressions it uses, but they are integrated over and do not play any essential part in the proof of the theorem. The key condition is factorizability.

does it deal with non-locality?

In the sense that "non-locality" means the locality condition above is violated (as it is in QM).

 

[entropy1]

Bell's Theorem defines "locality" as follows: the function that describes the correlation between the measurement results factorizes, i.e., it is the product of two functions, each of which depends on the measurement settings at only one of the two measurements.

So does that correspond with:

1. At A, we know if the particle is detected;
2. At A, we know the angle of the detector;
3. At B, we know if the particle is detected;
   
4. At B, we know the angle of the detector;
   
5. But the correlation is a function of the difference of both angles.

so to establish the correlation, if we adopt realism and take the outcome of the measurement at the instant it is measured, we need the difference between both angles, so it can't be a factorizable function?

 

[Zafa Pi]

That's because "find in" in two sentences have different meanings.
Let me rewrite your sentences like that:
I find that birds chirping takes place in the forest. I find that beauty takes place in the forest.
Now you can see that second sentence has lost it's meaning.

Sorry, I find it poetic and embued with meaning. Perhaps you've missed your calling.
Also the sentence "A disagreement has taken place between A and B. " is common place, unambiguous, with the same categories as the one above. Now perhaps you and @PeterDonis might want to ask where it takes place. Halfway between? Could the ratio of distances between be irrational? I have no good answers for such questions.

I hope you don't accuse me of being anti-Semantic, but I would prefer to check out of this topic and return to the happy place of QM.

 

[Jilang]

Bell's Theorem defines "locality" as follows: the function that describes the correlation between the measurement results factorizes, i.e., it is the product of two functions, each of which depends on the measurement settings at only one of the two measurements. It includes hidden variables in the mathematical expressions it uses, but they are integrated over and do not play any essential part in the proof of the theorem. The key condition is factorizability.

I am unsure why one would assume only one of the settings counts. Would not both play a part?

In the sense that "non-locality" means the locality condition above is violated (as it is in QM).

 

[PeterDonis]

does that correspond with

I think you're making this more difficult than it needs to be. Have you read Bell's paper?

Here is Bell's line of reasoning:

We have a measurement of particle A and a measurement of particle B. These measurements take place at spacelike separated events.

At each measurement, we have settings, call them $a$ and $b$.

If we do a large number of runs, we can construct a function $E(a, b)$ which gives the degree of correlation between the results as a function of the measurement settings.

Bell's locality condition is then that this function can be factorized: $E(a, b) = F(a) G(b)$. Notice that each factor is a function of only one of the two measurement settings. He calls this condition "locality" because it appears to capture our intuitive notion that the results at one measurement should not depend on the settings at the other measurement, since they are spacelike separated. However, in the end it's just a mathematical condition; whether or not you think "locality" is a proper name for it is a matter of words, not physics or math.

Bell then shows that if the function $E(a, b)$ factorizes in this way, it must obey certain inequalities.

We know, however, that the function $E(a, b)$ which is predicted by quantum mechanics violates these inequalities.

Therefore, the function $E(a, b)$ which is predicted by quantum mechanics cannot factorize in the way described above, which means it violates Bell's locality condition. Therefore, quantum mechanics is "nonlocal" in this sense.

 

[PeterDonis]

I am unsure why one would assume only one of the settings counts. Would not both play a part?

See my response to @entropy1 just now.

 

[entropy1]

@PeterDonis
I think we agree. Basicly we mean the same thing. Thanks.

 

[rubi]

Bell's locality condition is then that this function can be factorized: $E(a, b) = F(a) G(b)$.

Check the proof again. That's not Bell's locality condition. The locality condition is that $P$ factorizes, if conditioned on the hidden variable $\lambda$: $P(a,b|\lambda) = P(a|\lambda)P(b|\lambda)$. The assumption of a conditioning on $\lambda$ is a crucial part of the proof. In general, $E(a,b)$ doesn't even factorize in perfectly local classical scenarios such as Bertlmann's socks. QM violates the assumptions of Bell's theorem by not allowing for a conditioning on $\lambda$.

 

[PeterDonis]

The locality condition is that $P$ factorizes, if conditioned on the hidden variable $\lambda$: $P(a,b|\lambda) = P(a|\lambda)P(b|\lambda)$.

Yes, you're right; $\lambda$ is integrated over, but the factorization occurs inside the integral.

 

[rubi]

Yes, you're right; $\lambda$ is integrated over, but the factorization occurs inside the integral.

Right. I just wanted to point out that knowing the correlation $E(a,b)$ isn't enough to conclude anything about locality. One must also assume a specific class of underlying models that explain how $E(a,b)$ comes about. This class of models is given by the introduction of some arbitrary variables $\lambda$ and the integral you mentioned. Bell concludes that any such model must be non-local. However, he can't conclude something about models that don't belong to this class. In QM, the function $E(a,b)$ isn't given by an integral over some variables $\lambda$, but rather by the expectation value of a product of operators. One could conclude something about locality in general, if all models would belong to this class. However, QM is a counterexample that shows that Bell's class of models isn't completely general.

 

[Jilang]

I understand that the probabilities don’t factorise, but I am curious as to whether the probability amplitudes do. Is there a consensus on that?

 

[zonde]

Right. I just wanted to point out that knowing the correlation $E(a,b)$ isn't enough to conclude anything about locality. One must also assume a specific class of underlying models that explain how $E(a,b)$ comes about. This class of models is given by the introduction of some arbitrary variables $\lambda$ and the integral you mentioned. Bell concludes that any such model must be non-local. However, he can't conclude something about models that don't belong to this class. In QM, the function $E(a,b)$ isn't given by an integral over some variables $\lambda$, but rather by the expectation value of a product of operators. One could conclude something about locality in general, if all models would belong to this class. However, QM is a counterexample that shows that Bell's class of models isn't completely general.

Your argument that QM can be considered as a counterexample relies on subjective idea that QM actually explains how $E(a,b)$ comes about.
But certainly there are a lot of people who would consider only spacetime events based model as an explanation of $E(a,b)$.

 

[PeterDonis]

Bell concludes that any such model must be non-local.

More precisely, he concludes that any such model that can reproduce the predictions of QM must be non-local.

QM is a counterexample that shows that Bell's class of models isn't completely general

I think this depends on what interpretation of QM one adopts. According to the "shut up and calculate" interpretation, yes, QM does not belong to Bell's class of models. But in the de Broglie-Bohm interpretation, for example, it does--the hidden variables are the particle positions. But because the particle motions are affected by the quantum potential, they are nonlocal, so the function inside the integral over the hidden variables does not factorize, and the Bell inequalities are violated. In this interpretation, of course, the standard formalism of QM that uses operators and expectation values is not fundamental; it's emergent from the underlying model.

 

[rubi]

Your argument that QM can be considered as a counterexample relies on subjective idea that QM actually explains how $E(a,b)$ comes about.
But certainly there are a lot of people who would consider only spacetime events based model as an explanation of $E(a,b)$.

It has nothing to do with spacetime events. Bell's class of models doesn't mention spacetime events either and in QM, the detections are of course events in spacetime too. The question is only whether the detected values at each spacetime event arise as functions of a hidden variable or not.

More precisely, he concludes that any such model that can reproduce the predictions of QM must be non-local.

Yes, you're right of course.

I think this depends on what interpretation of QM one adopts. According to the "shut up and calculate" interpretation, yes, QM does not belong to Bell's class of models. But in the de Broglie-Bohm interpretation, for example, it does--the hidden variables are the particle positions. But because the particle motions are affected by the quantum potential, they are nonlocal, so the function inside the integral over the hidden variables does not factorize, and the Bell inequalities are violated. In this interpretation, of course, the standard formalism of QM that uses operators and expectation values is not fundamental; it's emergent from the underlying model.

That's also correct, but most people (even Einstein) don't consider dBB to be a serious solution. It's toy model and it serves it purpose as such, but it has way more conceptual problems than it solves and it doesn't generalize to QFT, so it is essentially falsified already as long as the Bohmians don't fix it. My point is that there exist models that explain the correlations without introducing hidden variables and since such models exist, we know that Bell's class of models isn't the most general class. Hence, the violation of the inequality doesn't imply anything about locality unless we accept extra assumptions.

 

[stevendaryl]

I understand that the probabilities don’t factorise, but I am curious as to whether the probability amplitudes do. Is there a consensus on that?

I posted a note about this a year or so ago. Here's the summary:

In a spin-1/2 anti-correlated EPR experiment, the joint probability that Alice will measure spin-up at angle \alpha and Bob will measure spin-up at angle \beta is given by:

P(A \hat B | \alpha, \beta) = \frac{1}{2} sin^2(\frac{\beta - \alpha}{2})

According to Bell, to explain this via local hidden variables would require a parameter \lambda and three probability distributions:

• P(\lambda)
• P(A | \alpha, \lambda)
• P(B | \beta, \lambda)

such that:

P(A \hat B | \alpha, \beta) = \sum_\lambda P(\lambda) P(A | \alpha, \lambda) P(B | \beta, \lambda)

(where the sum might be an integral, if \lambda takes on continuous values). If there were such probability distributions, then there would be a straight-forward local explanation of the EPR result:

1. Each pair of particles has a corresponding "hidden variable" \lambda selected according to the probability distribution P(\lambda)
2. When Alice measures her particle, she gets spin-up with probability P(A|\alpha, \lambda)
3. When Bob measures his particle, he gets spin-up with probability P(B|\beta, \lambda)
4. These probabilities are independent.

Alas, Bell showed that there were no such probability distributions. Now, you can ask the analogous question about amplitudes:

Let \psi(A \hat B | \alpha, \beta) be the amplitude (square-root of the probability, basically) that Alice and Bob will get spin-up at their respective angles. So \psi(A \hat B | \alpha, \beta) = \frac{1}{\sqrt{2}} sin(\frac{\beta - \alpha}{2}) (times a phase factor). Then the analogous question for amplitudes is:

Does there exist a parameter \lambda and amplitudes \psi(\lambda), \psi(A | \alpha, \lambda), \psi(B | \beta, \lambda) such that:

\psi(A \hat B | \alpha, \beta) = \sum_\lambda \psi(\lambda) \psi(A | \alpha, \lambda) \psi(B | \beta, \lambda) ?

The answer is straightforwardly "yes".

• Let \lambda have two possible values, \lambda_u and \lambda_d.
• Let the amplitude for these values be: \psi(\lambda_u) = \frac{1}{\sqrt{2}} and \psi(\lambda_d) = - \frac{1}{\sqrt{2}}
• Let the conditional amplitudes be:
  
  • \psi(A | \alpha, \lambda_u) = cos(\frac{\alpha}{2})
  • \psi(A | \alpha, \lambda_d) = sin(\frac{\alpha}{2})
  • \psi(B | \beta, \lambda_u) = sin(\frac{\alpha}{2})
  • \psi(B | \beta, \lambda_d) = cos(\frac{\beta}{2})

Then \psi(A \hat B | \alpha, \beta) = \sum_\lambda \psi(\lambda) \psi(A |\alpha, \lambda) \psi(B |\beta, \lambda)
= \psi(\lambda_u) \psi(A |\alpha, \lambda_u) \psi(B|\beta, \lambda_u) + \psi(\lambda_d) \psi(A |\alpha, \lambda_d) \psi(B|\beta, \lambda_d)
= \frac{1}{\sqrt{2}} (cos(\frac{\alpha}{2}) sin(\frac{\beta}{2}) - sin(\frac{\alpha}{2}) cos(\frac{\beta}{2})
= \frac{1}{\sqrt{2}} sin(\frac{\beta - \alpha}{2})

I don't know physically what it means that amplitudes, rather than probabilities factor, but it shows that quantum problems are often a lot simpler in terms of amplitudes.

 

[Zafa Pi]

In response to: @PeterDonis saying: More precisely, he [Bell] concludes that any such model that can reproduce the predictions of QM must be non-local.

Yes, you're right of course.

You guys are far from alone on this.
So after looking at Bell's 1964 paper I thought that he showed: {Locality & hidden variables} is incompatible with the predictions of QM.
I would naively conclude that that any model that can reproduce the predictions of QM must not have both locality & hidden variables. And also this seems to me not the same as being non-local. For example, have locality but not hidden variables.

Where have I gone wrong?

 

[rubi]

In response to: @PeterDonis saying: More precisely, he [Bell] concludes that any such model that can reproduce the predictions of QM must be non-local.

You guys are far from alone on this.
So after looking at Bell's 1964 paper I thought that he showed: {Locality & hidden variables} is incompatible with the predictions of QM.
I would naively conclude that that any model that can reproduce the predictions of QM must not have both locality & hidden variables. And also this seems to me not the same as being non-local. For example, have locality but not hidden variables.

Where have I gone wrong?

You're correct. "Such" in our sentences referred to hidden variables models, so we agree with you. No hidden variables model can be local, but other models can be.

 

[Nugatory]

Where have I gone wrong?

$\overline{A\cap{B}}$ implies $\bar{A}\cup\bar{B}$ but not $\bar{A}\cap\bar{B}$. However, the situation here is somewhat more complex because the assumptions Bell made in his original paper do not exactly correspond to "hidden variables" and "locality"; they're closer to "the properties that a theory must have to 'complete' (in the EPR sense) QM".

 

[OCR]

However, the situation here is somewhat more complex...

I rather think that Mr. Pi will not consider that fact appreciably more tolerable...

As a simpleton I appreciate simplicity.

 

[Zafa Pi]

You're correct. "Such" in our sentences referred to hidden variables models, so we agree with you. No hidden variables model can be local, but other models can be.

Thanks. Do you have a simple example of one of the other models? Does MWI qualify?

 

[Zafa Pi]

$\overline{A\cap{B}}$ implies $\bar{A}\cup\bar{B}$ but not $\bar{A}\cap\bar{B}$. However, the situation here is somewhat more complex because the assumptions Bell made in his original paper do not exactly correspond to "hidden variables" and "locality"; they're closer to "the properties that a theory must have to 'complete' (in the EPR sense) QM".

@OCR at #70 is almost right. Only I don't consider what you said to be a fact.
Bell defines locality on the first page, gets (2) as a consequence, has hidden variables λ throughout his proof. Many others think he assumes hidden variables and locality.
As a simpleton, I look forward to an explanation I can understand as to why they are not assumed.

 

[OCR]

Only I don't consider what you said to be a fact.

Fixed...!

 

[Jilang]

I posted a note about this a year or so ago. Here's the summary:

• P(A | \alpha, \la<br />

<br /> <br /> Does there exist a parameter \lambda and amplitudes \psi(\lambda), \psi(A | \alpha, \lambda), \psi(B | \beta, \lambda) such that:<br /> <br /> <ul> <li data-xf-list-type="ul">Let the conditional amplitudes be: <ul> <li data-xf-list-type="ul">\psi(A | \alpha, \lambda_u) = cos(\frac{\alpha}{2})</li> <li data-xf-list-type="ul">\psi(A | \alpha, \lambda_d) = sin(\frac{\alpha}{2})</li> <li data-xf-list-type="ul">\psi(B | \beta, \lambda_u) = sin(\frac{\alpha}{2})</li> <li data-xf-list-type="ul">\psi(B | \beta, \lambda_d) = cos(\frac{\beta}{2})</li> </ul></li> </ul><br /> .

<br /> Should the third one have a beta in it?

 

[Zafa Pi]

Fixed...!

So is my dog.
I agree with your edit.

 

[jk22]

Trying to reproduce the correlation we could try the following reasoning :

The model is : $$C(a,b)=\int \underbrace{A(a,x)\cdot B(b,x)}_{AB(a,b,x)}dx$$
more generally :

$$AB(a,b,x)=g(A(a,x),B(b,x))$$

numerically an exhaustive search over the functions : $$g,A,B : \{0,1\}^2->\{0,1\}$$ since a,b can take 2 values and x too for the spin 1/2 case (we could remap the names of the two angles)

The total number of functions AB is $$2^{(2^3)}=256$$

The decomposition above gave 192 functions generated.

Hence i came to the paradox : even if in this way 75% of the functions can be generated , the quantum real function shall be in the remaining.

But then this also lead to the fact that the decomposition $$AB(a,b,x)=g(h(A(a,x),B(b,x)),A'(a,x))$$
Gives 256 over 256 functions (but seems physically akward or unacceptable)?

I would like to try functions thst can take 3 values but on my pc it will last weeks...

 

[jk22]

Addendum, erratum :

In fact nonlocality is hidden here, since if the angles can take $$na,nb$$ values then the cube can be sliced in $$nb$$ functions of $$ a$$ and $$\lambda$$, hence it depends on $$a$$ and a property of b.