Understanding Bell Inequality Proof: Explained from a Probability Perspective

[cianfa72]

TL;DR Summary

About the proof of Bell inequality theorem

Starting from this link my understanding of Bell inequality proof goes as follows:
Suppose we have a model of local pre-determinate hidden variables for QM. This amounts to say QM objects are in pre-determinate given states even if we do not measure it. Locality just means that spacelike separated events cannot affect a given event like the measurement of a quantum state.

As explained in the link if the quantum states are preparated to give the expected result of QM applied to the measurement of the spin of a pair of entangled particles along different axis (probability $1/4$) then the Bell inequality for the sum of probabilities (always valid from a probability perspective) cannot be fullfilled.

Did I get it correctly? Thanks.

 

[PeterDonis]

Suppose we have a model of local pre-determinate hidden variables for QM. This amounts to say QM objects are in pre-determinate given states even if we do not measure it.

Yes, but these pre-determinate given states are not quantum states.

Locality just means that spacelike separated events cannot affect a given event

More precisely, it means that the probabilities for measurement results at each event can only depend on the settings at that event, not on the settings at the other spacelike separated event. The article you reference does not appear to mention that.

like the measurement of a quantum state.

You don't measure quantum states. You measure observables. In this particular case, you are measuring spin observables.

As explained in the link if the quantum states are preparated to give the expected result of QM applied to the measurement of the spin of a pair of entangled particles along different axis (probability $1/4$) then the Bell inequality for the sum of probabilities (always valid from a probability perspective) cannot be fullfilled.

This is true, but it's irrelevant to the derivation of the Bell inequalities or the proof of Bell's theorem. Bell's theorem itself was not about QM. It was about local hidden variable models; QM is not a local hidden variable model. The only relationship that Bell's theorem has to QM is that, since QM's predictions violate the Bell inequalities, QM must not satisfy at least one assumption that went into the proof of Bell's theorem; and that started a discussion in the literature, which is still ongoing, about which assumptions of the theorem QM violates--different QM interpretations make different claims about which one it is.

 

[cianfa72]

Bell's theorem itself was not about QM. It was about local hidden variable models; QM is not a local hidden variable model. The only relationship that Bell's theorem has to QM is that, since QM's predictions violate the Bell inequalities, QM must not satisfy at least one assumption that went into the proof of Bell's theorem

An example of QM prediction that violate Bell inequalities is the above mentioned measurement of the spin of entangled particles along different axis. According to local hidden variables models that probability on a long run of indipendent experiment/measurement (frequency of occurrences interpretation) has to be greater equal than $1/3$.

 

[PeterDonis]

An example of QM prediction that violate Bell inequalities is the above mentioned measurement of the spin of entangled particles along different axis.

Yes, that is well known. And, as I said, it is irrelevant to how the Bell inequalities are derived or how Bell's theorem is proved.

According to local hidden variables models that probability on a long run of indipendent experiment/measurement (frequency of occurrences interpretation) has to be greater equal than $1/3$.

Yes. But that has nothing to do with the QM prediction. See above.

 

[cianfa72]

Yes. But that has nothing to do with the QM prediction. See above.

Of course, the point is that instead the QM prediction for it is a probability of $1/4$ (therefore it violates the Bell inequality).

 

[PeroK]

An example of QM prediction that violate Bell inequalities is the above mentioned measurement of the spin of entangled particles along different axis. According to local hidden variables models that probability on a long run of indipendent experiment/measurement (frequency of occurrences interpretation) has to be greater equal than $1/3$.

Bell's inequality itself applies to possible correlations using classical probabilities. If we assume hidden variables then every entangled pair of particles must have one of the following eight configurations for the three angles:

		Particle 1				Particle 2
	A1	A2	A3		A1	A2	A3
P1	+	+	+		-	-	-
P2	+	+	-		-	-	+
P3	+	-	+		-	+	-
P4	+	-	-		-	+	+
P5	-	+	+		+	-	-
P6	-	+	-		+	-	+
P7	-	-	+		+	+	-
P8	-	-	-		+	+	+

In principle, these need not al be equally likely. In any case P1-P8 can be seen as the probability that a given entangled pair has each configuration. The key thing that Bell noted was:

If we measure the first particle's spin at angle A1 and the second particle at angle A2, then the probability that both are spin up is given by $P3 + P4$.

Likewise, if we measure the first particle's spin at angle A1 and the second particle at angle A3, then the probability that both are spin up is given by $P2 + P4$.

And, if we measure the first particle's spin at angle A3 and the second particle at angle A2, then the probability that both are spin up is given by $P3 + P7$.

Now, clearly, $P3 + P4 \le P2 + P4 + P3 + P7$. This is Bell's inequality. Which is fairly simple in itself.

The clever thing that Bell noticed is that for certain choices of angles A1, A2, A3, the predictions of QM violate this inequality. In fact, it's generally violated for a whole range of angles.

The required experiment, therefore, is to choose three such angles and repeat the experiment many times for each of the three options. If there were hidden variables, that would establish - with a certain statistical confidence - the probabilities $P3 + P4$, $P2 + P4$ and $P3 + P7$. The problem is that the experiment produces probabilities that do not satisfy that inequality and hence cannot be explained by hidden variables with positive real-valued probabilities.

Not only that, but the probabilities that are produced are in complete agreement with the predictions of QM. These were predictions that a) were on the table long before the experiment could be realised; and, b) were technically impossible given Einstein's view of nature. There is no fudge factor in QM predicting the outcome of this experiment.

At this point I would like to quote Scott Aaronson:

There’s the Bell experiment, which looks like Nature screaming the reality of ‘genuine indeterminism, as predicted by QM,’ louder than you might’ve thought it even logically possible for that to be screamed.

 

[PeterDonis]

Of course, the point is that instead the QM prediction for it is a probability of $1/4$ (therefore it violates the Bell inequality).

No, that's not the point, because this thread is not about the QM prediction, it's about the Bell inequalities and how they are derived and how Bell's theorem is proved. That's what you put in the title and the OP of this thread. The QM prediction is irrelevant to all of that, as I've already said. Please keep to the point of your own thread.

 

[cianfa72]

The required experiment, therefore, is to choose three such angles and repeat the experiment many times for each of the three options. If there were hidden variables, that would establish - with a certain statistical confidence - the probabilities $P3+P4$, $P2+P4$ and $P3+P7$. The problem is that the experiment produces probabilities that do not satisfy that inequality and hence cannot be explained by hidden variables with positive real-valued probabilities.

Therefore in principle, for a certain choice of angles A1, A2, A3, we perform the first experiment's type say a million of times (with a given distribution for probabilities P1-P8) calculating the resulting probability in terms of frequency of occurrences of the relevant event.

Next we repeat the second experiment's type another million of times with the same given distribution for probabilities P1-P8 calculating the relevant probability. The same for the third experiment's type.

As you pointed out the problem is that the probabilities resulting from these three runs of experiments do not satisfy the Bell's inequality.

 

[PeroK]

Therefore in principle, for a certain choice of angles A1, A2, A3, we perform the first experiment's type say a million of times (with a given distribution for probabilities P1-P8) calculating the resulting probability in terms of frequency of occurrences of the relevant event.

Next we repeat the second experiment's type another million of times with the same given distribution for probabilities P1-P8 calculating the relevant probability. The same for the third experiment's type.

As you pointed out the problem is that the probabilities resulting from these three runs of experiments do not satisfy the Bell's inequality.

Yes. In the experiments, the three tests (A1-A2, A1-A3, A3-A2) are themselves, randomly chosen, and eventually you have a large number of each. That shows that nature (whatever it is) cannot be based on any local hidden variables.

The second aspect of the experiment is that it directly confirms QM predictions.

 

[cianfa72]

Yes. In the experiments, the three tests (A1-A2, A1-A3, A3-A2) are themselves, randomly chosen, and eventually you have a large number of each. That shows that nature (whatever it is) cannot be based on any local hidden variables.

A point related to this: why we must use pairs of entangled particles (particle 1 and particle 2) to perform the experiments ? It is because even in local hidden variables model the measurement of the spin along an axis (say A1) affects the measurement of the spin along another axis (say A2) of the same particle ?

 

[PeroK]

A point related to this: why we must use pairs of entangled particles (particle 1 and particle 2) to perform the experiments ?

Quantum entanglement predicts non-classical results. Until Bell's theorem, however, local hidden variables couldn't be explicitly ruled out. That said, Bohr and Heisenberg etc. decided that QM must be correct long before Bell's inequality was thought of. Hidden variables already looks like a non-viable explanation for electron spin in the first place, IMO.

It is because even in local hidden variables model the measurement of the spin along an axis (say A1) affects the measurement of the spin along another axis (say A2) of the same particle ?

The correct term is correlation. In order to understand quantum entanglement and Bell's Theorem, you must understand correlation. The probabilities above are actually measurements of the amount of correlation between spin measurements about three different angles. Note that we cannot measure the same particle twice in the same state - because the first measurement destroys the original state. With entangled particles, we can measure the first about angle A1 and the second about angle A2. This gives us a measure of correlation between spin-up in the direction A1 and spin-up in the direction A2.

In fact, if you understand correlation and look at what QM predicts, then you may see immediately that the level of correlation is exceptionally (almost unbelievably) high. It looks impossible. And, in fact, this is what Bell managed to show: given classical local hidden variables, we have an impossibly high correlation.

 

[cianfa72]

Note that we cannot measure the same particle twice in the same state - because the first measurement destroys the original state. With entangled particles, we can measure the first about angle A1 and the second about angle A2.

Yes, this is the point I was making: even in the context of classical local hidden variables model is assumed that the first measurement destroys the original state (i.e. uncertainty principle applies).

 

[Nugatory]

A point related to this: why we must use pairs of entangled particles (particle 1 and particle 2) to perform the experiments ? It is because even in local hidden variables model the measurement of the spin along an axis (say A1) affects the measurement of the spin along another axis (say A2) of the same particle ?

Pretty much the exact opposite. A local theory is one in which the measurement of one particle does not affect the measurement of the other, and therefore the observed correlations must be the result of the two particles’ common origin.

 

[cianfa72]

Pretty much the exact opposite. A local theory is one in which the measurement of one particle does not affect the measurement of the other, and therefore the observed correlations must be the result of the two particles’ common origin.

I was referring to the measurements along different axis of the same particle.

 

[Nugatory]

I was referring to the measurements along different axis of the same particle.

Ah - right - never mind.

 

[PeroK]

I was referring to the measurements along different axis of the same particle.

That's basic electron spin. As I said, even that is hard to explain convincingly using hidden variables. Why do all the variables get reset in just the right way to mimic the QM spin states? It's technically possible, of course, but sounds quite artificial to me. The quantum spin states, on the other hand, quite naturally explain a sequence of spin measurements

 

[cianfa72]

That's basic electron spin. As I said, even that is hard to explain convincingly using hidden variables. Why do all the variables get reset in just the right way to mimic the QM spin states?

Could you be more specific on this point ? Thank you.

 

[vanhees71]

More importantly you can NOT measure more than one of the spin components of a particle, you can only measure the spin component along one direction. That's clear by thinking about, how you measure that component. Take the Stern-Gerlach experiment. There the spin measurement is through reflection in an appropriately taylored magnetic field (large homogeneous part in the direction you want to measure the spin and an inhomogeneous smaller part at the place of the electrons that makes the reflection), and you can choose the spin-determining large homogeneous part in one direction only, i.e., you can measure (or rather even select, i.e., prepare) the spin component along this so defined direction.

A very clear example for a Bell experiment with two spin 1/2 particles in the singlet state is explained in

J. J. Sakurai and S. Tuan, Modern Quantum Mechanics,
Addison Wesley (1993).

Of course you can only compare the statistical outcomes (in terms of the correlation function discussed there) of experiments that are possible in the real world as with classical probabilities (the "local realistic" model, where "realistic" means that all three spin components take alwasys determined values and "local" means that the measurements on the spin components at the two far distant particles do not influence each other) to the predictions of quantum mechanics, which lead to a violation of the Bell inequality valid for the correlation function when calculated in the "local realistic model" of classical probabilities.

That's the only important physics point of this famous paper(s) by Bell: that there is a contradiction between all arbitrary "local realistic theories of probabilities" and the "quantum theory of probabilities" when one experiments with highly correlated "Bell states" (i.e., entangled states like the spin-singlet state of a two-spin-1/2 systems or, in modern lingo, two qbits).

 

[PeroK]

Could you be more specific on this point ? Thank you.

If you measure the spin of an electron about some axis A1 and get a result of spin up, then the state of the electron is spin-up about axis A1. If you subsequently measure the spin about axis A2, which is set at an angle $\theta$ to A1, then the probability of getting spin -up (about A2) is $\cos^2(\dfrac \theta 2)$. This explains everything in terms of repeated spin measurements on a single particle and explains all the correlations in the tests of Bell's inequality.

This is also covered in Sakurai, as indicated above.

 

[Heidi]

I found the word proof in the title.
But i did not found the words wrong or false in the answers.

 

[PeroK]

I found the word proof in the title.
But i did not found the words wrong or false in the answers.

What does that mean?

 

[Heidi]

The Bell inequalties belong to a no-go theorem.
Mathematical tools are used: integrals, sums, products, complex numbers ans so on.

things like "suppose that there are hidden variables" or things like that are not enough to get inequalities. One has to define them in a mathematical model with operation that mix them (sums, products, integrals...)
Of course , as there is a violation of the inequalities, we have a wrong model for reality.
My question is to find what is the mathematical tool which is "wrongly" used
so that we get a false no-go theorem.

 

[PeroK]

Bell's theorem deals with natural phenomena. There need be no error in the mathematics for hidden variables to fail to predict the outcome of an experiment.

 

[Heidi]

Quantum phenomeana are natural phenomeana.
I did not say there is a mathematical error. i say that he uses a wrong model with rules that do not
agree with the quantum word.
Do you see where the flaw is ?

 

[PeterDonis]

a false no-go theorem

Bell's theorem is not a false no-go theorem. It's a valid theorem. QM simply does not satisfy at least one of the assumptions on which the theorem is based.

i say that he uses a wrong model with rules that do not
agree with the quantum word.

If you mean that QM violates the Bell inequalities, of course, we all know that. What's your point?

 

[Heidi]

Look a Bell's derivation:
https://en.wikipedia.org/wiki/CHSH_inequality
You say that of the assumtion is false (does not matches reality)
I think that it is when he uses
$E(a,b) = \int A(a,\lambda)B(b,\lambda) \rho (\lambda) d \lambda $
He sums over probabilities to get a mean value.
this model does not match the reality of a quantum word where amplitudes are summed.
Think of the double slit experiment. We would never get fringes

 

[PeterDonis]

You say that of the assumtion is false (does not matches reality)

Yes, that must be the case since reality matches the QM predictions, which violate the inequalities.

I think that it is...

There is an ongoing debate in the literature about exactly which assumptions QM violates. AFAIK there is no general resolution. In fact different QM interpretations give different answers.

 

[PeterDonis]

I think that it is when he uses
$E(a,b) = \int A(a,\lambda)B(b,\lambda) \rho (\lambda) d \lambda $
He sums over probabilities to get a mean value.

No, the integral there is not over probabilities, it's over results, which in the model he is using are deterministic, they depend on $\lambda$, so the expectation value is just the integral of the determined results as a function of $\lambda$, over all possible values of $\lambda$.

One of the common responses in the literature to which assumption QM violates is indeed this one, but not for the reason you give: for the reason that the results factorize: $A(a, \lambda)$ does not depend on $b$, the measurement settings at B, and $B(b, \lambda)$ does not depend on $a$, the measurement settings at A. On its face this is obviously violated by QM, since in QM the results A and B depend on the angle between the two measurement settings, which obviously cannot factorize.

 

[DrChinese]

Quantum phenomena are natural phenomena.
I did not say there is a mathematical error. i say that he uses a wrong model with rules that do not
agree with the quantum world.
Do you see where the flaw is ?

You have to look at things within the full context of the EPR Paradox (1935) and Bell's paper (1964). The proofs really assume you know this. And in today's world, they also assume you are aware of the experimental support going back to the era of Aspect (1981).

First, EPR successfully argued that entangled particles would indicate that the result of a measurement by Alice (say) would tell you the outcome of a later measurement (same basis) by Bob. That demonstrates an element of reality - perfect correlations. They (EPR) made several assumptions in their conclusion, not Bell! Those assumptions were: a) locality (obviously Einstein took this for granted); and b) that those "perfect correlations" would apply to all possible basis measurements simultaneously, even if they could not be experimentally demonstrated simultaneously. From EPR:

"One could object to this conclusion on the grounds that our criterion of reality is not sufficiently restrictive. Indeed, one would not arrive at our conclusion if one insisted that two or more physical quantities can be regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted. On this point of view, since either one or the other, but not both simultaneously, of the quantities P and Q can be predicted, they are not simultaneously real. This makes the reality of P and Q depend upon the process of measurement carried out on the first system, which does not disturb the system in any way. No reasonable definition of reality could be expected to permit this."

Bell challenged this very "reasonable" assumption mathematically. He asked the question: what happens if you attempt to assign values to more basis measurements than could be performed experimentally? We can make a prediction and test it for 2 values (say a separation of 120 degrees for entangled spin components), but we cannot measure all 3 simultaneously with only 2 particles. But if you follow EPR's assumption, as Bell did, you will see quickly that there are almost not values for such situations that are internally consistent.

Specifically, see after his (14), where Bell says "it follows that c is another unit vector". That c is in addition to a and b, which makes 3 possible observables. They can't all 3 produce a consistent average of results in agreement with QM under the EPR assumption(s). Not even if you give them values by hand. So it is Bell's introduction of more than one "element of reality" - usually called realism - that does EPR in.

You can try all you like to hand wave away the Bell conclusion ("No physical theory of local realism can ever reproduce all of the predictions of Quantum Mechanics"). But those arguments are only convincing to those who make them. The rest of the scientific world is well aware of both the insight and the limitations of Bell and Bell tests. Pretty well all so-called "loopholes" have been cleanly eliminated, although there was never much doubt there. And then there is GHZ from about 1989, which requires no statistics to disprove local realism.

 

[vanhees71]

Yes, that must be the case since reality matches the QM predictions, which violate the inequalities.There is an ongoing debate in the literature about exactly which assumptions QM violates. AFAIK there is no general resolution. In fact different QM interpretations give different answers.

I think there's a unique mathematical answer, independent of any interpretational sophistry.

The violation is due to what's quite vaguely called "realism" of Bell's class of "local realistic theories". It's the assumption that all observables always take determined values. In a probablistic context this implies that the statistical outcomes for measurements on any observables follow the Kolmogorov axioms, while in quantum theory the Kolmogorov axioms are valid only for measurements of compatible sets of obervables. See the proof for the most simple example for two qbits prepared in the singlet state, as demonstrated in Sakurai and Tuan, modern QM.

 

[PeterDonis]

I think there's a unique mathematical answer, independent of any interpretational sophistry.

As far as QM's predictions go, of course there is a "unique mathematical answer". All QM interpretations use the same math. Different QM interpretations are about different verbal descriptions of what the math is telling you, or why the math's predictions are correct.

The fact that you do not want to pay any attention to what you call "interpretational sophistry" does not mean the vast literature on QM interpretations does not exist or that nobody else wants to discuss it. If you have nothing substantive to say about QM interpretations, please just refrain from posting in this subforum.

 

[vanhees71]

As far as QM's predictions go, of course there is a "unique mathematical answer". All QM interpretations use the same math. Different QM interpretations are about different verbal descriptions of what the math is telling you, or why the math's predictions are correct.

The math has a unique meaning, i.e., it provides probabilities for the outcomes of measurements given the preparation of the measured system in the state.

I use a very specific interpretation here, i.e., the minimal statistical interpretation, which simply says that all there is we can know about a quantum system are the probabilities for the outcomes of measurements, predicted by the uniquely defined mathematical formalism. Why do you want to forbid discussing this single interpretation, which clearly says what you really observe in the lab?

The fact that you do not want to pay any attention to what you call "interpretational sophistry" does not mean the vast literature on QM interpretations does not exist or that nobody else wants to discuss it. If you have nothing substantive to say about QM interpretations, please just refrain from posting in this subforum.

Why do you again want to forbid me to post in this subforum, only because I critisize the one or the other interpretation? Isn't it the purpose of this subforum to discuss about interpretations? Why should then the minimal statistical interpretation be forbidden and all kinds of metaphysical additions be allowed?

 

[physika]

The math has a unique meaning, i.e., it provides probabilities for the outcomes of measurements given the preparation of the measured system in the state.

I use a very specific interpretation here, i.e., the minimal statistical interpretation, which simply says that all there is we can know about a quantum system are the probabilities for the outcomes of measurements, predicted by the uniquely defined mathematical formalism. Why do you want to forbid discussing this single interpretation, which clearly says what you really observe in the lab?

Why do you again want to forbid me to post in this subforum, only because I critisize the one or the other interpretation? Isn't it the purpose of this subforum to discuss about interpretations? Why should then the minimal statistical interpretation be forbidden and all kinds of metaphysical additions be allowed?

The point is you want to proclamate that the scheme (or interpretación or X or Y) that YOU propose is the definitive truth.
...By Fiat.

 

[PeterDonis]

The math has a unique meaning

With a particular interpretation, yes, as you acknowledge:

I use a very specific interpretation here, i.e., the minimal statistical interpretation

But there are other interpretations. You don't like them; we know that. But they exist.

Why do you again want to forbid me to post in this subforum, only because I critisize the one or the other interpretation?1

Because criticizing other interpretations is not what this subforum is for. It is for discussing what they say, not for claiming that interpretations other than your preferred one are wrong or misguided or whatever. The subforum guidelines state this explicitly.

Isn't it the purpose of this subforum to discuss about interpretations?

Discuss, yes. Criticize, no. See above.

Why should then the minimal statistical interpretation be forbidden

It's not. You are perfectly free to discuss what this interpretation says. Nobody has ever told you otherwise.

and all kinds of metaphysical additions be allowed?

Because discussion of any interpretation that is in the literature is allowed. And yet you keep protesting against any intepretation that is not your preferred one. That is not allowed.

 

[vanhees71]

Ok, but than you cannot forbid the minimal statistical interpretation to be discussed in this subforum.

If you forbid a discussion about why you follow the one but not the other interpretation, then you don't need to discuss interpretations at all.