50th anniversary of Bell's theorem

[DrChinese]

I might not have gotten everything right above, because I do find the terminology confusing. But my main point is that I think there is overall very little disagreement. As carllooper said in a different thread, once you know what they mean, it doesn't matter even if they say the wave function is a chicken. :)

Pretty good job of it I thought!

We all accept Bell anyway. So we can't be that far off from each other. :-) And in the end, that is actually what this thread is about... IE 50 years of Bell and we are still debating the details!

 

[stevendaryl]

I said "independent of observation".

EPR's "elements of reality" might be independent of observation, in the sense that they exist whether or not we observe anything, but they are not independent of observation in the sense that we posit elements of reality in order to account for our observations. You know, Occam's razor, you don't make up elements of reality above and beyond those necessary to explain what we observe.

So if you ask the question of whether a photon's polarization is an element of reality, whether we hypothesize that there is such a thing as photon polarization depends on what the experimental evidence is. To give an analogy: you flip a coin, and you slap the coin to the ground, and you get a result "heads" or "tails". Its being "heads" might not correspond to any "element of reality" prior to slapping the coin to the ground, because the result might cooperatively be created by the state of the coin and also by the details of how you slap it to the ground. Photon polarization could conceivably be the same sort of thing, that whether the photon passes a filter at angle $\vec{a}$ might depend on details of the filter, as well as details of the photon. A different filter might have produced a different result. But that's where entanglement provides evidence to the contrary: the predictability of a polarization measurements for one photon from measurements on the other photon imply that there is nothing about the details of the detector that are relevant (at least if we ignore exotic causality, such as FTL, back-in-time, superdeterminism, etc).

Does a single photon have simultaneous definite polarization at 0/12/240? I am not asking whether you believe one way or the other, I am saying EPR saw this as the essential question. They thought a particle - any particle, entangled or not - had all attributes at all times. And the idea of perfect correlations for entangled pairs supports this idea very strongly.

Surely, EPR realized the possibility that a measurement result could depend on details of the measuring device, as well as details of the object being measured? That's a possibility classically for things such as the result of a coin flip.

 

[DrChinese]

EPR's "elements of reality" might be independent of observation, in the sense that they exist whether or not we observe anything, but they are not independent of observation in the sense that we posit elements of reality in order to account for our observations. You know, Occam's razor, you don't make up elements of reality above and beyond those necessary to explain what we observe.

So if you ask the question of whether a photon's polarization is an element of reality, whether we hypothesize that there is such a thing as photon polarization depends on what the experimental evidence is. ...

Surely, EPR realized the possibility that a measurement result could depend on details of the measuring device, as well as details of the object being measured? That's a possibility classically for things such as the result of a coin flip.

EPR didn't say what was real, just that something was real IF you could predict with certainty the result of a measurement (and as you say, the measuring device could be a factor too). We all agree that you can predict the result of ANY measurement on an entangled photon by observing its partner. But you can only do that for 1 non-commuting observable at a time. Even in 1935, this point was not in serious doubt (although admittedly it was early on).

So they used what everyone agreed as the basis for their argument, and then asked: if they are separately elements of reality, are they collectively so?

No need to jump ahead and talk about how to explain entanglement or what experiments show. The question EPR & Bell were addressing was specifically: whether a, b and c existed simultaneously (Bell used these 3, that's why I'm so certain of this) and if so, was their relationship consistent with QM's predictions? You do not need to consider locality at all. Locality is an issue to discuss as to mechanism as to explain experimental results. The reference in Bell to Bohmian Mechanics is strictly superfluous, and ditto for any comments about locality. If you assume a, b and c are observer independent (which is the proposition), then you are saying that Alice's outcome will always be independent of Bob's setting.

So Bell proves observer independence false. Even for non-local observers. That is where locality comes in, because non-local observer Bob could potentially change the outcome of a measurement by Alice that would have yielded a different result if Bob had done something else. EPR specified that the prediction (with certainty) had to be made without disturbing the system.

Bohr would object anyway (I guess) that there is no way to have a quantum system of 2 entangled particles and expect a measurement on one NOT to have an effect on the other. In other words, that there are no elements of reality until a measurement is performed. :-)

 

[stevendaryl]

No need to jump ahead and talk about how to explain entanglement or what experiments show. The question EPR & Bell were addressing was specifically: whether a, b and c existed simultaneously (Bell used these 3, that's why I'm so certain of this) and if so, was their relationship consistent with QM's predictions?

But it seems to me that locality is the reason for believing that they all exist simultaneously. The reasoning goes: either (1) the element of reality existed all along (independently of what axis Alice or Bob chooses to measure), or (2) it came into existence at the moment Alice (or Bob) made their respective choices. Case (2) is not consistent with locality: If it came into existence when Alice made her choice, then how could it affect Bob, who is far away?

So, to me, it's not simply a matter of blithely assuming that if they exist separately, then they all must exist simultaneously. The fact that they must exist simultaneously follows from locality (plus the perfect correlations predicted by quantum mechanics).

 

[DrChinese]

But it seems to me that locality is the reason for believing that they all exist simultaneously. The reasoning goes: either (1) the element of reality existed all along (independently of what axis Alice or Bob chooses to measure), or (2) it came into existence at the moment Alice (or Bob) made their respective choices. Case (2) is not consistent with locality: If it came into existence when Alice made her choice, then how could it affect Bob, who is far away?

So, to me, it's not simply a matter of blithely assuming that if they exist separately, then they all must exist simultaneously. The fact that they must exist simultaneously follows from locality (plus the perfect correlations predicted by quantum mechanics).

Locality is not the reason for believing they exist simultaneously. The reason for that is that you can predict ANY attribute in advance (perfect correlations). Sort of a Bertlmann's socks viewpoint led them to your (1). EPR was never suspecting that measuring Bob (to predict Alice) would change Alice's outcome to match anyway - your (2). (Of course entanglement was not well understood at that point anyway so everyone was taking a bit of a leap.)

Note that depending on where you want to place the emphasis (or where I want to), it seems like both of our viewpoints are quite reasonable . :)

What we all are asking is: once you know about Bell's Theorem, where do we go from here? We cannot expect an EPR-like "more complete" solution as this would be ruled out. OK, we all agree on that. Where do we look next? We can dissect Bell a lot of ways, but the important point is once we know about it, our viewpoint is forever changed.

Note that Bohr and others, in 1935, denied the EPR conclusion (QM is incomplete) without the benefit of Bell. And their answer was not "let's talk about locality/nonlocality now". Perhaps someone could have come up with an EPR-like argument that proved QM must be completed by a nonlocal theory. In fact, perhaps all this didn't sit well with Bohm, and so he developed the non-local Mechanics named after him - I don't know much about that. But the overall point is that QM was already "quantum nonlocal" and perhaps it took a while longer for that to become clear.

 

[atyy]

http://dare.uva.nl/document/2/104604

Even here, I don't think there is any big disagreement. The suggestion here is that Bell assumes that the measurement settings and measurement outcomes do not influence the state that is prepared. It is generally agreed that this is an assumption. If we get rid of this assumption, can one come up with a local deterministic model that is consistent with experimental data? The question is discussed in these papers. (The focus here is different from the Maudlin/Werner discussion. Maudlin/Werner are talking about whether quantum mechanics is nonlocal, whereas here the question is whether reality is nonlocal given experimental constraints.)

http://arxiv.org/abs/quant-ph/0110137
Accardi contra Bell (cum mundi): The Impossible Coupling
Richard D. Gill

http://arxiv.org/abs/quant-ph/0301059
Time, Finite Statistics, and Bell's Fifth Position
Richard D. Gill

http://arxiv.org/abs/quant-ph/0205016
Quantum nonlocality, Bell inequalities and the memory loophole
Jonathan Barrett, Daniel Collins, Lucien Hardy, Adrian Kent, Sandu Popescu

http://arxiv.org/abs/1001.1750
The statistical strength of experiments to reject local realism with photon pairs and inefficient detectors
Yanbao Zhang, Emanuel Knill, Scott Glancy

http://arxiv.org/abs/1108.2468
Asymptotically optimal data analysis for rejecting local realism
Yanbao Zhang, Scott Glancy, Emanuel Knill

 

[bohm2]

Even here, I don't think there is any big disagreement.

I see a big difference:

1. One group holds that one can have both locality and realism.
2. Another group holds that nature is fundamentally non-local, irrespective of all other issues (e.g. "realism", determinism, CFD, etc.)
3. Another group holds that we are driven to a choice between non-locality versus non-realism (e.g. local/non-local non-realism versus non-local realism).

I see these 3 views as quite different.

 

[atyy]

I see a big difference:

1. One group holds that one can have both locality and realism.
2. Another group holds that nature is fundamentally non-local, irrespective of all other issues (e.g. "realism", determinism, CFD, etc.)
3. Another group holds that we are driven to a choice between non-locality versus non-realism (e.g. local/non-local non-realism versus non-local realism).

I see these 3 views as quite different.

All three views can be correct in some sense. I'm not sure whether all proponents of the various views will agree with me here, but here are senses in which the three views can all be correct.

1. Local realism of nature is always in play, because real experiments have a finite number of trials, so experiments can only make local realism unlikely, not impossible. Also, it is true that the memory loophole makes local realism of nature less unlikely. It is also true that one cannot rule out that quantum mechanics may have a local explanation in some sense, if one goes beyond theories consistent with Kolmogorov's definition of probability. The references for this are http://arxiv.org/abs/quant-ph/0301059 and others listed in post #111.

2 and 3. If one defines "local" as "local causality" following Bell 1976 and "La Nouvelle Cuisine", then quantum mechanics does not have a local explanation. If one defines "local" following Wiseman's http://arxiv.org/abs/1402.0351 Definition 9 (Eq 2), then the Copenhagen-style interpretation of quantum mechanics is local, and provides a counterexample to the idea that locality is sufficient to derive a Bell inequality. Other references for this are in post #104.

 

[Ilja]

The question EPR & Bell were addressing was specifically: whether a, b and c existed simultaneously (Bell used these 3, that's why I'm so certain of this) and if so, was their relationship consistent with QM's predictions? You do not need to consider locality at all.

You do not need to consider Bell's theorem at all, if you don't like it.

And, without doubt, you do not need to consider the first part of Bell's proof that, given locality and the EPR criterion, they have to exist simultaneously. Ignore the first part, the remaining part remains a valuable contribution to science, suggests experimental tests if the resulting inequalities are really violated.

But the whole really important conclusion - that one has to give up Einstein causality - is, then, forgotten.

I have, by the way, always suggested that "locality" is a very wrong and misleading name. Simply imagine a theory with preferred frame and a maximal speed of information transfer in this preferred frame of $C=10^300 c$. This theory would be unable to create violations of Bell's inequality for pairs of events which are space-like separated in the Minkowski-metric corresponding to C instead of c, but it could be Einstein-causal in the C-metric, and the theory could be realistic and local in any meaningful meaning of these notions but nonetheless in agreement with observation, and approximately (except for pairs of C-spacelime-separated events) in agreement with quantum theory.

Einstein causality is a much better name for what is impossible. Because "signal locality" is only a fact about correlations, not about causation. Causality is something which goes beyond operational observation of correlation, and, thus, closely connected with realism. To use "causality" presupposes the acceptance of Reichenbach's principle of common cause, it implies the idea that there is something like a causal explanation of observable correlations, else it makes no sense.

 

[Demystifier]

Well you got me. :)

It was my intention. By observing you for a long time, I thought you might be an easy prey and designed a logical trap for you, for which I was quite confident you might fall in it. :D

Now seriously.

But most Bohmians insist that there are values for a, b and c at all times.

Actually they don't. All they insist is that there are values for positions of all particles at all times.

So rather than press the point, I thought this would be a reasonable compromise.

By contrast, below I will press my point, without compromise.

But honestly, I don't know enough about BM to really argue the point one way or the other.

I believe I know enough about BM and about all other interpretations, to say the following:
According to your definition of reality, no viable interpretation of QM is realistic, either local or non-local. With such a definition of reality, the Bell theorem can be restated as "QM is non-realistic, period!", without any mentioning of locality or non-locality.

On the other hand, your signature contains a very different formulation of Bell theorem, so your signature is not compatible with your definition of reality.

So if the formulation of Bell theorem "QM is non-realistic, period!" sounds too uncompromising and you want to make a reasonable compromise, then what can you do? My final point is: Abandon your definition of "reality" and adopt a more conventional one! If you do that, the definition of reality you will adopt will become compatible with your signature.

 

[DrChinese]

Einstein causality is a much better name for what is impossible. Because "signal locality" is only a fact about correlations, not about causation. Causality is something which goes beyond operational observation of correlation, and, thus, closely connected with realism.

Yes, I would say that's a goner. :-) A good description indeed.

 

[DrChinese]

1. It was my intention. By observing you for a long time, I thought you might be an easy prey and designed a logical trap for you, for which I was quite confident you might fall in it. :D

Now seriously.

2. Actually they don't. All they insist is that there are values for positions of all particles at all times.3. I believe I know enough about BM and about all other interpretations, to say the following:
According to your definition of reality, no viable interpretation of QM is realistic, either local or non-local. With such a definition of reality, the Bell theorem can be restated as "QM is non-realistic, period!", without any mentioning of locality or non-locality.

4. On the other hand, your signature contains a very different formulation of Bell theorem, so your signature is not compatible with your definition of reality.

1. I trapped you into trapping me. :-)

2. Ah, that sounds more Bohmian-like than what I said.

3. It would not surprise me that ultimately, realism and locality can't be separated into completely distinct concepts - and we rule out both. To explain Bell correlations, you need a non-local mechanism. On the other hand, no interpretation has the ability to postulate more information than the HUP allows (i.e. realism, hidden variables, a/b/c, etc.).

4. My signature is good, certainly a solid representation of Bell's Theorem. It may seem inconsistent to what I have said in this thread in some respects, but I don't think the overall effect of Bell hinges on the meaning of a word. We use Bell to weed out candidate theories left and right. And after all this time, new interpretations and all, QM sits there as it has for a long time. And we still debate things it tells us, just as in 1927 - but with even greater appreciation.

 

[DrChinese]

A timely 50th anniversery item appeared in arxiv today I think you will enjoy:

http://arxiv.org/abs/1411.5322

If you think our views vary a bit but are in essential agreement, here are some quotes from the SAME paper:

1. Amazingly, just by relying on conditions (3) and (4) one can construct so-called No-Go Theorems that arrive at a contradiction. [Note: 3 is
f(A;B;C;...) = 0 and 4 is similar. I.e. these are requirements of realism.]

2. Bell's profound discovery was that the requirement of locality is incompatible with the statistical predictions ofquantum mechanics

3. Local realistic theories are incompatible with quantum mechanics!

All of the above from Bertlmann, who knew Bell quite well. As we know, Bell was also an expert in Bertlmann's socks. This paper covers a lot of ground, and has some cool anecdotes about Bell.

 

[atyy]

2. Another group holds that nature is fundamentally non-local, irrespective of all other issues (e.g. "realism", determinism, CFD, etc.)
3. Another group holds that we are driven to a choice between non-locality versus non-realism (e.g. local/non-local non-realism versus non-local realism).

As Wiseman points out, this depends on definitions. One new way of defining locality that makes locality alone insufficient, and requires one to add determinism is to use signal locality. Signal local theories include Bell local theories, and the Bell local theories can be excluded by requiring "intrinsic randomness". This seems to be an old idea, but apparently only proved by Masanes, Acin and Gisin in http://arxiv.org/abs/quant-ph/0508016.

So this means there are two different ways to define locality so that it is insufficient to define Bell locality. The first definition of local is that local actions cannot affect distant local observations. The second definition is local in the sense of no faster than light communication. Of course there is still another way of defining locality so that locality alone is sufficient is to use the concept of local causality, or Ilja's term "Einstein causality", to derive the mathematical condition of separability. If I were to put it in words, I think local causality or Einstein causality is the idea that nonlocal correlations have local explanations.

Basically, once you reach the mathematical condition of separability you can derive the Bell inequalities. So the question is what are your definitions and assumptions in reaching separability.

 

[atyy]

Einstein causality is a much better name for what is impossible. Because "signal locality" is only a fact about correlations, not about causation. Causality is something which goes beyond operational observation of correlation, and, thus, closely connected with realism. To use "causality" presupposes the acceptance of Reichenbach's principle of common cause, it implies the idea that there is something like a causal explanation of observable correlations, else it makes no sense.

Another group that often mentions Reichenbach's principle are computer scientists and statisticians who use Bayesian networks, causal networks or graphical models. These are very popular in biology, so I instinctively picture the separability condition as a graphical model. It's interesting how similar Bell's description of local causality and Pearl's description of causality in Bayesian networks is. Norsen http://arxiv.org/abs/0707.0401 quotes Bell: "completely shields off", while in Bayesian networks http://en.wikipedia.org/wiki/Markov_blanket one says "The Markov blanket of a node contains all the variables that shield the node from the rest of the network." A paper discussing Bell's theorem using Bayesian networks is http://arxiv.org/abs/1208.4119, which mentions Reichenbach's principle.

I wonder whether Bell and Pearl came up with the language independently, or whether they knew each other's work. Nowadays, Bell is well known in Bayesian networks, and is mentioned in textbooks, but what is the history of it?

 

[atyy]

2. Another group holds that nature is fundamentally non-local, irrespective of all other issues (e.g. "realism", determinism, CFD, etc.)
3. Another group holds that we are driven to a choice between non-locality versus non-realism (e.g. local/non-local non-realism versus non-local realism).

Here is yet another way of seeing things. http://arxiv.org/abs/1208.4119 mentions the idea that Bell's theorem a choice between some form of "locality" and "common cause". In this case, quantum mechanics is nonlocal in order to save the idea of common cause. This is in contrast to the other definitions of "locality" in which quantum mechanics is local but discards some other idea like determinism.

 

[billschnieder]

This is the part that confuses me. It isn't only DrC who uses sees contextuality as implying non-realism. There are a number of other authors like Nieuwenhuizen, Hess, Krennikov, Accardi, Pitowsky, Rastal , Kupczynski, de Raedt, etc. who see contextuality in a somewhat similar light but draw different conclusions:
...

From the basics, Bell's inequality can be written as $P(a, c) - P(b, a) - P(b, c) \le 1$ Which we compare with QM/experiment. Experimentally, we are measuring three corresponding averages $\langle AC\rangle, \langle BA \rangle, \langle BC \rangle$ for which we assume that $P(a, c) ≈ \langle AC\rangle, P(b, a) ≈ \langle BA \rangle, P(b, c) ≈ \langle BC \rangle$ for a large number of measurements.
According to Bell's local realistic prescription (equation 2), we then have three quantities:

$$P(a,b) = \int_{[\lambda _{1..n}]} A(a,\lambda )B(b,\lambda ) \rho (\lambda ) d\lambda$$
$$P(b,c) = \int_{[\lambda _{n+1..m}]} A(b,\lambda )B(c,\lambda ) \rho (\lambda ) d\lambda$$
$$P(a,c) = \int_{[\lambda _{m+1..l]}]} A(a,\lambda )B(c,\lambda ) \rho (\lambda ) d\lambda$$
Reflecting the fact that experimentally, we can only measure each expectation value on a different set of particle pairs. Why is this relevant to contextuality, realism, locality, loophole etc. In a simple way:

Realism: Bell's inequality can not be violated if $[\lambda _{1..n}] = [\lambda _{n+1..m}] = [\lambda _{m+1..l]}]$. This is equivalent to measuring each particle pair at three angles simultaneously (a practical impossibility). No spreadsheet of three columns of outcomes, one for each pair of particles can ever violate Bell's inequality. On the other hand, it is very easy to violate the inequality if $[\lambda _{1..n}] \neq [\lambda _{n+1..m}] \neq [\lambda _{m+1..l]}]$ (if you are interested I can show you examples). Therefore the derivation of the inequality must include (even implicitly) the assumption that 3 outcomes exist simultaneously for each particle pair. There is one problem though, you can't then rely on experimental data measured on different particle pairs to rule out realism. You can't claim 3 values do not exist simultaneously based on experiments which can never measure the 3 values simultaneously even if they existed.

Locality: If you assume that the same process is generating the particles $[\lambda _{1..n}], [\lambda _{n+1..m}], [\lambda _{m+1..l]}]$ used for each correlation, for large enough particle pairs, even if the sets are not the same, the probability distributions may be so similar that you should still obtain the same expectation values as you would have obtained from a single set as in the realism case. Then you can violate the inequality is if there is some communication between the sides. Another way to see this is to create a new "non-local" variable and assign it to the sets. Then you end up with $[\lambda _{1..n}, \eta_1], [\lambda _{n+1..m}, \eta_2], [\lambda _{m+1..l]}, \eta_3]$ Then you end up with different probability distributions which can violate the inequality. While we do not assume that the particles are the same set, we still assume that we should have had the same distribution, unless there is a non-local influence which changes the effective distribution after emission of the particle pairs. Note, if the non-local variables are the same value, you still won't be able to violate the inequality. You need non-local variables which change value in an angle dependent way.

Contextuality: This is very similar to the Locality case in that, we could encapsulate the context in another variable such that we now have sets $[\lambda _{1..n}, \beta_1], [\lambda _{n+1..m}, \beta_2], [\lambda _{m+1..l]}, \beta_3]$ If the contexts are different, then we have a way to introduce differences between the distributions and the inequality can be violated. Hess, De Raedt, Accardi and others argue that when measuring on different sets of particles at different times, it is natural to expect differences in context that are angle dependent. This is what loopholes are about, they are just ways of introducing differences in context and thus different distributions. For example:
- detection loophole: particle pairs are less likely to be detected at certain angles than others
- coincidence loophole: the likelihood of matching a pair varies with angle difference.
- "Superdeterminism": Same thing. Alice and Bob do not have the free to control the experiment such that $[\lambda _{1..n}, \beta_1] = [\lambda _{n+1..m}, \beta_2] = [\lambda _{m+1..l]}, \beta_3]$.

There isn't a whole lot of difference between them.

 

[bohm2]

For completeness, B. J. Hiley also just published a paper on the 50th anniversary of Bells 'theorem:
Some Personal Reflections on Quantum Non-locality and the Contributions of John Bell.
http://arxiv.org/pdf/1412.0594.pdf

 

[bohm2]

More papers (all free access) devoted to the 50th anniversary of Bell's theorem in another new physics journal. Some interesting papers include a critical paper by Norsen on Wiseman's paper: "The two Bell’s theorems of John Bell”:

Are there really two different Bell's theorems?
http://www.ijqf.org/wps/wp-content/uploads/2014/12/Norsen-on-Wiseman.pdf

There also contributing papers by Bernard d'Espagnat, Tumulka, Bricmont, Zeh, Stapp, Healey, etc.:

John Bell Workshop 2014
http://www.ijqf.org/groups-2/bells-theorem/forum/

 

[atyy]

More papers (all free access) devoted to the 50th anniversary of Bell's theorem in another new physics journal. Some interesting papers include a critical paper by Norsen on Wiseman's paper: "The two Bell’s theorems of John Bell”:

Are there really two different Bell's theorems?
http://www.ijqf.org/wps/wp-content/uploads/2014/12/Norsen-on-Wiseman.pdf

It's good to see that Norsen and Wiseman don't disagree on physics, their disagreement is literary - there is more than one set of assumptions form which separability can be derived, and their disagreement is over exactly which set Bell used in 1964.

 

[stevendaryl]

More papers (all free access) devoted to the 50th anniversary of Bell's theorem in another new physics journal. Some interesting papers include a critical paper by Norsen on Wiseman's paper: "The two Bell’s theorems of John Bell”:

Are there really two different Bell's theorems?
http://www.ijqf.org/wps/wp-content/uploads/2014/12/Norsen-on-Wiseman.pdf

There also contributing papers by Bernard d'Espagnat, Tumulka, Bricmont, Zeh, Stapp, Healey, etc.:

John Bell Workshop 2014
http://www.ijqf.org/groups-2/bells-theorem/forum/

I'm not 100% sure I understand what the argument is about, in the paper by Norsen. But I appreciate that he makes a point that I think a lot of people miss:

Bell, in his model of a locally realistic theory, assumes a deterministic local theory. Many people assume that this is because either Bell, or Einstein, who inspired Bell's analysis had a prejudice in favor of deterministic theories. Einstein may have had a preference for deterministic theories, but the reason for Bell making his theory deterministic was not because of this preference, but simply because the perfect correlations predicted by quantum mechanics cannot possibly be reproduced by a local, nondeterministic theory. In the spin-1/2 version of the EPR experiment, when Alice measures the spin of one particle in one direction, she knows exactly what result Bob will get if he measures the spin of the other particle in that direction. So if the states of the two particles factor and evolve separately, then Bob's particle's state must be deterministic.

Bell could have started out with a more general form for hidden variables model; instead of being deterministic, it could be stochastic, so that the outcomes of a measurement are probabilistically related to the value of the hidden variable, instead of deterministically. But the answer would have been the same--no local hidden variables theory, deterministic or not, can reproduce the predictions of QM.