Binney's interpretation of Violation of Bell Inequalities

[stevendaryl]

Would you say that a geometrical feature like say equality of right angles implying isotropy of the space and that figures may be moved to any location and still be congruent is a local or a nonlocal property? Does the fact that figures with arbitrary spacelike separation can be measured to be congruent if they are put together again mean that measuring one figure is affecting the measure of the other figure making it congruent with itself?
Now this particular geometrical property is commutative so it would not violate the inequalities, but a non-commutative geometrical property like euclidean rotations is not commutative and does violate the inequalities, the spatial separation(local-vs nonlocal) is not the important feature for such properties because they are pervasive(ubiquitous) and independent of spatial separation of the objects on which it is observed.
Is this more understandable?

It isn't to me. As I said, Bell's inequalities are about a very specific situation: Alice has a device that is capable of producing a measurement result of \pm 1. The device has (at least) two possible settings, a or a'. Bob similarly has a device with two possible settings, b or b'. For many rounds, you perform the following procedure and collect statistics:

On round number n,

1. Alice chooses a setting a_n.
2. She performs a measurement, and gets a result A_n
3. Bob chooses a setting b_n.
4. Bob gets result B_n

Then we compute: E(a,b) = the average, over all rounds n such that a_n = a and b_n = b, of A_n \cdot B_n

That's the context for which Bell derived his inequality. If you allow for Alice's choice a_n to affect Bob's result, B_n, and vice-verse, then there is no reason to expect the inequality to hold. That's where locality comes in: without assuming locality, there is no reason to assume that the inequality holds. Locality in the context of Bell's theorem means a very specific thing: that Alice's choice cannot affect Bob's result, and vice-verse. So your musings about whether congruence of geometric figures is local or nonlocal don't seem to be related to Bell's notion of locality.

Bell defined a hidden variables theory to be an explanation of the correlations along the following lines:

1. Each round, there is a hidden variable \lambda_n influencing the results. The variable may take on different values on different rounds (hence the subscript).
   
2. There are deterministic functions F_A(a, b, \lambda) and F_B(a, b, \lambda) such that A_n = F_A(a_n, b_n, \lambda_n) and B_n = F_B(a_n, b_n, \lambda_n)
3. Each round, \lambda_n is chosen randomly according to some probability distribution P(\lambda)

Those three define what Bell means by a "hidden variables theory". The special case of a "local" hidden variables theory makes the additional assumption that F_A does not depend on b and F_B does not depend on a. That is,

A_n = F_A(a_n, \lambda_n) and B_n = F_B(b_n, \lambda_n)

That's the critical assumption that allows him to derive his inequality. Right off the bat, I don't see how his derivation has anything to do with whether a and b are described by a commutative or noncommutative algebra.

Now, what someone has shown is that QM only predicts a violation of Bell's inequality in the case where the two measurements corresponding to Alice's settings a and a' are described by noncommuting operators, and the two measurements corresponding to Bob's settings b and b' are described by noncommuting operators. But that's a fact about quantum mechanics. Bell's derivation doesn't (as far as I can see) assume anything at all about whether things commute or not. The noncommutativity is about the two choices that Alice (or Bob) might make, not about Alice's measurements versus Bob's. Alice's measurements do commute with Bob's measurements.

 

[stevendaryl]

Bell defined a hidden variables theory to be an explanation of the correlations along the following lines:

1. Each round, there is a hidden variable \lambda_n influencing the results. The variable may take on different values on different rounds (hence the subscript).
   
2. There are deterministic functions F_A(a, b, \lambda) and F_B(a, b, \lambda) such that A_n = F_A(a_n, b_n, \lambda_n) and B_n = F_B(a_n, b_n, \lambda_n)
3. Each round, \lambda_n is chosen randomly according to some probability distribution P(\lambda)

Those three define what Bell means by a "hidden variables theory". The special case of a "local" hidden variables theory makes the additional assumption that F_A does not depend on b and F_B does not depend on a. That is,

A_n = F_A(a_n, \lambda_n) and B_n = F_B(b_n, \lambda_n)

I should say that the above form for F_A and F_B are not the most general form compatible with local realism. More generally, the outcome A might also depend on an additional hidden variable \alpha_n that is local to Alice, and Bob's outcome B might depend on a hidden variable \beta_n that is local to Bob. But the fact that in an EPR-type experiment, there are perfect correlations at some settings implies that there can't be any dependency on additional local variables.

 

[TrickyDicky]

It isn't to me. As I said, Bell's inequalities are about a very specific situation: Alice has a device that is capable of producing a measurement result of \pm 1. The device has (at least) two possible settings, a or a'. Bob similarly has a device with two possible settings, b or b'. For many rounds, you perform the following procedure and collect statistics:

On round number n,

1. Alice chooses a setting a_n.
2. She performs a measurement, and gets a result A_n
3. Bob chooses a setting b_n.
4. Bob gets result B_n

Then we compute: E(a,b) = the average, over all rounds n such that a_n = a and b_n = b, of A_n \cdot B_n

That's the context for which Bell derived his inequality. If you allow for Alice's choice a_n to affect Bob's result, B_n, and vice-verse, then there is no reason to expect the inequality to hold. That's where locality comes in: without assuming locality, there is no reason to assume that the inequality holds. Locality in the context of Bell's theorem means a very specific thing: that Alice's choice cannot affect Bob's result, and vice-verse. So your musings about whether congruence of geometric figures is local or nonlocal don't seem to be related to Bell's notion of locality.

Bell defined a hidden variables theory to be an explanation of the correlations along the following lines:

1. Each round, there is a hidden variable \lambda_n influencing the results. The variable may take on different values on different rounds (hence the subscript).
   
2. There are deterministic functions F_A(a, b, \lambda) and F_B(a, b, \lambda) such that A_n = F_A(a_n, b_n, \lambda_n) and B_n = F_B(a_n, b_n, \lambda_n)
3. Each round, \lambda_n is chosen randomly according to some probability distribution P(\lambda)

Those three define what Bell means by a "hidden variables theory". The special case of a "local" hidden variables theory makes the additional assumption that F_A does not depend on b and F_B does not depend on a. That is,

A_n = F_A(a_n, \lambda_n) and B_n = F_B(b_n, \lambda_n)

That's the critical assumption that allows him to derive his inequality. Right off the bat, I don't see how his derivation has anything to do with whether a and b are described by a commutative or noncommutative algebra.

Now, what someone has shown is that QM only predicts a violation of Bell's inequality in the case where the two measurements corresponding to Alice's settings a and a' are described by noncommuting operators, and the two measurements corresponding to Bob's settings b and b' are described by noncommuting operators. But that's a fact about quantum mechanics. Bell's derivation doesn't (as far as I can see) assume anything at all about whether things commute or not. The noncommutativity is about the two choices that Alice (or Bob) might make, not about Alice's measurements versus Bob's. Alice's measurements do commute with Bob's measurements.

Is F_A× F_B the same as F_B× F_A for the inequalities ?

 

[stevendaryl]

Is F_A× F_B the same as F_B× F_A for the inequalities ?

Yes, because they are real numbers, not operators.

 

[TrickyDicky]

Yes, because they are real numbers, not operators.

Got it, thanks!

 

[Pat71]

No, it counters Popper's original claim that QM's indeterminacy(HUP) was falsified by a certain outcome of his thought experiment. Both Shih's 1999 and more recent 2015(see http://phys.org/news/2015-01-popper-againbut.html) papers agreed on this too.

Interesting...observations suggesting nonlocal interference between randomly paired - i.e. not pre-entangled - photons??

 

[Derek Potter]

Do you understand the experiment, Pat? I can't make sense of the article.

 

[morrobay]

It isn't to me. As I said, Bell's inequalities are about a very specific situation: Alice has a device that is capable of producing a measurement result of \pm 1. The device has (at least) two possible settings, a or a'. Bob similarly has a device with two possible settings, b or b'. For many rounds, you perform the following procedure and collect statistics:

On round number n,

1. Alice chooses a setting a_n.
2. She performs a measurement, and gets a result A_n
3. Bob chooses a setting b_n.
4. Bob gets result B_n

Then we compute: E(a,b) = the average, over all rounds n such that a_n = a and b_n = b, of A_n \cdot B_n

That's the context for which Bell derived his inequality.

Those three define what Bell means by a "hidden variables theory". The special case of a "local" hidden variables theory makes the additional assumption that F_A does not depend on b and F_B does not depend on a. That is,

A_n = F_A(a_n, \lambda_n) and B_n = F_B(b_n, \lambda_n)

That's the critical assumption that allows him to derive his inequality.

In summery then the above applies to this inequality : (AB) + (AB') + (A'B) - (A'B') ≤ 2
where A,A',B,B' = ± 1
With assumptions that p (a,b) depend only on past variable λ and local measurements at x and y
p(ab|xy,λ) = p(a|x,λ) p(b|y,λ)

 

[Pat71]

Do you understand the experiment, Pat? I can't make sense of the article.

If you can't, Derek, I don't think there's much chance I can!

It does say:

"The randomly paired photons do not have any pre-prepared entanglement, which means they are considered to be a classical system. This raises the question, how could a classical system produce the same result as a quantum system?"

"...The observations suggest that the photon pair is interfering with itself instantaneously in a phenomenon called nonlocal interference. In the new experiment, the randomly paired photons have two different yet indistinguishable properties to be simultaneously annihilated at two distant photodetectors. The observations are the result of the superposition of these two probability amplitudes."


Quote from the abstract of the paper itself (I can't access the full text without buying it):

"Although the observation cannot be considered as a violation of the uncertainty relation as Popper believed, this experiment reveals a concern about nonlocal interference of a random photon pair, which involves the superposition of multi-photon amplitudes, and multi-photon detection events at a distance."

 

[TrickyDicky]

That's where locality comes in: without assuming locality, there is no reason to assume that the inequality holds. Locality in the context of Bell's theorem means a very specific thing: that Alice's choice cannot affect Bob's result, and vice-verse. So your musings about whether congruence of geometric figures is local or nonlocal don't seem to be related to Bell's notion of locality.Those three define what Bell means by a "hidden variables theory". The special case of a "local" hidden variables theory makes the additional assumption that F_A does not depend on b and F_B does not depend on a. That is,

A_n = F_A(a_n, \lambda_n) and B_n = F_B(b_n, \lambda_n)

That's the critical assumption that allows him to derive his inequality. Right off the bat, I don't see how his derivation has anything to do with whether a and b are described by a commutative or noncommutative algebra.

I still see something confusing about what it implies to assume locality, if we agree on defining locality for two spacelike separated subsystems as the absence of influence of measurement of one subsystem on the other, I'm not sure how the expression above exhausts this definition.
It is obvious that assuming that F_A does not depend on b and F_B does not depend on a assures us locality, but that is not the same as saying that assumptions that violate it are automatically not obeying the locality principle, because there might be ways to assure locality independent of that assumption that we are not aware of.

In other words to say that a theory is local only if and only if it assumes the inequalities is a very strong assertion and assumes that any correlated measurement of spatially separated systems implies instantaneous influence of measurements.

Is there any logical obstruction to the possibility that there are conditions independent from A_n = F_A(a_n, \lambda_n) and B_n = F_B(b_n, \lambda_n)
that are compatible with locality? The fact that no one has come up with them shouldn't be a definitive impediment. In general in science statistical correlation does not necessarily imply direct influence, why should it be so in physics?

Now, what someone has shown is that QM only predicts a violation of Bell's inequality in the case where the two measurements corresponding to Alice's settings a and a' are described by noncommuting operators, and the two measurements corresponding to Bob's settings b and b' are described by noncommuting operators. But that's a fact about quantum mechanics. Bell's derivation doesn't (as far as I can see) assume anything at all about whether things commute or not. The noncommutativity is about the two choices that Alice (or Bob) might make, not about Alice's measurements versus Bob's. Alice's measurements do commute with Bob's measurements.

The experiments usually have the measurements being simultaneous so Alice's measurements versus Bob's commutativity is not what matters here. The commutativity (or lack of) is indeed about the choice(the angles) that Alice and Bob make, i.e, the measurements each one of them performs, in the inequalities the choices must commute . Why should all theories obeying locality (absence of influence at a distance) have this feature about the choice a and a'? Are all possible theories describing the measurements settings with operators automatically incompatible with absence of instantaneous influences?

 

[stevendaryl]

In other words to say that a theory is local only if and only if it assumes the inequalities is a very strong assertion and assumes that any correlated measurement of spatially separated systems implies instantaneous influence of measurements.

No, it absolutely does not assume that. That was the whole point of the inequality, was to distinguish nonlocality from mere correlation.

For example, suppose that you have a process that takes a pair of shoes and randomly selects one shoe out of a pair to send to Alice and the other one to send to Bob. When Alice sees that she has a left shoe, she immediately knows that Bob received a right shoe, and vice-verse. So the observations made by Alice and Bob are perfectly correlated. But that correlation does not imply that FTL signals go from Alice to Bob or vice-verse.

The whole point of Bell's inequality was to be able to distinguish between correlations that (classically) require distant influences from those that do not.

 

[stevendaryl]

No, it absolutely does not assume that. That was the whole point of the inequality, was to distinguish nonlocality from mere correlation.

For example, suppose that you have a process that takes a pair of shoes and randomly selects one shoe out of a pair to send to Alice and the other one to send to Bob. When Alice sees that she has a left shoe, she immediately knows that Bob received a right shoe, and vice-verse. So the observations made by Alice and Bob are perfectly correlated. But that correlation does not imply that FTL signals go from Alice to Bob or vice-verse.

The whole point of Bell's inequality was to be able to distinguish between correlations that (classically) require distant influences from those that do not.

Bell absolutely does not assume that correlation implies a causal influence. What he does assume is that correlation between two measurements implies something about their common past. Actually, it's not so much that he assumes that, but that it's part of what he means by a "hidden variable theory". The whole point of a hidden variable theory is to explain correlations by invoking a shared variable whose value was determined in their shared past (the overlap of their past light-cones, according to SR causality).

Let me try another way to describe it, in terms of flow of information. Roughly speaking, information is knowing the answer (or probabilities for possible answers) to some question about the history of the universe. Bell's concept of "local realism" basically amounts to the assumption that information about the history of the universe propagates at the speed of light (or slower). If information is available in some localized region of space, then that information must either have been created there, or it must have propagated there from some source in the backwards lightcone. If Alice at time t+\delta knows something (say, about Bob's measurement results), then that information in principle follows from

1. Information that was already available to Alice at time t
   
2. Information that came into existence in the region near Alice (a distance of less than or equal to c \delta from her)
3. Information that flowed into that region from elsewhere.

When I say information that was already available at time t, I mean information that could be deduced, in principle, from detailed knowledge of the little region near Alice. Classically, that would mean that the information is deducible from the positions and momenta of the particles near Alice, as well as the values of fields in that region. Similarly, "information that flowed into that region from elsewhere" classically would mean either particles that flowed into the region, or energy/momentum that flowed into the region, or waves that propagated into that region. When I say "information that came into existence", that is allowing for intrinsic nondeterminism. If I flip a coin and the coin is intrinsically nondeterministic, then the result is new information that didn't exist prior to flipping the coin.

Quantum mechanics seems to violate this concept of information as being created locally and flowing at the speed of light or slower.

 

[TrickyDicky]

No, it absolutely does not assume that. That was the whole point of the inequality, was to distinguish nonlocality from mere correlation.

For example, suppose that you have a process that takes a pair of shoes and randomly selects one shoe out of a pair to send to Alice and the other one to send to Bob. When Alice sees that she has a left shoe, she immediately knows that Bob received a right shoe, and vice-verse. So the observations made by Alice and Bob are perfectly correlated. But that correlation does not imply that FTL signals go from Alice to Bob or vice-verse.

The whole point of Bell's inequality was to be able to distinguish between correlations that (classically) require distant influences from those that do not.

But your example doesn't violate the inequalities, and for that case I said they are fine, they discern correlation from action at a distance, the problem is in assuming that there is not locality from violation of the inequalities.

 

[TrickyDicky]

Bell absolutely does not assume that correlation implies a causal influence. What he does assume is that correlation between two measurements implies something about their common past. Actually, it's not so much that he assumes that, but that it's part of what he means by a "hidden variable theory". The whole point of a hidden variable theory is to explain correlations by invoking a shared variable whose value was determined in their shared past (the overlap of their past light-cones, according to SR causality).

Let me try another way to describe it, in terms of flow of information. Roughly speaking, information is knowing the answer (or probabilities for possible answers) to some question about the history of the universe. Bell's concept of "local realism" basically amounts to the assumption that information about the history of the universe propagates at the speed of light (or slower). If information is available in some localized region of space, then that information must either have been created there, or it must have propagated there from some source in the backwards lightcone. If Alice at time t+\delta knows
something (say, about Bob's measurement
results), then that information in principlefollows from Information that was already available to Alice at time t

1. Information that came into existence in the region near Alice (a distance of less than or equal to c \delta from her)
2. Information that flowed into that region from elsewhere.

When I say information that was already available at time t, I mean information that could be deduced, in principle, from detailed knowledge of the little region near Alice. Classically, that would mean that the information is deducible from the positions and momenta of the particles near Alice, as well as the values of fields in that region. Similarly, "information that flowed into that region from elsewhere" classically would mean either particles that flowed into the region, or energy/momentum that flowed into the region, or waves that propagated into that region. When I say "information that came into existence", that is allowing for intrinsic nondeterminism. If I flip a coin and the coin is intrinsically nondeterministic, then the result is new information that didn't exist prior to flipping the coin.

Quantum mechanics seems to violate this concept of information as being created locally and flowing at the speed of light or slower.

You are apparently talking about determinism rather than about locality.
Does a non-deterministic theory(I don't mean non-determinism only in the stochastic or probabilistic sense that you use in your post, I mean for instance in the nonlinear sense) by definition violate the inequalities?

 

[stevendaryl]

You are apparently talking about determinism rather than about locality.

No, I'm not. Determinism, in the context of information, would mean that information never increases. So the answers to any question is already present in the initial conditions of the universe. But if there is nondeterminism, information can be created. For example, if flipping a coin is nondeterministic, then the information about what the result of a coin flip will be doesn't exist until the moment the coin is flipped.

Does a non-deterministic theory(I don't mean non-determinism in the stochastic or probabilistic sense that you use in your post, I mean for instance in the nonlinear sense) by definition violate the inequalities?

I don't know what you mean by "in the nonlinear sense". I use nondeterminism to mean that the state of the universe at a future time is not uniquely determined by the state at earlier times.

 

[TrickyDicky]

No, I'm not. Determinism, in the context of information, would mean that information never increases. So the answers to any question is already present in the initial conditions of the universe. But if there is nondeterminism, information can be created. For example, if flipping a coin is nondeterministic, then the information about what the result of a coin flip will be doesn't exist until the moment the coin is flipped.
I don't know what you mean by "in the nonlinear sense". I use nondeterminism to mean that the state of the universe at a future time is not uniquely determined by the state at earlier times.

But you surely muat have heard about nonlinear equations not being deterministic? There is more to non-determinism than the coin flip probabilistic concept.

 

[stevendaryl]

But you surely muat have heard about nonlinear equations not being deterministic? There is more to non-determinism than the coin flip probabilistic concept.

No, I don't know what you mean by that. Why would a nonlinear equation be nondeterministic? Are you talking about chaos?

 

[Derek Potter]

But you surely muat have heard about nonlinear equations not being deterministic? There is more to non-determinism than the coin flip probabilistic concept.

I'm sure stevendaryl will have heard of it, but others here, like me for instance, may not. In fact I wouldn't be at all surprised if some people are wondering whether you're confusing non-computability with indeterminacy.

 

[TrickyDicky]

No, I don't know what you mean by that. Why would a nonlinear equation be nondeterministic? Are you talking about chaos?

Well, for example, I know they are considered deterministic traditionally but they aren't precisely in the increase of information sense you mentioned above.

 

[stevendaryl]

Well, for example, I know they are considered deterministic traditionally but they aren't precisely in the increase of information sense you mentioned above.

If you are talking about chaotic systems, then they are still deterministic. The information about future conditions is present in the initial conditions, it's just computationally infeasible to extract it.

 

[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!