Assumptions of the Bell theorem

[PeterDonis]

we are at a point that we have to define "locality"

We shouldn't for this thread, since in the context of Bell's theorem, we already have a definition of locality: the factorizability assumption. That is "locality" as far as Bell's theorem is concerned. Discussions of other definitions of "locality" belong in another thread, not this one.

 

[PeterDonis]

there is no longer a single Bell's theorem, but a variety of related theorems that all go under the heading of Bell's theorem

Please give references for this "variety of related theorems". If we are going to discuss something, we should know what we are discussing.

 

[PeterDonis]

Please give references

And to follow my own advice, here is the original paper by Bell on his theorem:

https://cds.cern.ch/record/111654/files/vol1p195-200_001.pdf

What I am calling the factorizability assumption is equation (2) in that paper.

 

[Nullstein]

Discussions based on the EPR argument are pretty unproductive, because the argument is known to be flawed. Contrary to their postulate, the absence of interactions really does not imply pre-existing values. Bohmian mechanics for example serves as a counterexample. Since Bell, there is no more need to appeal to the EPR argument. All assumptions are pretty neatly exposed in the theory of probabilistic causality, i.e. the causal Markov condition (which generalizes such assumptions as Bell factorizability or the Reichenbach principle).

 

[vanhees71]

However, the word "locality" is not really unclear. It may mean slightly different things in different context, but it is still clear in the "you know it when you see it" sense. So for Bohmian mechanics, the way the trajectories influence each other is nonlocal. For Copenhagen, the way the Born rule updates the wavefunction is nonlocal. Not in the sense of a judgment, but in the sense that this is what we mean when we say that the interpretation is nonlocal. And in the sense of "conservation of difficulty", the expectation is that you will find similar elements of nonlocality in any complete and valid interpretation of quantum mechanics. As an example, consistent histories in its basic form (as a logical framework) does not contain elements of nonlocality. But if you complete it to a full interpretation, then nonlocal elements appear.

If it were so clear, you could give a clear definition. The only really clear language you can discuss about physics is math. So can you give a clear mathematical definition of what you mean by "local"? E.g., you say the "wave function" is "not local". I've no clue what you mean. Also you are then obviously discussing non-relativistic quantum theory. Then it's a function of time and arguments with the meaning of the eigenvalues of a complete set of compatible self-adjoint operators (e.g., $3N$ position-vector components and spins of a of an $N$-particle system). What specifies this function to be "local" or "non-local"? I've no idea what it means to say a quantum state is local or nonlocal.

Before the scientific side of the question, what we mean by "local" or "non-local" is clarified, it doesn't make sense to discuss interpretations or other philosophy concerning this question.

 

[vanhees71]

Discussions based on the EPR argument are pretty unproductive, because the argument is known to be flawed. Contrary to their postulate, the absence of interactions really does not imply pre-existing values. Bohmian mechanics for example serves as a counterexample. Since Bell, there is no more need to appeal to the EPR argument. All assumptions are pretty neatly exposed in the theory of probabilistic causality, i.e. the causal Markov condition (which generalizes such assumptions as Bell factorizability or the Reichenbach principle).

In this sense QT is not local, because the time evolution of quantum systems is non-Markovian. E.g., Markovian master equations in the theory of open quantum systems are for sure an approximation.

 

[PeterDonis]

the word "locality" is not really unclear. It may mean slightly different things in different context, but it is still clear in the "you know it when you see it" sense.

Since this thread is about Bell's Theorem, the definition of "locality" for this thread should be the one Bell used in his theorem, which, as I have already pointed out, is the factorizability assumption.

can you give a clear mathematical definition of what you mean by "local"?

We already have one for this thread; see above.

 

[vanhees71]

We shouldn't for this thread, since in the context of Bell's theorem, we already have a definition of locality: the factorizability assumption. That is "locality" as far as Bell's theorem is concerned. Discussions of other definitions of "locality" belong in another thread, not this one.

Good, can you give a reference with that precise definition. Then at least we have finally one!

 

[PeterDonis]

can you give a reference with that precise definition.

I already did. See post #178.

 

[Nullstein]

In this sense QT is not local, because the time evolution of quantum systems is non-Markovian. E.g., Markovian master equations in the theory of open quantum systems are for sure an approximation.

Well, now you're confusing Markov with Markov . The "causal Markov condition" is actually completely unrealated to the "Markov condition" in stochastic processes. Unfortunately, we are stuck with this terminology.

 

[vanhees71]

Ok, thanks. Then I'm completely lost.

 

[vanhees71]

I already did. See post #178.

Sorry, I've overlooked this.

 

[Nullstein]

The causal Markov condition is just a more elaborate version of the screening off conditions that are used in the derivation of Bell's theorem. In the absence of superdeterminism and similarly fancy loopholes, e.g. under the assumption that there is just one common cause and no conspiracies, it reduces to the Reichenbach principle, which leads to the Bell factorization condition. The CMC is basically the state of the art in causality research. However, it still uses classical probability and might not apply to quantum systems, as has been acknowledged by Pearl.

 

[Demystifier]

Discussions based on the EPR argument are pretty unproductive, because the argument is known to be flawed.

No it isn't. The EPR conclusions follow from the EPR assumptions. The assumptions could be wrong, but it doesn't make the argument flawed.

the absence of interactions really does not imply pre-existing values.

Yes it does, given their other assumptions.

Bohmian mechanics for example serves as a counterexample.

It does not, because in Bohmian mechanics particles interact with each other. That's most explicitly seen when Bohmian mechanics is formulated in terms of the quantum potential.

Since Bell, there is no more need to appeal to the EPR argument.

But Bell himself used EPR argument as a part of his argument. First he used EPR to show that the assumption of locality implies preexisting values. Then he has shown that preexisting values lead to another contradiction. Therefore, he concluded, the initial assumption (locality) has been wrong.

 

[Nullstein]

No it isn't. The EPR conclusions follow from the EPR assumptions. The assumptions could be wrong, but it doesn't make the argument flawed.Yes it does, given their other assumptions.It does not, because in Bohmian mechanics particles interact with each other. That's most explicitly seen when Bohmian mechanics is formulated in terms of the quantum potential.But Bell himself used EPR argument as a part of his argument. First he used EPR to show that the assumption of locality implies preexisting values. Then he has shown that preexisting values lead to another contradiction. Therefore, he concluded, the initial assumption (locality) has been wrong.

The EPR argument is wrong, because it missed the possibility for contextuality or less desirable explanations like superdeterminism. Their conclusion just doesn't follow from their assumptions if one doesn't exclude these possibilities. EPR certainly didn't mention them at all. That's why a precise formulation such as Bell's is required. Bell's theorem is completely independent of the EPR argument. However, he also missed contextuality and superdeterminism in his first paper. He corrected himself later on. The derivations of his theorem given in his later papers basically resemble exactly what a derivation from the CMC would yield.

 

[PeterDonis]

First he used EPR to show that the assumption of locality implies preexisting values.

Note that Bell's $\lambda$ doesn't necessarily have to be "preexisting values", although Bell does say that that is the kind of thing Einstein was envisioning when he talked about a "more complete specification of the state".

 

[PeterDonis]

He corrected himself later on.

Can you give a reference to a later paper that shows this?

 

[Nullstein]

Can you give a reference to a later paper that shows this?

In his original paper, which you cited above, he assumed pre-existing values for all observables, because he assumed that functions like $A(\alpha,\lambda)$ exist and they indeed give values for every choice of $\alpha$ and $\lambda$ (i.e. given $\lambda$, there is simultaneously a value $A$ for each choice of $\alpha$). He was led to this assumption by the EPR argument.

In his paper "The theory of local beables," he relaxed this assumption and gave a derivation only based on probabilities, which opens up the possibility for contextuality. (If you want to express this in terms of observables, you would have to introduce two observables $A(\lambda)$ and $\alpha(\lambda)$, which depend only on $\lambda$ and not on each other. In order to still derive the inequality after this relaxation, you must additionally postulate certain probabilistic independence relations as he did in that paper.)

Then there was a paper called "La nouvelle cuisine," where he gave the same derivation again without tacitly excluding the possibility of superdeterminism.

You can find all of these papers in a nice little book called "Speakable and unspeakable in quantum mechanics." (And just to emphasize it: No trace of the EPR argument is left in his latter derivations. Instead, they show that the EPR argument was flawed and exlcuded certain possibilities. They arrive at their conclusion only due to tacit assumptions they weren't aware of.)

 

[PeterDonis]

In his paper "The theory of local beables,"

This one (at least in draft form) is also on the CERN website:

https://cds.cern.ch/record/980036/files/197508125.pdf

a paper called "La nouvelle cuisine,"

This one I have not been able to find online, although it is in the book you mention (which I have but don't have my copy handy right now).

 

[gentzen]

there is no longer a single Bell's theorem, but a variety of related theorems that all go under the heading of Bell's theorem

Please give references for this "variety of related theorems". If we are going to discuss something, we should know what we are discussing.

The article on Bell's theorem in the Stanford Encyclopedia of Philosophy starts as follows:

Bell’s Theorem is the collective name for a family of results, all of which involve the derivation, from a condition on probability distributions inspired by considerations of local causality, together with auxiliary assumptions usually thought of as mild side-assumptions, of probabilistic predictions about the results of spatially separated experiments that conflict, for appropriate choices of quantum states and experiments, with quantum mechanical predictions.

But that SEP article is just a proof that I am not the only one who believes that Bell's theorem has become a heading for a variety of theorems. What I had in mind personally were reformulations in terms of games, or presentations like Mermin's that try to make the counter intuitiveness of the quantum behavior even more concrete for a general audience. And in terms of different assumptions (instead of separability), I was thinking of papers like Asher Peres' Unperformed experiments have no results which goes with "Let us assume that the outcome of an experiment performed on one of the systems is independent of the choice of the experiment performed on the other."

 

[Nullstein]

The failure of the EPR argument can be demonstrated quite easily:
1. We know that the QM predictions can in principle be reproduced by a local, superdeterministic, contextual theory, i.e. such a theory can violate Bell's inequality.
2. However, EPR claim that the assumption of locality alone implies non-contextuality (pre-existing values). But non-contextuality (+ locality) implies that Bell's inequality holds, as shown in Bell's first paper.
Thus, we arrive at a contradiction.

Basically, EPR were asking the right questions but giving the wrong answers. They just missed the possibility of contextual theories and other undesirable loopholes like superdeterminism. Thus, all discussions based on the EPR argument are bound to fall into the same trap. Today, there is no need to appeal to it, because we have crystal clear quantitative conditions such as the CMC. Arguments based on EPR always tacitly assume all the postulates that are encompassed in the CMC. (And in my experience, more often than not, they do so in bad faith, e.g. Maudlin.)

 

[Fra]

I'm not sure exactly what @Demystifier is meaning by the distinction, but to me, FTL causation is in terms of a proposed law of physics. If the state of one object at a future time depends on the state of distant objects in the recent past (too recently for light to propagate), then that implies FTL causation.

FTL signaling is a special case in which the conditions can be manipulated.

Here's a made-up law of physics that might illustrate the difference. Suppose that there is some weird object, a will-o-the-wisp, which just randomly appears at various locations, and then disappears, only to re-appear at some random spot. Suppose that there is a force $F_{wow}$ which acts instantaneously on electrons everywhere in the universe, and is constant in magnitude, and is directed toward the will-o-the-wisp.

This would imply FTL causation: the will-o-the-wisp affects electrons instantaneously. But it couldn't be used for FTL signaling, since there is no way to control where the will-o-the-wisp appears.

No. Causation is influence of one physical object (or phenomenon) on another. Signaling is deliberate influence of a subject (human or intelligent animal) on something else (another subject or a physical object or phenomenon). Signaling is a very anthropomorphic concept, causation is not so much. In deterministic interpretations of QM such as Bohmian mechanics, there is FTL causation (the position of one particle influences the velocity of another particle), but there is no FTL signalling (a human does not have a control over Bohmian particle positions, i.e. she cannot deliberately put the particle here rather than there).

Thanks for the explanation. I have a similar association of causation and law, as Stevendaryl suggests. But due to the way I think about the nature and emergence of physical laws (my problem of course) in an observer/agent perspective, I tend to classify mentioned "FTL-causations" instead as FTL-correlation - for this reason:

On short time scale:

I think of causation as simply manifesting how the action of an agent/observer is chosen (This reflects the "physical law").

I see no fundamental meaning for an agent to rationally speak about "causation" in an observational record, this is just correlations. (This does not reflect law, it just is what it is, as there are no "choices" to be made)

(The contrast is on evolutionary time scales: I envision a more speculative connection between observed correlations and future causation, and thus formation of law.)

It seems to me what you describe as uncontrollable happenings are merely correlated. Although you imply a "hidden law" that imples the correlation - thus it's a causation (I get and appreciate this). I guess what I fail to see, is how such "hidden law" can be intrinsic to the observer in my own understanding. I agree this is not a hard argument at this point, I just reflects by strange bias.

But this is indeed deeply a question for the nature of causation and law, which was my point, that this seems to be that main troublesome point in the EPR discussion.

It's hard for me to to understand the presumed explanatory logic of the notion of laws, if they are hidden to those who are supposed to follow them? ie. to the inside observers? It seems as strange as to suggest that you need extrinsic curvatures to infer intrinsic geometry.

/Fredrik

 

[atyy]

The failure of the EPR argument can be demonstrated quite easily:
1. We know that the QM predictions can in principle be reproduced by a local, superdeterministic, contextual theory, i.e. such a theory can violate Bell's inequality.
2. However, EPR claim that the assumption of locality alone implies non-contextuality (pre-existing values). But non-contextuality (+ locality) implies that Bell's inequality holds, as shown in Bell's first paper.
Thus, we arrive at a contradiction.

Basically, EPR were asking the right questions but giving the wrong answers. They just missed the possibility of contextual theories and other undesirable loopholes like superdeterminism. Thus, all discussions based on the EPR argument are bound to fall into the same trap. Today, there is no need to appeal to it, because we have crystal clear quantitative conditions such as the CMC. Arguments based on EPR always tacitly assume all the postulates that are encompassed in the CMC. (And in my experience, more often than not, they do so in bad faith, e.g. Maudlin.)

Is the causal markov condition the same argument as given in Woods and Spekkens (Fig. 19)?
https://arxiv.org/abs/1208.4119

Why do you say Maudlin tacitly assumes postulates equivalent to CMC in bad faith?

 

[Demystifier]

And to follow my own advice, here is the original paper by Bell on his theorem:

https://cds.cern.ch/record/111654/files/vol1p195-200_001.pdf

What I am calling the factorizability assumption is equation (2) in that paper.

Bell does not call it "factorizability assumption". Indeed, it can be misleading to call it so because it involves a $\lambda$-integral over products, which is not a product itself. (In fact, I was misled myself when I asked you whether Bertlmann socks violate it.) Is it your own invention to call it "factorizability assumption", or is there another reference where it is called so?

 

[stevendaryl]

Bell does not call it "factorizability assumption". Indeed, it can be misleading to call it so because it involves a $\lambda$-integral over products, which is not a product itself.

I don’t know the answer to the question of who has called it “the factorizability assumption”, but I would argue that it’s appropriate, if you formulate it like this:

For any joint probability distribution $P(A \wedge B)$ describing distant measurement results, there is some collection of facts $\Lambda$ about conditions in the intersection of their backwards lightcones such that

$P(A \wedge B | \Lambda) = P(A|\Lambda) P(B|\Lambda)$

In other words, the assumption is that the conditional probabilities factor, if you condition on their mutual influences.

 

[Fra]

I don’t know the answer to the question of who has called it “the factorizability assumption”, but I would argue that it’s appropriate, if you formulate it like this:

For any joint probability distribution $P(A \wedge B)$ describing distant measurement results, there is some collection of facts $\Lambda$ about conditions in the intersection of their backwards lightcones such that

$P(A \wedge B | \Lambda) = P(A|\Lambda) P(B|\Lambda)$

In other words, the assumption is that the conditional probabilities factor, if you condition on their mutual influences.

I also agree the term was confusing, because the bell equation (2) references in a previous post actually contains two non-trivial assumptions.

1) Partition assumption: $P(A,B |O_ {A}) = \sum_{\lambda} P(A,B|\lambda|O_ {A}) P(\lambda|O_ {A})$
This holds only if we stick to the same probability space and that the lamba set is a mutually disjoint sample space (this is IMO what is wrong).

2) Indpendence assumption, (this is what should be called factorization assumption i suppose), that Steven wrote.

/Fredrik

 

[Demystifier]

I don’t know the answer to the question of who has called it “the factorizability assumption”, but I would argue that it’s appropriate, if you formulate it like this:

For any joint probability distribution $P(A \wedge B)$ describing distant measurement results, there is some collection of facts $\Lambda$ about conditions in the intersection of their backwards lightcones such that

$P(A \wedge B | \Lambda) = P(A|\Lambda) P(B|\Lambda)$

In other words, the assumption is that the conditional probabilities factor, if you condition on their mutual influences.

I don't see how is that related to the Bell's Eq. (2). Note that in (2), the quantities P, A and B are not probabilities. The only probability function in (2) is $\rho(\lambda)$.

 

[WernerQH]

I would add this to the assumptions if there was at least one interpretation of QM which claims to avoid nonlocality by not dealing with individual objects. But I am not aware of any such interpretation.

You are unable to perceive the existence of objects as an assumption because it is "hardwired" into physics as you know it. :-)
By contrast, in condensed matter physics "objects" like phonons, magnons, plasmons, Cooper pairs ... are not viewed as fundamental constituents of matter, but just a useful way of describing correlations.

 

[Demystifier]

You are unable to perceive the existence of objects as an assumption because it is "hardwired" into physics as you know it. :-)
By contrast, in condensed matter physics "objects" like phonons, magnons, plasmons, Cooper pairs ... are not viewed as fundamental constituents of matter, but just a useful way of describing correlations.

I don't see any contrast here. Sure, a phonon is a collective effect, but condensed matter physics still distinguishes single phonon from an ensemble of phonons.

 

[vanhees71]

Free phonons as well as free photons are described by a Fock space. Both phonons and photons are not localized in any classical-particle sense.

 

[Demystifier]

Free phonons as well as free photons are described by a Fock space. Both phonons and photons are not localized in any classical-particle sense.

Yes. It's often stated in the literature that phonon is a quasiparticle, while photon is a true particle. I think it's misleading. Both are collective excitations. Phonon is a collective excitation of a lattice made of atoms. Photon is a collective excitation of the EM field. The math formalisms describing those two kinds of collective excitations are almost identical, especially when one regularizes the field on a lattice.

 

[WernerQH]

I don't see any contrast here. Sure, a phonon is a collective effect, but condensed matter physics still distinguishes single phonon from an ensemble of phonons.

Yes, there is a difference between an individual particle and an ensemble. But the original discussion was about properties of a particle versus properties of an ensemble, and Bell's proof really is about the former. One must not talk about ensembles without being explicit about its members. I agree with your assessment that the statistical interpretation is vague when it talks about particles. For me it makes more sense to think not of pre-existing values, but of values that come into existence through interaction ("measurement"). That's not statistics of particles, but of events.

 

[Demystifier]

Yes, there is a difference between an individual particle and an ensemble. But the original discussion was about properties of a particle versus properties of an ensemble, and Bell's proof really is about the former. One must not talk about ensembles without being explicit about its members. I agree with your assessment that the statistical interpretation is vague when it talks about particles. For me it makes more sense to think not of pre-existing values, but of values that come into existence through interaction ("measurement"). That's not statistics of particles, but of events.

I'm fine with that, I can talk about events without talking about particles. But in the context of statistical interpretation, one then has to distinguish a single event from an ensemble of events. And I think you and me agree that the notion of ensemble of events does not make sense without individual events. But @martinbn seems to have a problem with that.

 

[stevendaryl]

I don't see how is that related to the Bell's Eq. (2). Note that in (2), the quantities P, A and B are not probabilities. The only probability function in (2) is $\rho(\lambda)$.

Sorry, I wasn't looking at the paper when I wrote my note. The connection is a little long, but it's the same assumption, at its root.

In Bell's formula, he has:

$P(\vec{a}, \vec{b}) = \int d\lambda \rho(\lambda) A(\vec{a}, \lambda) B(\vec{b}, \lambda)$

Let me rewrite this using $E$ instead of $P$, so that I can use $P$ for a probability. Then we have:

$E(\vec{a}, \vec{b}) = \int d\lambda \rho(\lambda) A(\vec{a}, \lambda) B(\vec{b}, \lambda)$

In this formula, $A(\vec{a}, \lambda)$ is the result ($\pm 1$) of one measurement conducted using setting $\vec{a}$, and $B(\vec{b}, \lambda)$ is the result of the other measurement conducted using setting $\vec{b}$. $E(\vec{a}, \vec{b})$ is the correlation, which is the average of the products $A B$ over many measurements, for particular choices of $\vec{a}$ and $\vec{b}$. $\rho(\lambda)$ is the probability (or probability density) for the hidden variable $\lambda$.

Instead of correlations, we can use joint probabilities. Let $P(A=x, B = y \ |\ \vec{a}, \vec{b})$ be the probability of the first measurement producing result $x$ and the second measurement producing result $y$ given settings $\vec{a}$ and $\vec{b}$. Then (under the assumption of complete symmetry between the results +1 and -1 and the symmetry between measurements A and B):

$E(\vec{a}, \vec{b}) = 4 P(A=1, B = 1 \ |\ \vec{a}, \vec{b}) - 1$

Bell's form for $E$ can be obtained under the assumptions that:

1. $P(A = x, B = y \ |\ \vec{a}, \vec{b}) = \int d\lambda \rho(\lambda) P(A = x \ |\ \vec{a}, \lambda) P(B = y \ |\ \vec{b}, \lambda)$
2. $P(A = x \ |\ \vec{a}, \lambda) = 0\ \text{or}\ 1$ (deterministic function of settings and $\lambda$)
3. $P(B = y \ |\ \vec{b}, \lambda) = 0\ \text{or}\ 1$ (deterministic function of settings and $\lambda$)

Assumptions 2 and 3 are actually derivable from the assumption of perfect correlations/anti-correlations for the case of identical settings for the two measurements.

So Bell's assumed form of the correlation function is equivalent to the assumption that conditional probabilities factor.

 

[Demystifier]

Then (under the assumption of complete symmetry between the results +1 and -1 and the symmetry between measurements A and B):

$E(\vec{a}, \vec{b}) = 4 P(A=1, B = 1 \ |\ \vec{a}, \vec{b}) - 1$

I don't think that this symmetry assumption is justified, but I also think that you don't actually need it in the rest of your analysis.