Binney's interpretation of Violation of Bell Inequalities

[Derek Potter]

Have you considered multivalued branching functions? There is, for example, The Free-Will Function. Nobody knows how it arrives at a value and it changes each time you use it.

 

[stevendaryl]

Have you considered multivalued branching functions? There is, for example, The Free-Will Function. Nobody knows how it arrives at its value.

Well, this is a little off the track. The fact is that whether things are deterministic or not doesn't really affect Bell's inequality, if we assume that any nondeterminism is localized. A coin flip, or a choice made by a person using free will are localized. It doesn't make any difference for Bell's theorem whether the results are nondeterministic, or are just very difficult to predict.

 

[TrickyDicky]

Well, this is a little off the track. The fact is that whether things are deterministic or not doesn't really affect Bell's inequality, if we assume that any nondeterminism is localized. A coin flip, or a choice made by a person using free will are localized. It doesn't make any difference for Bell's theorem whether the results are nondeterministic, or are just very difficult to predict.

It is off the track, which wasn't really my intention, So back to Bell and determinism, you are saying that violating the inequalities doesn't imply non-determinism?

 

[Derek Potter]

Even if they are easy to predict the theorem doesn't say anything about the correlations unless the system fulfills the theorem's conditions. But to revert to the previous point, the mathematical representation of locality is not obvious. TrickyDicky seems to have sidetracked himself into determinism but I'm still not sure whether we got to the bottom of his previous question, how BI violation implies that F(A) must depend on b. Or conversely if F(A) does not depend on b then the BI must hold. It's probably elementary but I seem to have a blind spot to it. Or perhaps the way the Wikipedia[/PLAIN] article suddenly plunges into rather formal language like this:
"Implicit in assumption 1) above, the hidden parameter space Λ has a probability measure ρ and the expectation of a random variable X on Λ with respect to ρ is written

where for accessibility of notation we assume that the probability measure has a density ρ that therefore is nonnegative and integrates to 1"
... is a clue that it's not trivial (and perhaps that I should chew on it a bit more).

 

[stevendaryl]

It is off the track, which wasn't really my intention, So back to Bell and determinism, you are saying that violating the inequalities doesn't imply non-determinism?

Definitely not. The Bohm interpretation of QM shows that--it is deterministic.

 

[TrickyDicky]

Definitely not. The Bohm interpretation of QM shows that--it is deterministic.

I thought QM was not a deterministic theory.

 

[TrickyDicky]

It is quite possible that determinism-indeterminism is a semantic issue leading to dead ends if it depends on interpretations of a theory, so perhaps we can concentrate on the questions on #110 that have not been adressed as Derek suggests.

 

[atyy]

I thought QM was not a deterministic theory.

It is quite possible that determinism-indeterminism is a semantic issue leading to dead ends if it depends on interpretations of a theory, so perhaps we can concentrate on the questions on #110 that have not been adressed as Derek suggests.

I'm not sure I'm going to get this right, because it is tricky. But here is my try.

It's unclear whether there is such a thing as a fundamentally indeterministic theory. QM itself is not deterministic, but if it can be embedded in a deterministic theory, then the determinism is not fundamental. Similarly, it is unclear if there is such a thing as a fundamentally deterministic theory, since stochastic theories can be well approximated by deterministic theories in certain regimes. Bohmian mechanics constructs an explicit embedding of non-relativistic QM into a classical indeterministic theory which can be embedded into a deterministic theory.

However, there is a different version of Bell's theorem in which it can be shown that if a theory does not allow faster than light communication and violates the Bell inequalities, then the theory must be indeterministic in some sense.
http://arxiv.org/abs/quant-ph/0508016
http://arxiv.org/abs/0911.2504

As far as I can tell, there are 3 different definitions of locality in Wiseman's recent papers.

(1) signal locality
http://arxiv.org/abs/0911.2504 (Eq 8)

(2) locality
http://arxiv.org/abs/0911.2504 (Eq 7)
http://arxiv.org/abs/1402.0351 (Eq 2)

(3) local causality
http://arxiv.org/abs/1402.0351 Eq (4)

 

[TrickyDicky]

I'm not sure I'm going to get this right, because it is tricky. But here is my try.

It's unclear whether there is such a thing as a fundamentally indeterministic theory. QM itself is not deterministic, but if it can be embedded in a deterministic theory, then the determinism is not fundamental. Similarly, it is unclear if there is such a thing as a fundamentally deterministic theory, since stochastic theories can be well approximated by deterministic theories in certain regimes. Bohmian mechanics constructs an explicit embedding of non-relativistic QM into a classical indeterministic theory which can be embedded into a deterministic theory.

However, there is a different version of Bell's theorem in which it can be shown that if a theory does not allow faster than light communication and violates the Bell inequalities, then the theory must be indeterministic in some sense.
http://arxiv.org/abs/quant-ph/0508016
http://arxiv.org/abs/0911.2504

As far as I can tell, there are 3 different definitions of locality in Wiseman's recent papers.

(1) signal locality
http://arxiv.org/abs/0911.2504 (Eq 8)

(2) locality
http://arxiv.org/abs/0911.2504 (Eq 7)
http://arxiv.org/abs/1402.0351 (Eq 2)

(3) local causality
http://arxiv.org/abs/1402.0351 Eq (4)

Thanks for the references, the second paper, by Wiseman asserts: "Bell’s seminal 1964 paper shows that quantum correlations violate the conjunction of locality1 and determinism. However, there are quantum models that violate locality but maintain determinism (Bohmian mechanics is an example), and models that maintain locality but violate determinism (standard operational quantum theory is an example). Thus nothing can be concluded from Bell’s theorem about locality or determinism independently of each other."
The bolded part is specifically what I had in mind with my questions. Certainly this is not so clearly stated in the usual rendition of Bell's theorem where this subtle distinction about the relation of locality with the theorem is never made. If determinism is not fundamental it adds another layer of ambiguity to the already not clear cut meaning of the violation of the BE. Again I think that labeling a theory as "nonlocal" because it violates the BE without further qualifications as it's almost universally done when talking about Bell's theorem is highly misleading.

 

[atyy]

Thanks for the references, the second paper, by Wiseman asserts: "Bell’s seminal 1964 paper shows that quantum correlations violate the conjunction of locality1 and determinism. However, there are quantum models that violate locality but maintain determinism (Bohmian mechanics is an example), and models that maintain locality but violate determinism (standard operational quantum theory is an example). Thus nothing can be concluded from Bell’s theorem about locality or determinism independently of each other."
The bolded part is specifically what I had in mind with my questions. Certainly this is not so clearly stated in the usual rendition of Bell's theorem where this subtle distinction about the relation of locality with the theorem is never made. If determinism is not fundamental it adds another layer of ambiguity to the already not clear cut meaning of the violation of the BE. Again I think that labeling a theory as "nonlocal" because it violates the BE without further qualifications as it's almost universally done when talking about Bell's theorem is highly misleading.

Let's use the 3 definitions in post #58. Quantum mechanics is local and signal local, but it violates local causality, so it is in the third sense in which the violation of the Bell inequalities by quantum mechanics that renders it "nonlocal".

 

[TrickyDicky]

Let's use the 3 definitions in post #58. Quantum mechanics is local and signal local, but it violates local causality, so it is in the third sense in which the violation of the Bell inequalities by quantum mechanics that renders it "nonlocal".

Agreed, but let's not say "nonlocal" as it lends itself to confusion, maybe "nonlocally causal"?
I highly recommend the last article by Wiseman:"The two Bell's theorems of John Bell", it really clarifies things.

 

[TrickyDicky]

Then again we are faced with the problem that the elementary particles of the standard model are locally causal objects by definition while quantum field excitations are not, but everybody seems happy with this flagrant contradiction.

 

[atyy]

Agreed, but let's not say "nonlocal" as it lends itself to confusion, maybe "nonlocally causal"?
I highly recommend the last article by Wiseman:"The two Bell's theorems of John Bell", it really clarifies things.

Yes. Even "local causality" can be confusing, since "signal locality" also provides a notion of causality, eg. http://arxiv.org/abs/quant-ph/9508009v1.

Older terms are "local determinism" or "local realism", but those are also confusing since it is not clear how indeterminism or non-realism can save local causality after the Bell inequalities are violated - non-realism can save locality before (but not after) the Bell inequalities are violated.

The one I like best is "local explainability", but that is not so common, although it is mentioned by http://arxiv.org/abs/0909.0015, and fits in with http://arxiv.org/abs/1311.6852.

 

[atyy]

Then again we are faced with the problem that the elementary particles of the standard model are locally causal objects by definition while quantum field excitations are not, but everybody seems happy with this flagrant contradiction.

Well, terminology varies, but quantum mechanics and rigourous relativistic quantum field theory is simply not locally causal (unless one is using the less common definition of the term). The Bell theorem excludes this.

 

[TrickyDicky]

Well, terminology varies, but quantum mechanics and rigourous relativistic quantum field theory is simply not locally causal (unless one is using the less common definition of the term). The Bell theorem excludes this.

Well, rigourous relativistic quantum field theory will be not locally causal when(or if) it comes to existence some day.

 

[atyy]

Well, rigourous relativistic quantum field theory will be not locally causal when(or if) it comes to existence some day.

It already exists in 1+1 and 2+1 spacetime dimensions. The hunt is on in 3+1D.

 

[stevendaryl]

Let's use the 3 definitions in post #58. Quantum mechanics is local and signal local, but it violates local causality, so it is in the third sense in which the violation of the Bell inequalities by quantum mechanics that renders it "nonlocal".

I looked at the paper by Wiseman (here: http://arxiv.org/pdf/1402.0351v2.pdf) where he makes the distinction between locality and local causality, but I didn't say a statement of the definitions that showed how they were different. Equation (2) (Page 6) gives a definition of local, and equation (4) (Page 14) gives a "criterion" (not a definition, because it's not if and only if) for local causality. But they're the same equation! So what's the difference?

 

[atyy]

I looked at the paper by Wiseman (here: http://arxiv.org/pdf/1402.0351v2.pdf) where he makes the distinction between locality and local causality, but I didn't say a statement of the definitions that showed how they were different. Equation (2) (Page 6) gives a definition of local, and equation (4) (Page 14) gives a "criterion" (not a definition, because it's not if and only if) for local causality. But they're the same equation! So what's the difference?

Locality (Eq 2) doesn't have Alice's measurement outcome on the LHS, whereas local causality (Eq 4) does. So Eq 2 is just the reduced density matrix, which means that Bob's results don't depend on what Alice does. But Eq 4 is the nonlocal correlation, which means that if Bob knows Alice's results, he can sort his results and find perfect correlation.

 

[TrickyDicky]

I looked at the paper by Wiseman (here: http://arxiv.org/pdf/1402.0351v2.pdf) where he makes the distinction between locality and local causality, but I didn't say a statement of the definitions that showed how they were different. Equation (2) (Page 6) gives a definition of local, and equation (4) (Page 14) gives a "criterion" (not a definition, because it's not if and only if) for local causality. But they're the same equation! So what's the difference?

The difference is what I was referring to in the second paragraph of #110, it is explained in the paper since the second equation is <if> instead of <iff> like the first one, so local causality is weaker than locality. I think this distinction is obscured in many popular and also textbook(since people apparently knowledgeable about the theorem ignores this distinction) accounts of the theorem leading to empty debates and lots of unnecessary confusion. The discussions about BT I've witnessed so far (in PF and elsewhere) use the the strong definition without qualification in the place of the weak one which is the appropriate according to scholars experts in the theorem.
It would be interesting to know if Binney is using the stron or weak sense of locality when making his assertions, maybe in the light of it his view would be less controversial.

 

[Derek Potter]

It would be interesting to know if Binney is using the stron or weak sense of locality when making his assertions, maybe in the light of it his view would be less controversial.

What difference would it make? His theory doesn't work because his geometrical calculation is wrong, not because of any assumptions about locality.

 

[TrickyDicky]

What difference would it make? His theory doesn't work because his geometrical calculation is wrong, not because of any assumptions about locality.

I haven't found anything about Binney's geometrical calculations in the thread and Binney has no theory that I know of. Reading again the first posts Binney's quotes seem perfectly compatible with the indeterministic and local QM view of Bell's theorem, and all the comments about his "not even considering nonlocality" look like misled over-reactions, did you not read the Wiseman paper linked by atyy?

 

[atyy]

I haven't found anything about Binney's geometrical calculations in the thread. Reading again the first posts Binney's quotes seem perfectly compatible with the indeterministic and local QM view of Bell's theorem, and all the comments about his "not even considering nonlocality" look like misled over-reactions, did you not read the Wiseman paper linked by atyy?

To be honest, I didn't join the discussion till now because I have no idea what Binney is saying, let alone whether it is right or wrong, so I don't know if Binney is simply using locality as defined by Wiseman. Locality as defined by Wiseman is uncontroversially a property of quantum mechanics.

 

[Derek Potter]

I haven't found anything about Binney's geometrical calculations in the thread and Binney has no theory that I know of. Reading again the first posts Binney's quotes seem perfectly compatible with the indeterministic and local QM view of Bell's theorem, and all the comments about his "not even considering nonlocality" look like misled over-reactions, ...

I agree he does not set out his calculation, but he does spell out his model and asserts that it produces a result. So it is charitable to assume that there is a calculation and it is is based on the situation he describes:

Hence the most Alice can know about the orientation of the spin vector is that it lies in a particular hemisphere. Whatever hemisphere Alice determines, she can argue that the positron's spin lies in the opposite hemisphere. So if Alice finds the electron's spin to lie in the northern hemisphere, she concludes that the positron's spin lies in the southern hemisphere. This knowledge excludes only one result from the myriad of possibilities open to Bob: namely he cannot find Sz = +1/2. He is unlikely to find +1/2 if he measures the component of spin along a vector b that lies close to the z axis because the hemisphere associated with this result has a small overlap with the southern hemisphere, but since there is an overlap, the result +1/2 is not excluded.

Post #25 is my calculation based on what he says:
Construct a plane through a and b. Draw a circle and split it for the two "given" hemispheres. Choose a direction for a. This determines Alice's hemisphere and thus her result. Choose a direction for b, pointing the other way. Obviously b goes into the other hemisphere. Now allow b to swing round to some other angle. Recall that the electron spins are evenly distributed around 360 degrees. So the probabilty of a and b both lying in the same semicircle is proportional to the angle between them. No trigonometrical functions involved. Hence the anti-correlation will follow a linear rule, not the required [cosine] rule.

Unfortunately, although it produces a result, it is the wrong result. For small angles, the overlap of two hemispheres is proportional to the overlap. The quantum correlation is proportional to the overlap squared. Of course, if his words admit of another geometrical interpretation the exact kind of locality might be relevant but I cannot see how any other meaning is possible so it's pretty pointless discussing the finer details of a model which is simply wrong.

 

[Derek Potter]

Post #25

not the required cos^((b-a)/2) rule

Oops! Just say "cosine rule"

 

[TrickyDicky]

Here is my take on what Binney means just by the quotes in the OP, (I must say I don't care if he is right or wrong, had never heard about him):
There are two key points in what he says that I'm guiding my opinión on, he says literallythat no hidden variables can explain violations of BI, which I take as a clear rejection of determinism, and in the example used by Derek in #25 he doesn't say one can predict or calculate anything based simply on anything fixed at the outset, if one did like Derek did would find wrong results but he only talks about "consistence", by which I understand he means that the indeterminism he endorses allows the principle of locality to hold.

 

[Derek Potter]

Here is my take on what Binney means just by the quotes in the OP, (I must say I don't care if he is right or wrong, had never heard about him):
There are two key points in what he says that I'm guiding my opinión on, he says literallythat no hidden variables can explain violations of BI, which I take as a clear rejection of determinism, and in the example used by Derek in #25 he doesn't say one can predict or calculate anything based simply on anything fixed at the outset, if one did like Derek did would find wrong results but he only talks about "consistence", by which I understand he means that the indeterminism he endorses allows the principle of locality to hold.

They're consistent with there being a g in my middle name too but that is just as irrelevant.

 

[TrickyDicky]

They're consistent with there being a g in my middle name too but that is just as irrelevant.

Do you see the difference between claiming one can calcule the results of Bob from knowing that the hemisphere containing the positron's spin is fixed at the outset vs claiming that "a posteriori" (after the fact) the results are consistent with it being fixed at the outset and unaffected by Alice's measurement?

 

[stevendaryl]

Locality (Eq 2) doesn't have Alice's measurement outcome on the LHS, whereas local causality (Eq 4) does. So Eq 2 is just the reduced density matrix, which means that Bob's results don't depend on what Alice does. But Eq 4 is the nonlocal correlation, which means that if Bob knows Alice's results, he can sort his results and find perfect correlation.

Thanks! I stared at the two equations and didn't see that difference.

But now I'm having the opposite problem. It seems to me that if locality is violated that that would imply the possibility of signaling. If Bob's result depends on Alice's detector setting, then wouldn't Alice be able to signal to Bob by varying her setting?

Ah, nevermind. I see the difference: Equation 2 is this:

P_\theta(B|a,b,c,\lambda) = P_\theta(B|b,c,\lambda)

If that is violated, then Alice's setting affects Bob's result, but only if \lambda is kept fixed. Since \lambda is a hidden variable, Alice has no way of taking \lambda into account in signalling to Bob, and Bob has no way of taking \lambda into account in interpreting Alice's signal. So as far as signalling, it's sort of like cryptography using a one-time pad of random bits. If you have an independent way of knowing the pad, then you could use it to send messages. But if you don't know the pad, then messages will look like random noise.

 

[Derek Potter]

Do you see the difference between claiming one can calcule the results of Bob from knowing that the hemisphere containing the positron's spin is fixed at the outset vs claiming that "a posteriori" (after the fact) the results are consistent with it being fixed at the outset and unaffected by Alice's measurement?

Yes.

 

[Derek Potter]

the overlap of two hemispheres is proportional to the overlap. The quantum correlation is proportional to the overlap squared.

I mean the Binney correlation is proportional to the overlap!