Spin difference between entangled and non-entangled

[DrChinese]

Would be fair to say that the differences between realistic and QM models in the Bell's theorem is that realistic model is equivalent to the photon that already interacted with a polarizer while for the QM model photon didn't have any interaction yet?

No. Realistic models assume there are values for observables at all times. For QM, observables take a value at the time of observation and depending on the nature of the observation.

It is difficult to give more specifics on realistic models because a) they don't match experiment; and b) you can make up as many wrong ones as you like.

 

[miosim]

Einstein died before Bell published. He never knew, and would have been forced to re-assess his position had he known.

Unfortunately he didn't have opportunity to respond to Bell's theorem.

 

[miosim]

It is difficult to give more specifics on realistic models because a) they don't match experiment ...

If the incorrect model was chosen it should not be a surprise that it doesn't match experiment.
I would like to have a better understanding of how the realistic models was derived, probably from the EPR paper and I believe this derivation was sufficiently scrutinized. Do you have any references for this topics?

Thanks

 

[miosim]

DrChinese,
I found online the article about Bell’s theorem that talk about issues I concern with.

http://www.scholarpedia.org/article/Bell's_theorem#factorizability

What is your opinion about this article? Does it represent the mainstream view on Bell’s theorem or it tilts toward the “realism” in QM (opposite to Copenhagen interpretation) so I should take it as one point of view only?

 

[stevendaryl]

If the incorrect model was chosen it should not be a surprise that it doesn't match experiment.
I would like to have a better understanding of how the realistic models was derived, probably from the EPR paper and I believe this derivation was sufficiently scrutinized. Do you have any references for this topics?

Thanks

Well, in this thread, we've discussed two different local hidden variables models, and neither one matches the predictions of QM. Two examples doesn't prove anything, which is why it is so important that Bell proved a theorem showing that there are no locally realistic models at all that reproduce the predictions of quantum mechanics.

 

[DrChinese]

DrChinese,
I found online the article about Bell’s theorem that talk about issues I concern with.

http://www.scholarpedia.org/article/Bell's_theorem#factorizability

What is your opinion about this article? Does it represent the mainstream view on Bell’s theorem or it tilts toward the “realism” in QM (opposite to Copenhagen interpretation) so I should take it as one point of view only?

I don't consider this a useful article. It re-writes history and emphasis to be consistent with Norsen's well-known views on the matter. The authors are Bohmians and it is written from an unabashed perspective to push that intepretation.

It's sad really, because the science of Bohmian Mechanics does not need a distorted historical derivation to give it relevance. Bell himself was an advocate, probably their best advocate in the long run.

 

[atyy]

I don't consider this a useful article. It re-writes history and emphasis to be consistent with Norsen's well-known views on the matter. The authors are Bohmians and it is written from an unabashed perspective to push that intepretation.

It's sad really, because the science of Bohmian Mechanics does not need a distorted historical derivation to give it relevance. Bell himself was an advocate, probably their best advocate in the long run.

Why do you consider the article inaccurate?

 

[DrChinese]

Why do you consider the article inaccurate?

I didn't use the word "inaccurate", preferring to indicate it is not useful to any understanding of the subject. I wouldn't read it to learn about QM, history of EPR/Bell, or even BM. An example: "The new strategy also sheds some light on the meaning of locality." Locality is all that this article indicates is at the root of Bell, which is a denial of the role of realism.

That is consistent for Norsen (I am quite sure he wrote most of the historical part as I am well familiar with his writing style). In his mind, violation of a Bell Inequality equates to a proof of non-locality. That view is generally rejected by the community in favor of one in which realism may alternately (or additionally) be rejected. You will find few in the scientific community who advocate a realistic view of QM regardless of the locality issue.

But hey, read away and judge for yourself! :)

 

[atyy]

Does it represent the mainstream view on Bell’s theorem or it tilts toward the “realism” in QM (opposite to Copenhagen interpretation) so I should take it as one point of view only?

One should not take Copenhagen to be necessarily anti-realist. Copenhagen assumes a commonsense realism by virtue of the Heisenberg cut. It is agnostic about the realism of the wave function, and takes an operational or instrumental approach to the wave function as a useful tool for calculating the probabilities of events. Most versions of Copenhagen assume enough reality to agree that quantum mechanics predicts the violation of Bell inequalities by systems at spacelike separation. There are versions of Copenhagen such as Quantum Bayesianism which try to avoid this, but these are not the only flavours of Copenhagen. Historically, some versions of Copenhagen have denied the existence of hidden variables, because of von Neumann's purported proof against the existence of hidden variables, which was not widely known to be flawed before Bell. Modern versions of Copenhagen do not necessarily deny the possibility of hidden variables.

 

[atyy]

I didn't use the word "inaccurate", preferring to indicate it is not useful to any understanding of the subject. I wouldn't read it to learn about QM, history of EPR/Bell, or even BM. An example: "The new strategy also sheds some light on the meaning of locality." Locality is all that this article indicates is at the root of Bell, which is a denial of the role of realism.

That is consistent for Norsen (I am quite sure he wrote most of the historical part as I am well familiar with his writing style). In his mind, violation of a Bell Inequality equates to a proof of non-locality. That view is generally rejected by the community in favor of one in which realism may alternately (or additionally) be rejected. You will find few in the scientific community who advocate a realistic view of QM regardless of the locality issue.

But hey, read away and judge for yourself! :)

I see. But perhaps the definition of "realism" is debated? I suspect Norsen would consider Many-Worlds to be realistic, whereas I think you would not?

 

[DrChinese]

I see. But perhaps the definition of "realism" is debated? I suspect Norsen would consider Many-Worlds to be realistic, whereas I think you would not?

Not sure about that (MWI). But it would not be fair to say the definition of "realism" is debated so much as it is distorted. Norsen's position is quite clear in his paper "Against 'Realism'":

http://arxiv.org/abs/quant-ph/0607057

His non-mainstream position is obvious from the extract:

"Carefully surveying several possible meanings, we argue that all of them are flawed in one way or another as attempts to point out a second premise (in addition to locality) on which the Bell inequalities rest... We thus suggest that the phrase `local realism' should be banned from future discussions of these issues, and urge physicists to revisit the foundational questions behind Bell's Theorem."

He and I have discussed this ad infinitum, in fact our discussions may have spurred him to write that paper. :) Norsen is regarded as brilliant in the area, but has been harshly reviewed by Shimony and others. I would be happy to refute him any day of the week, as it is not that hard.

 

[wle]

Locality is all that this article indicates is at the root of Bell, which is a denial of the role of realism.

Well what is the role of "realism"? I know that the default catchphrase many authors use is that Bell's theorem rules out "local realism", but I've never seen a good explanation of what "realism" actually means in this context or why it's a necessary part of the argument. If it just means measurement outcomes in an experiment being predetermined, then that's not necessary as an assumption in order to derive Bell inequalities (Bell himself was quite explicit about this in later essays, starting at least as early as the mid 1970s).

That is consistent for Norsen (I am quite sure he wrote most of the historical part as I am well familiar with his writing style). In his mind, violation of a Bell Inequality equates to a proof of non-locality. That view is generally rejected by the community in favor of one in which realism may alternately (or additionally) be rejected. You will find few in the scientific community who advocate a realistic view of QM regardless of the locality issue.

I'm not aware of such a consensus, at least among people who actually do research on the topic. There's certainly some disagreement on the terminology and the finer points of what Bell's theorem is about, but as far as I'm aware, Norsen's expositions on Bell's theorem are known about and at at least reasonably well regarded in the community. I'm certainly not aware of any overwhelming consensus that "realism", "determinism", "counterfactual definiteness", etc., is a necessary or important ingredient in Bell's theorem. For instance, the terminology that I'm most familiar with is that the correlations that satisfy Bell inequalities are just called the "local set" or the "local polytope".

 

[DrChinese]

Well what is the role of "realism"? I know that the default catchphrase many authors use is that Bell's theorem rules out "local realism", but I've never seen a good explanation of what "realism" actually means in this context or why it's a necessary part of the argument.

How many quotes from folks like Aspect and Zeilinger would it take to convince you that "local realism" is what is ruled out by Bell? EPR is all about realism (defined as simultaneous elements of reality there, as well as by Bell). Locality is an afterthought to EPR, as they assumed there would be no spooky action at a distance. As to why it is necessary to the Bell argument, simply look after Bell's (14) and you will see realism introduced as an assumption (let c be a unit vector...).

Honestly, I was asked my opinion and gave it. After hours of discussing this with Travis, I am not likely to change my opinion any more than he is likely to change his. If we want to continue this discussion, we should do it outside of this thread as I think we have strayed off target.

 

[wle]

How many quotes from folks like Aspect and Zeilinger would it take to convince you that "local realism" is what is ruled out by Bell? EPR is all about realism (defined as simultaneous elements of reality there, as well as by Bell). Locality is an afterthought to EPR, as they assumed there would be no spooky action at a distance. As to why it is necessary to the Bell argument, simply look after Bell's (14) and you will see realism introduced as an assumption (let c be a unit vector...).

Honestly, I was asked my opinion and gave it. After hours of discussing this with Travis, I am not likely to change my opinion any more than he is likely to change his. If we want to continue this discussion, we should do it outside of this thread as I think we have strayed off target.

This isn't about me personally convincing you or vice versa. If you want to hold the opinion that Bell's theorem rules out something called "local realism", that's one thing and it can be debated. It's certainly how a lot of physicists and textbooks would describe Bell's theorem. But if you're going to insist that this is how 99% of theorists working in the field today would explain Bell's theorem and Norsen represents a 1% anti-mainstream fringe stance, then that's not my impression based on my exposure to what's going on in the field. For instance, there was a review article published on the topic earlier this year [1] (which, incidentally, I'd recommend to anyone looking for a modern overview of the field) that hardly mentions realism at all.

[1] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, "Bell nonlocality", Rev. Mod. Phys. 86, 419 (2014), arXiv:1303.2849 [quant-ph].

 

[DrChinese]

[1] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, "Bell nonlocality", Rev. Mod. Phys. 86, 419 (2014), arXiv:1303.2849 [quant-ph].

You are correct that "realism" is not mentioned. This definitely follows Norsen's reasoning. I am surprised to see Cavalcanti in the list of authors, as he had recently written about "local realism" in the same vein as I. So you may be correct that the tide has changed.

 

[Alien8]

What exactly makes QM compatible with non-locality or non-realism? Is there an example of QM non-locality or non-realism before or other than Bell test experiments and inequalities?

 

[DrChinese]

What exactly makes QM compatible with non-locality or non-realism? Is there an example of QM non-locality or non-realism before or other than Bell test experiments and inequalities?

Quantum systems are not always localized as point particles. Entangled pairs are but one example of that. That alone causes locality to be a suspect idea.

Quantum systems obey the HUP. That alone causes realism to be a suspect idea, since non-commuting observables do not seem to have definite values at all times.

This was known in 1935, but the full implications were not clear at that time.

 

[Alien8]

Quantum systems are not always localized as point particles. Entangled pairs are but one example of that. That alone causes locality to be a suspect idea.

I understand that position of quantum particles is given as probability function or average in QM equations, but what does that have to do with interaction or connection between two particles over distance? I see that entangled pairs are example of non-locality, but can you name any other example?

Quantum systems obey the HUP. That alone causes realism to be a suspect idea, since non-commuting observables do not seem to have definite values at all times.

I can see connection between uncertainty principle and the idea it might be due to actual undefined reality, but I don't see that can explain non-locality.

 

[Point Conception]

You are correct that "realism" is not mentioned. This definitely follows Norsen's reasoning. I am surprised to see Cavalcanti in the list of authors, as he had recently written about "local realism" in the same vein as I. So you may be correct that the tide has changed.

1] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, "Bell nonlocality", Rev. Mod. Phys. 86, 419 (2014), arXiv:1303.2849 [quant-ph].

You are right realism is not mentioned. But their definition of locality seems to apply and be interchangeable with realism : Top page 3 : ' Let us formalize the idea of a local theory more precisely: The assumption of locality implies that we should be able to identify a set of past factors, described by some variables lambda having a joint causal influence on both outcomes'

 

[Nugatory]

I understand that position of quantum particles is given as probability function or average in QM equations, but what does that have to do with interaction or connection between two particles over distance? I see that entangled pairs are example of non-locality, but can you name any other example?

It is impossible to assign a precise position to any particle, and it is rather difficult to define "locality" without talking about the positions of the particles involved.

 

[bohm2]

Well what is the role of "realism"? I know that the default catchphrase many authors use is that Bell's theorem rules out "local realism", but I've never seen a good explanation of what "realism" actually means in this context or why it's a necessary part of the argument. If it just means measurement outcomes in an experiment being predetermined, then that's not necessary as an assumption in order to derive Bell inequalities (Bell himself was quite explicit about this in later essays, starting at least as early as the mid 1970s).

Actually, even Norsen himself argues in his paper that a particular notion of 'realism' is required for Bell's theorem; that is, the notion of "metaphysical realism" or the existence of an external world “out there” whose existence and identity is independent of anyone’s awareness:

So it should not be surprising that Bell’s Theorem (a specific instance of, among other things, using certain words with their ordinary meanings) rests on Metaphysical Realism. This manifests itself most clearly in Bell’s use of the symbol λ to refer to a (candidate theory’s) complete description of the state of the relevant physical system – a usage which obviously presupposes the real existence of the physical system possessing some particular set of features that are supposed to be described in the theory. Putting it negatively, without Metaphysical Realism, there can be no Bell’s theorem. Metaphysical Realism can (thus) be thought of as a premise that is needed in order to arrive at a Bell-type inequality.

And so it seems we may have finally discovered the meaning of the ‘realism’ in ‘local realism’. One cannot, as suggested earlier, derive a Bell-type inequality from the assumption of Locality alone; one needs in addition this particular Realism assumption. This therefore explains the ‘local realism’ terminology and explains precisely the nature of the two assumptions we are entitled to choose between in the face of the empirical violations of Bell’s inequality. On this interpretation, we must either reject Locality or reject Metaphysical Realism.

http://arxiv.org/pdf/quant-ph/0607057v2.pdf

 

[atyy]

I think "metaphysical realism" is not what most people have in mind when they say "local realism". Most people mean counterfactual definiteness. Here is an explanation by Gill, but with a hint of why this may be a subtle issue: "Instead of assuming quantum mechanics and deriving counterfactual deniteness, Bell turned the EPR argument on its head. He assumes three principles which Einstein would have endorsed anyway, and uses them to get a contradiction with quantum mechanics; and the first is counterfactual deniteness. We must first agree that though, say, only A and B are actually measured in one particular run, still, in a mathematical sense, A' and B' also exist (or at least may be constructed) alongside of the other two; and moreover they may be thought to be located in space and time just where one would imagine." http://arxiv.org/abs/1207.5103

Bell's theorem only assumes that A' and B' "may be constructed". Then the question is whether one wants to go from "may be constructed" to terms like "exist", "counterfactual definiteness" and "realism".

Scarani makes a similar comment: "Therefore LV statistics can always be explained by a deterministic model. Of course, this does not mean that such an explanation must necessarily be adopted: your favorite explanation, as well as the “real” phenomenon, may not involve determinism. For instance, as we shall see soon, measurement on separable quantum states leads to LV statistics, but this does not make quantum theory deterministic (if that is your favorite explanation), nor forces us to believe that the physical phenomenon “out there” is deterministic." http://arxiv.org/abs/1303.3081

 

[DrChinese]

Bell's theorem only assumes that A' and B' "may be constructed". Then the question is whether one wants to go from "may be constructed" to terms like "exist", "counterfactual definiteness" and "realism".

To me, the locality requirement is tied up with the requirement that A and B are separable.

The realism requirement is the requirement that there is a counterfactual C in addition to A and B which can be measured. You need that too for Bell, and it is introduced after his (14). "Let c be a unit vector..." This assumption was originally made explicit in EPR, which says that it is not reasonable to require each element of reality to be predictable simultaneously. So you aren't REQUIRED to accept that, but if you do, that's your "realism". Bell built on that by picking his a/b/c and saying: a and b are separable, so b and c are separable, and a and c are separable.

You can't get the Bell result with a counterfactual to go with the 2 you actually measure. And that part has nothing to do with locality.

 

[atyy]

The realism requirement is the requirement that there is a counterfactual C in addition to A and B which can be measured.

Yes, I think everyone agrees that a violation of the Bell inequalities are incompatible with a theory that has a local deterministic explanation. I think everyone would also agree that a deterministic theory can be written in a counterfactual definite way.

So you aren't REQUIRED to accept that, but if you do, that's your "realism".

I think that is the question. For example, Bell's theorem can be used to rule out an unentangled state, but not everyone would be comfortable with saying that an unentangled state has to be real, because otherwise we can't apply Bell's theorem to it.

Maybe an analogy is that for a free Gaussian wave function, the results of experiments on position and momentum are consistent with particles that had definite position and momentum at all times. However, I am not comfortable from within Copenhagen saying that this means that the particles described by a free Gaussian wave function had real trajectories with definite position and momentum.

 

[DrChinese]

For example, Bell's theorem can be used to rule out an unentangled state, but not everyone would be comfortable with saying that an unentangled state has to be real, because otherwise we can't apply Bell's theorem to it.

Maybe an analogy is that for a free Gaussian wave function, the results of experiments on position and momentum are consistent with particles that had definite position and momentum at all times. However, I am not comfortable from within Copenhagen saying that this means that the particles described by a free Gaussian wave function had real trajectories with definite position and momentum.

I am not asserting anything (and certainly not trajectories) is real or realistic outside of what can be predicted with certainty. I am simply saying that is one of the 2 key Bell assumptions: locality and realism (as I showed). As far as I am concerned, you could say both are contradicted by QM/Bell tests.

 

[atyy]

I am not asserting anything (and certainly not trajectories) is real or realistic outside of what can be predicted with certainty. I am simply saying that is one of the 2 key Bell assumptions: locality and realism (as I showed). As far as I am concerned, you could say both are contradicted by QM/Bell tests.

How about if I have a pure state, say two unentangled photons of the same definite vertical polarization (0$^{\circ}$)? If both polarizers are set vertical (0$^{\circ}$) or horizontal (45$^{\circ}$), then each photon will definitely pass or not pass. But in a Bell test, the polarizer angles used may be 0$^{\circ}$, -45$^{\circ}$, and 22.5$^{\circ}$, so not all angles have results that are predicted with certainty. Would you consider this to be a state that is excluded by a Bell inequality violation?

 

[DrChinese]

How about if I have a pure state, say two unentangled photons of the same definite vertical polarization (0$^{\circ}$)? If both polarizers are set vertical (0$^{\circ}$) or horizontal (45$^{\circ}$), then each photon will definitely pass or not pass. But in a Bell test, the polarizer angles used may be 0$^{\circ}$, -45$^{\circ}$, and 22.5$^{\circ}$, so not all angles have results that are predicted with certainty. Would you consider this to be a state that is excluded by a Bell inequality violation?

Not sure if we are on different sides of this or not. :)

EPR says something is real (an "element of reality") if it can be predicted with certainty without previously disturbing it. That definition is used by them as a building block to conclude QM is incomplete. Bell used that same idea, along with the assumption that the elements of reality not be required to be simultaneously predictable, to conclude that you could not match the QM predictions.

My own viewpoint is that reality is shaped by the nature of the observation, and therefore we do not exist in an objective reality. So going back to your question, the Bell Inequality does not apply because I don't assert the existence of a reality independent of observation. Any observation predicable with certainty is merely redundant. Everything else is up to chance.

 

[atyy]

Not sure if we are on different sides of this or not. :)

EPR says something is real (an "element of reality") if it can be predicted with certainty without previously disturbing it. That definition is used by them as a building block to conclude QM is incomplete. Bell used that same idea, along with the assumption that the elements of reality not be required to be simultaneously predictable, to conclude that you could not match the QM predictions.

My own viewpoint is that reality is shaped by the nature of the observation, and therefore we do not exist in an objective reality. So going back to your question, the Bell Inequality does not apply because I don't assert the existence of a reality independent of observation. Any observation predicable with certainty is merely redundant. Everything else is up to chance.

I think we are on different "sides" of a circle. :D

Unlike you, I would say the Bell inequality applies, because the inequality holds as long as the counterfactuals exist "in principle" in the sense that they "can be constructed", even if they don't exist "in reality". So for the two unentangled photons with the same definite vertical polarization, I would say that they are excluded by a Bell violation, because the counterfactuals exist "in principle", even though they may not exist "in reality".

Maybe this is why the Brunner et al review doesn't use "local realism", because they wish to use the violation of the Bell inequalities to also certify things like entanglement, ie. they want to be able to consider quantum states as the hidden variable $\lambda$.

 

[wle]

How about if I have a pure state, say two unentangled photons of the same definite vertical polarization (0$^{\circ}$)? If both polarizers are set vertical (0$^{\circ}$) or horizontal (45$^{\circ}$), then each photon will definitely pass or not pass. But in a Bell test, the polarizer angles used may be 0$^{\circ}$, -45$^{\circ}$, and 22.5$^{\circ}$, so not all angles have results that are predicted with certainty. Would you consider this to be a state that is excluded by a Bell inequality violation?

Separable states don't lead to a Bell violation. The most general situation in quantum mechanics is that the two parties in a Bell-type experiment (Alice and Bob) can perform POVM measurements on a shared mixed state. Unentangled mixed states are generally defined as those that can be decomposed in the form $$\rho_{\mathrm{AB}} = \sum_{\lambda} p_{\lambda} \rho_{\mathrm{A}}^{(\lambda)} \otimes \rho_{\mathrm{B}}^{(\lambda)} \,,$$
in which $p_{\lambda}$ are a set of probability coefficients and $\rho_{\mathrm{A}}^{(\lambda)}$ and $\rho_{\mathrm{B}}^{(\lambda)}$ are density operators defined on Alice's and Bob's Hilbert spaces respectively. If Alice has a set $\{M_{a}^{(x)}\}$ of POVM measurements she can perform (indicated by an index $x$ denoting the choice of measurement, with the index $a$ indicating the result) and Bob similarly can perform the set $\{N_{b}^{(y)}\}$ of POVMs, then the joint probabilities predicted by quantum mechanics just reduce to the definition of a local model: $$\begin{eqnarray}
P(ab \mid xy) &=& \mathrm{Tr} \bigl[ M_{a}^{(x)} \otimes N_{b}^{(y)} \rho_{\mathrm{AB}} \bigr] \\
&=& \sum_{\lambda} p_{\lambda} \, \mathrm{Tr} \bigl[\bigl( M_{a}^{(x)} \otimes N_{b}^{(y)} \bigr) \, \bigl( \rho_{\mathrm{A}}^{(\lambda)} \otimes \rho_{\mathrm{B}}^{(\lambda)} \bigr) \bigr] \\
&=& \sum_{\lambda} p_{\lambda} \, \mathrm{Tr}_{\mathrm{A}} \bigl[ M_{a}^{(x)} \rho_{\mathrm{A}}^{(\lambda)} \bigr] \, \mathrm{Tr}_{\mathrm{B}} \bigl[ N_{b}^{(y)} \rho_{\mathrm{B}}^{(\lambda)} \bigr] \\
&=& \sum_{\lambda} p_{\lambda} \, P_{\mathrm{A}}(a \mid x; \lambda) \, P_{\mathrm{B}}(b \mid y; \lambda) \,,
\end{eqnarray}$$ with $P_{\mathrm{A}}(a \mid x; \lambda) = \mathrm{Tr}_{\mathrm{A}} \bigl[ M_{a}^{(x)} \rho_{\mathrm{A}}^{(\lambda)} \bigr]$ and $P_{\mathrm{B}}(b \mid y; \lambda) = \mathrm{Tr}_{\mathrm{B}} \bigl[ N_{b}^{(y)} \rho_{\mathrm{B}}^{(\lambda)} \bigr]$ according to the Born rule. So for nonentangled states, you always trivially have a local model that makes the same predictions as quantum mechanics which, of course, won't violate any Bell inequality.

An unentangled pure state is just the special case of a density operator of the form $\rho_{\mathrm{AB}} = \lvert \psi \rangle \langle \psi \rvert_{\mathrm{A}} \otimes \lvert \phi \rangle \langle \phi \rvert_{\mathrm{B}}$. In that case, the quantum predictions factorise completely: $$\begin{eqnarray}
P(ab \mid xy) &=& \langle \psi \rvert M_{a}^{(x)} \lvert \psi \rangle_{\mathrm{A}} \, \langle \phi \rvert N_{b}^{(y)} \lvert \phi \rangle_{\mathrm{B}} \\
&=& P_{\mathrm{A}}(a \mid x) \, P_{\mathrm{B}}(b \mid y) \,.
\end{eqnarray}$$

As far as I know, the converse isn't so clear. Specifically, I think it's known that all entangled pure states can predict a Bell violation, but I don't think it's known for arbitrary entangled mixed states (though this isn't a topic I know much about, so don't quote me on this).

(EDIT: Section III of the review I linked to covers all of this.)

 

[atyy]

Separable states don't lead to a Bell violation.

Yes, that was the point of my question. To me, a Bell violation excludes separable states. However, if I understand DrChinese correctly, although we know from QM that a Bell violation excludes separable states, we don't know from "Bell's theorem" that a Bell violation excludes separable states.

The issue is that if counterfactual definiteness "in reality" is an assumption of Bell's theorem, then it doesn't apply to separable states since a separable state like the 2 unentangled photons with the same vertical polarization won't give 100% certain results at more than 2 of the angles used in a Bell test.

On the other hand one cannot say that counterfactual definiteness is not used at all in Bell's theorem. This is because a local variable theory that is excluded by Bell's theorem can be rewritten as a local deterministic theory. So by excluding local deterministic theories, one also excludes local variable theories. So the counterfactual definiteness is there "in principle", although not necessarily "in reality".