How does Bell make this step in his proof?

[N88]

From drchinese website http://www.drchinese.com/David/Bell_Compact.pdf on page 406 there are 2 equations at the top. How do you get from the top one to the second one? There is a hint about using (1) but I think it cannot be done. You might be able to do it with other assumptions but I think they will be wrong too. Any suggestions?

 

[Nugatory]

I have taken the liberty of changing the thread title to reflect the question that's actually being asked.

 

[zonde]

There is a hint about using (1) but I think it cannot be done.

$A(\vec{a},\lambda)$ is either 1 or -1 so $[1-A(\vec{b},\lambda)A(\vec{c},\lambda)]$ is either 0 or 2. If for particular $\lambda$ it's 0 then expression in the first integral for the same $\lambda$ is 0 as well. If it's 2 then in first integral it's 2 or -2. So there is no way how module of the first integral can get bigger than the second integral.

 

[N88]

$A(\vec{a},\lambda)$ is either 1 or -1 so $[1-A(\vec{b},\lambda)A(\vec{c},\lambda)]$ is either 0 or 2. If for particular $\lambda$ it's 0 then expression in the first integral for the same $\lambda$ is 0 as well. If it's 2 then in first integral it's 2 or -2. So there is no way how module of the first integral can get bigger than the second integral.

Are you discussing how to get from the second equation to the third equation? My question is how to get from the first to the second.

 

[zonde]

My question is how to get from the first to the second.

You mean how to see that $A(\vec{a},\lambda)A(\vec{b},\lambda)-A(\vec{a},\lambda)A(\vec{c},\lambda)=-A(\vec{a},\lambda)A(\vec{b},\lambda)[A(\vec{b},\lambda)A(\vec{c},\lambda)-1]$ ?

 

[N88]

You mean how to see that $A(\vec{a},\lambda)A(\vec{b},\lambda)-A(\vec{a},\lambda)A(\vec{c},\lambda)=-A(\vec{a},\lambda)A(\vec{b},\lambda)[A(\vec{b},\lambda)A(\vec{c},\lambda)-1]$ ?

Thanks. Yes.

 

[zonde]

Thanks. Yes.

It's easy to see in opposite direction. Just notice that from (1) you get that $A(\vec{b},\lambda)^2=1$

 

[N88]

It's easy to see in opposite direction. Just notice that from (1) you get that $A(\vec{b},\lambda)^2=1$

But in the second equation, the first $A(\vec{b},\lambda)$ was tested by Bob when Alice's detector was set to $\vec{a}$. The second $A(\vec{b},\lambda)$ in the second equation can be tested by Bob when Alice's detector is at setting $\vec{c}$. How does Bob come to have the same particles to test to get $A(\vec{b},\lambda)^2=1$? Because doesn't that relation $A(\vec{b},\lambda)^2=1$ only hold for tests on the same λ?

Looks like there's another assumption somewhere?

 

[zonde]

But in the second equation, the first $A(\vec{b},\lambda)$ was tested by Bob when Alice's detector was set to $\vec{a}$. The second $A(\vec{b},\lambda)$ in the second equation can be tested by Bob when Alice's detector is at setting $\vec{c}$. How does Bob come to have the same particles to test to get $A(\vec{b},\lambda)^2=1$? Because doesn't that relation $A(\vec{b},\lambda)^2=1$ only hold for tests on the same λ?

Looks like there's another assumption somewhere?

$\lambda$ is independent of measurement settings. If follows from assumption that is stated at the top of page 404 (Alice's result does not depend on Bob's measurement setting and vice versa).

 

[DrChinese]

If (and this sequence works in either direction):
$-[A(\vec{a},\lambda)A(\vec{b},\lambda)-A(\vec{a},\lambda)A(\vec{c},\lambda)]=A(\vec{a},\lambda)A(\vec{b},\lambda)[A(\vec{b},\lambda)A(\vec{c},\lambda)-1]$

Then:
$-[A(\vec{a},\lambda)A(\vec{b},\lambda)-A(\vec{a},\lambda)A(\vec{c},\lambda)]=A(\vec{a},\lambda)A(\vec{b},\lambda)[-1 + A(\vec{b},\lambda)A(\vec{c},\lambda)]$

Then:
$A(\vec{a},\lambda)A(\vec{c},\lambda)=A(\vec{a},\lambda)A(\vec{b},\lambda)[A(\vec{b},\lambda)A(\vec{c},\lambda)]$

Then we are finally saying:
$1[A(\vec{a},\lambda)A(\vec{c},\lambda)]=A(\vec{b},\lambda)A(\vec{b},\lambda)[A(\vec{a},\lambda)A(\vec{c},\lambda)]=A(\vec{a},\lambda)A(\vec{b},\lambda)[A(\vec{b},\lambda)A(\vec{c},\lambda)]$

Which I think Bell took for granted we would follow, I guess with some reference to (1) - not sure. As zonde already mentioned, $A(\vec{b},\lambda) A(\vec{b},\lambda)$ is 1 classically. The above is a relationship that should be true if $\vec{a}, \vec{b}, \vec{c}$ are all simultaneously well defined. That assumption is essentially the assumption of realism - that $\vec{a}, \vec{b}, \vec{c}$ are well-defined at all times. If they are not, it is like dividing by zero and you get results that themselves are suspect. Ergo a Bell Inequality, which does not always hold.

 

[N88]

If (and this sequence works in either direction):
$-[A(\vec{a},\lambda)A(\vec{b},\lambda)-A(\vec{a},\lambda)A(\vec{c},\lambda)]=A(\vec{a},\lambda)A(\vec{b},\lambda)[A(\vec{b},\lambda)A(\vec{c},\lambda)-1]$
Then:

But here you have assumed on the RHS the equality that I want to derive from the LHS. My earlier post remains valid:

On RHS, the first $A(\vec{b},\lambda)$ was tested by Bob when Alice's detector was set to $\vec{a}$. The second $A(\vec{b},\lambda)$ in the second equation can be tested by Bob when Alice's detector is at setting $\vec{c}$. How does Bob come to have the same particles to test to get $A(\vec{b},\lambda)^2=1$? Because doesn't that relation $A(\vec{b},\lambda)^2=1$ only hold for tests on the same λ?

Looks like there's another assumption somewhere?[/QUOTE]

 

[N88]

$\lambda$ is independent of measurement settings. If follows from assumption that is stated at the top of page 404 (Alice's result does not depend on Bob's measurement setting and vice versa).

I agree. But to make sense of their test results, they need to be comparing paired results from tests on the same particle pair. And Bell and QM do not allow that you can test the same pair twice.

 

[zonde]

But to make sense of their test results, they need to be comparing paired results from tests on the same particle pair. And Bell and QM do not allow that you can test the same pair twice.

We can do that in Bell's model. In his model outcome of measurement is fully determined by measurement angle and $\lambda$ and of course we can copy value (or set of values) of lambda.
Of course we can't rewind back reality, change something and then look what is different. But our models are not copies of reality, they are just models. And we can do such things with models.

 

[rubi]

Of course we can't rewind back reality, change something and then look what is different. But our models are not copies of reality, they are just models. And we can do such things with models.

We can do it only in classical models. In quantum models, non-commuting observables don't simultaneously have well-defined outcomes, so that it's not possible to consistently assign values to unobserved quantities. This phenomenon is called contextuality. It's very apparent in the GHZ state, where any assignment of values to unbserved quantities is inconsistent.

 

[stevendaryl]

On RHS, the first $A(\vec{b},\lambda)$ was tested by Bob when Alice's detector was set to $\vec{a}$. The second $A(\vec{b},\lambda)$ in the second equation can be tested by Bob when Alice's detector is at setting $\vec{c}$. How does Bob come to have the same particles to test to get $A(\vec{b},\lambda)^2=1$? Because doesn't that relation $A(\vec{b},\lambda)^2=1$ only hold for tests on the same λ?

I'm not sure I understand your objection. Bell is assuming (for the purpose of showing that the assumption leads to a contradiction with the predictions of quantum mechanics) that when a twin-pair of particles is created, a value of $\lambda$ is associated with each particle, and that Alice's measurement result is a deterministic function of $\lambda$. She deterministically gets the result $A(\vec{a}, \lambda)$, where $\lambda$ is the hidden variable associated with the twin pair, and $\vec{a}$ is the axis she uses for the measurement. Similarly, Bob's measurement result is a different function of $\lambda$: He gets the result $B(\vec{b}, \lambda)$, where $\vec{b}$ is the axis he uses.

So $A(\vec{a}, \lambda)$ and $B(\vec{b}, \lambda)$ are assumed to be ordinary functions of two parameters. His proof then just amounts to showing that there are no such functions reproducing the predictions of quantum mechanics.

In reasoning about $A$ and $B$, you can forget that they correspond to measurements, and just think of them as mathematical functions.

 

[zonde]

We can do it only in classical models. In quantum models, non-commuting observables don't simultaneously have well-defined outcomes, so that it's not possible to consistently assign values to unobserved quantities. This phenomenon is called contextuality.

Quantum contextuality basically means that certain measurement results of individual particles can't be predetermined by hidden variables. This is the same conclusion that Bell reaches with his theorem.
However it does not mean that we can't calculate statistical predictions for alternative scenarios.

 

[rubi]

However it does not mean that we can't calculate statistical predictions for alternative scenarios.

It means that we cannot calculate statistical predictions for scenarios in which non-commuting observables have simultaneously well-defined outcomes. Hence the answer to post #12 is that this is where Bell's assumptions contradict QM, which eventually allows for the inequality violation.

 

[zonde]

It means that we cannot calculate statistical predictions for scenarios in which non-commuting observables have simultaneously well-defined outcomes. Hence the answer to post #12 is that this is where Bell's assumptions contradict QM, which eventually allows for the inequality violation.

OP has doubts whether Bell reasoning does not make additional assumption. So what is your answer? Is there additional assumption that does not follow from assumptions he has stated?

 

[rubi]

OP has doubts whether Bell reasoning does not make additional assumption. So what is your answer? Is there additional assumption that does not follow from assumptions he has stated?

Yes, Bell's paper is not very explicit about the assumptions he makes. This is the reason for the big confusion about his assumptions. Bell never states explicitely that the existence of values for unobserved variables is an assumption. On the contrary, he believes that this follows from the EPR argument. That's why I was pointing at the GHZ state, which shows that the EPR argument itself is inconsistent.

 

[zonde]

Yes, Bell's paper is not very explicit about the assumptions he makes. This is the reason for the big confusion about his assumptions. Bell never states explicitely that the existence of values for unobserved variables is an assumption. On the contrary, he believes that this follows from the EPR argument. That's why I was pointing at the GHZ state, which shows that the EPR argument itself is inconsistent.

Right at the start Bell says: "It is the requirement of locality, or more precisely that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past, that creates the essential difficulty."
Then later on he provides his reasoning how he arrives at "more complete specification of the state" described as lambda.
Then again he gives his assumption at the top of page 404 (that B's result does not depend on A's measurement setting and vice versa).

But as I understand you maintain position that Bell's reasoning is faulty and there can be independent measurements on two distant particles that with certainty will give (UP,DOWN) or (DOWN,UP) but not (UP,UP) nor (DOWN,DOWN) and yet it does not follow that we can meaningfully talk about UP or DOWN measurement for each particle separately. So it is reasonable option that combination of two independent parts taken together is meaningful while each independent part taken separately is not.

 

[DrChinese]

But here you have assumed on the RHS the equality that I want to derive from the LHS. My earlier post remains valid:

On RHS, the first $A(\vec{b},\lambda)$ was tested by Bob when Alice's detector was set to $\vec{a}$. The second $A(\vec{b},\lambda)$ in the second equation can be tested by Bob when Alice's detector is at setting $\vec{c}$. How does Bob come to have the same particles to test to get $A(\vec{b},\lambda)^2=1$? Because doesn't that relation $A(\vec{b},\lambda)^2=1$ only hold for tests on the same λ?

Looks like there's another assumption somewhere?

$\lambda$ is the same set of parameters or functions for settings a, b c. The "other" assumption is one I spelled out: realism. That a, b and c yield simultaneously well defined outcome functions. That assumption is the heart of the Bell proof, as can be seen.

You also say: "But to make sense of their test results, they need to be comparing paired results from tests on the same particle pair. And Bell and QM do not allow that you can test the same pair twice."

That statement is incorrect. Bell assumes you could have made a counterfactual measurement. That assumption leads to a contradiction with experiment (and with the predictions of QM). Yours is a red herring argument that is frequently made when discussing Bell. Bell assumes the classical position - in which case your objection wouldn't matter. The QM position is irrelevant to that view other than the fact that it makes a different prediction. But QM could be wrong and Bell would still set limits (the inequality), and that would still be contradicted by experiment. (Of course Bell didn't know that for sure back in the day, but we do.)

 

[rubi]

Right at the start Bell says: "It is the requirement of locality, or more precisely that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past, that creates the essential difficulty."

Yes, this is where he misses the possibility of contextuality. It was later shown by Stapp and Eberhard that this is an essential assumption of every Bell-type theorem. Theorems like GHZ or Kochen-Specker show that it is violated by QM. From Bell's theorem, we can only conclude that contextuality or locality (or both) is violated by QM.

But as I understand you maintain position that Bell's reasoning is faulty

No, it's not faulty. He just doesn't spell out the assumptions very explicitely. Original papers are rarely a good source for learning something.

and there can be independent measurements on two distant particles that with certainty will give (UP,DOWN) or (DOWN,UP) but not (UP,UP) nor (DOWN,DOWN) and yet it does not follow that we can meaningfully talk about UP or DOWN measurement for each particle separately. So it is reasonable option that combination of two independent parts taken together is meaningful while each independent part taken separately is not.

Bell assumes that there is a value up/down for each particle and each value of the angle $\vec a$, even for those that aren't measured. That's the assumption that isn't made very explicit in the paper and which is violated by QM, independently of Bell's result.

 

[DrChinese]

Yes, Bell's paper is not very explicit about the assumptions he makes. This is the reason for the big confusion about his assumptions. Bell never states explicitely that the existence of values for unobserved variables is an assumption. On the contrary, he believes that this follows from the EPR argument. That's why I was pointing at the GHZ state, which shows that the EPR argument itself is inconsistent.

I agree with rubi on this. Although the 2 equations at the top page of page 406 is where the realism assumption is introduced, Bell hardly makes note of it. He casually says "let c be another unit vector" when Bell might have said:

"Let's assume there are an infinite number of other unit vectors, c, d, e, etc. that are also defined by $\lambda$. Those would correspond to elements of reality as defined in the EPR paper. Using vector c, we would have:"

Had he done that, it would have clarified things for many a subsequent reader. However, at the time he wrote there were only a relatively small group of persons he was making his argument to, and he expected they would see his reasoning as presented. Apparently that was correct.

 

[DrChinese]

Right at the start Bell says: "It is the requirement of locality, or more precisely that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past, that creates the essential difficulty."
Then later on he provides his reasoning how he arrives at "more complete specification of the state" described as lambda.
Then again he gives his assumption at the top of page 404 (that B's result does not depend on A's measurement setting and vice versa).

But as I understand you maintain position that Bell's reasoning is faulty and there can be independent measurements on two distant particles that with certainty will give (UP,DOWN) or (DOWN,UP) but not (UP,UP) nor (DOWN,DOWN) and yet it does not follow that we can meaningfully talk about UP or DOWN measurement for each particle separately. So it is reasonable option that combination of two independent parts taken together is meaningful while each independent part taken separately is not.

As mentioned, I agree with rubi's take on Bell. Forget Bell's statement "It is the requirement of locality ... that creates the essential difficulty." The requirement of locality is not the essential difficulty, if anything it is the requirement of realism that is essential. That can be better seen in the GHZ proof, which does not depend on locality. The requirement of locality is of course needed for the Bell conclusion, as the EPR formulation required 2 separated particles and you must assume there is no interaction between them that allows $\lambda$ to include a remote measurement setting.

 

[zonde]

No, it's not faulty. He just doesn't spell out the assumptions very explicitely. Original papers are rarely a good source for learning something.

Bell assumes that there is a value up/down for each particle and each value of the angle $\vec a$, even for those that aren't measured. That's the assumption that isn't made very explicit in the paper and which is violated by QM, independently of Bell's result.

I will just quote Bell (emphasis mine): "Since we can predict in advance the result of measuring any chosen component of $\vec\sigma_2$, by previously measuring the same component of $\vec\sigma_1$, it follows that the result of any such measurement must actually be predetermined."

 

[DrChinese]

I will just quote Bell (emphasis mine): "Since we can predict in advance the result of measuring any chosen component of $\vec\sigma_2$, by previously measuring the same component of $\vec\sigma_1$, it follows that the result of any such measurement must actually be predetermined."

Exactly. The classical view is that a function of $\lambda$ at any and all vector settings are predetermines the outcome. That was the (wrong) EPR conclusion. They too assumed realism and locality. But they assumed a theory was possible that respected these and would replicate the predictions of QM. Obviously they couldn't know what we now know.

 

[zonde]

Exactly. The classical view is that a function of $\lambda$ at any and all vector settings are predetermines the outcome. That was the (wrong) EPR conclusion. They too assumed realism and locality. But they assumed a theory was possible that respected these and would replicate the predictions of QM. Obviously they couldn't know what we now know.

Well, Bell assumes locality and from that follows that measurements are independent no matter what measurement angle A or B chooses. And QM prediction says that perfect anti-correlations will be observed whenever A and B chooses the same measurement angle. There is no lambda yet.

 

[rubi]

Well, Bell assumes locality and from that follows that measurements are independent no matter what measurement angle A or B chooses. And QM prediction says that perfect anti-correlations will be observed whenever A and B chooses the same measurement angle. There is no lambda yet.

Forget the EPR argument. It might be intriguing, but it is falsified experimentally by GHZ. There are no elements of reality for unobserved spin directions, not even if you don't introduce $\lambda$. However, Bell clearly uses them in his proof.

 

[DrChinese]

And QM prediction says that perfect anti-correlations will be observed whenever A and B chooses the same measurement angle. There is no lambda yet.

Right. That is not the classical view (EPR) that Bell was writing about. QM does not posit predetermination of all outcomes. Just that a few will be perfectly correlated. The classical view goes much further. Specifically, that outcomes are predetermined at all possible measurement settings.

 

[zonde]

Forget the EPR argument. It might be intriguing, but it is falsified experimentally by GHZ. There are no elements of reality for unobserved spin directions, not even if you don't introduce $\lambda$. However, Bell clearly uses them in his proof.

Of course we all know that Bell inequalities are violated. As far as I know there is no loophole free experiment performed for GHZ but it's not very important.
Important question is from what assumptions one can construct Bell inequality. So that we know what assumptions have to be questioned when we learn about experimental violation of Bell inequalities. And this question can be discussed without much reference to experiments.