Bell's derivation; socks and Jaynes

PeterDonis

A good question, to which I don't have a ready answer. Neither did Jaynes, for that matter. But pointing out that nobody has (yet) thought of a way to answer this question is not the same as *proving* that local realism *requires* the correlations to work the way Bell assumed they did. Bell didn't prove this; he just assumed it. The assumption certainly looks reasonable; it may even be true. But that's not the same as proving it true.

Another point that Jaynes makes is worth mentioning. We don't even understand why quantum measurements work the way they do for spin measurements on *single* particles. I take a stream of electrons all of which have come from the "up" beam of a Stern-Gerlach measuring device. I put them all through a second Stern-Gerlach device oriented left-right. As far as I can tell, all the electrons in the beam are the same going in to the second device, yet they split into two beams coming out. Why? What is it that makes half the "up" electrons go left and half go right? Nobody knows.

One response to this, which has been the standard response in QM, is to redefine what counts as a physical explanation. Physics no longer has to explain why particular events happen in particular ways; in QM, it's now sufficient to explain probabilities over ensembles of "similar" events, without even pretending to explain why the individual events themselves turn out the way they do.

Another response, which is Jaynes' response, is to say that our physical knowledge is simply insufficient at this point, and what we ought to be doing, rather than lowering our standards of explanation, is to look harder for underlying mechanisms. Such a search may not yield any results; but Jaynes' claim is basically that since QM was adopted physicists haven't really been trying very hard. Perhaps if we looked harder, we would figure out an underlying mechanism that explained why half the "up" electrons go left and half go right; and once we had that mechanism, we might find that it also explained the EPR correlations in a way that showed how local realism can be true even if the probabilities don't factorize.

I'm not saying Jaynes' response is necessarily right; but it doesn't seem to me that it can just be rejected out of hand either.

DrChinese

It is safe to say that Bell used a definition of realism that Einstein would have appreciated. Specifically, Einstein stated that there IS a reality independent of the act of observation. In addition, EPR defines something called elements of reality and clearly Bell was trying to model that. And nicely he did!

So I would happily say that Jaynes and others may have different definitions of realism, and under their definitions, local realism is quite possibly not ruled out.

lugita15

I think you misunderstood me. I was asking a rhetorical question. Clearly, if the correlation depends on something other than local variables, it is definitionally not a local theory, end of story. That was the point I was trying to make.

PeterDonis

I agree, Bell did a good job of capturing what EPR were getting at.

DrChinese

If Bell's is wrong (which can be stated many different ways, and has already in this thread: What is a better definition of realism?

If p(x,y,z)>=0 for any x, y, z doesn't work, I think we have a bigger problem. I keep asking this, and so far, not a single local realist will give me a satisfactory *alternative* definition. All the while rejecting Bell's. And Einstein's!

PeterDonis

It's not that simple. First of all, remember that Jaynes views probabilities as expressing our knowledge about reality, not reality itself. When he writes conditional probabilities that condition on "other than local variables", he's not saying there's any "action at a distance" that occurs physically; he's merely saying that, *logically*, knowledge of those "other than local variables" can in principle change your posterior probability estimates.

Second, however, Jaynes hiimself points out that, actually, the probabilities [itex]P(A|ab\lambda)[/itex] and [itex]P(B|ab\lambda)[/itex] (i.e., the ones that apparently depend on *both* sets of measurement settings, but *not* on either measurement result--each of these appears as one of two factors in the two versions of the "factorized" equation that I took from Jaynes' equation 15) can actually be simplified, because it's easy to show that knowledge only of the *direction* of the "a" measurement, for example, gives no additional information about the probabilities of possible results of the "b" measurement. So the two conditional probabilities above can actually be simplified to [itex]P(A|a\lambda)[/itex] and [itex]P(B|b\lambda)[/itex]--meaning that the probabilities that condition only on the measurement settings (not on the results) *are* "local" in the sense you are using the term.

The additional information that *does* change the posterior probabilities is knowledge of the *results* of the measurements, A and B. But the observer at the "a" measurement doesn't know the result of the "b" measurement until it reaches him via a light signal, and vice versa. So the actual correlations that are observed could, in principle, be explained entirely by information traveling at light speed or less; there is nothing in the probability functions themselves, once simplified as above, that rules that out.

PeterDonis

I'm not sure what you're driving at. Do you see something in a viewpoint like Jaynes' that appears to violate this condition?

PeterDonis

I wasn't questioning Bell's definition of "realism", only of "local realism", and only the "local" part.

As far as alternative definitions, I don't have any ready-packaged one, but I do have an observation: in Quantum Field Theory, the definition of "causality" is that field operators have to commute at spacelike separations. There is nothing in there about lack of correlations or what variables correlations can depend on; the only requirement is that, if two measurements are spacelike separated, the results can't depend on which one occurs first. The QM probabilities in EPR experiments certainly meet that requirement. Would something along these lines count as an alternative definition of "local realism"?

IsometricPion

I assumed local meant that each outcome was only a function of properties at the measuring device (i.e., the settings of the device and the values of any hidden variables evaluated at that event in space-time). Note that without the sufficiency gauranteed by realism one cannot factor the joint probability as I did above (the last equality in the last line is no longer valid in general).

I assumed realism to mean that a thing's state exists independent of measurement. I took this to imply that the measurement is determined completely by the other properties in the system (i.e., A=A(B,a,b,λ); B=B(A,a,b,λ) and since this must hold for the states of both objects, A=A(a,b,λ); B=B(a,b,λ)).

I made no reference to correlations anywhere except when calculating the joint probability at the end. Do you mean (in "questioning... only the 'local' part") that the analysis is no longer local when one is examining correlations between events separated by a spacelike interval? If so, I would say that locality does not apply to analyses, only to interactions between things modelled by the theory in question (though I could be wrong). As an aside, Wikipedia- Principle of locality indicates that QFT obeys locality.

lugita15

I've always felt the whole debate about whether quantum mechanics is local to be largely semantics. To my mind, entanglement makes it rather clear that quantum mechanics is nonlocal, but of course many people have the positon that entanglement means that QM possesses "quantum nonlocality" (spooky action at a distance), to be distinguished from "classical nonlocality".

Delta Kilo

Yes, this is exactly the assumption of local realism: the outcome P(A|a,λ) does not depend on the choice of parameter b and vice versa. Let's see how Bell explains it:
(eq (10) is the same as Bell's eq(11) and Jayne's eq (14) except M and N are renamed to A and B)

The factorization in eq. (10) means P1(...) and P2(...) are independent. Now the reason they are independent is because Bell chose it to be that way, by encapsulating all common factors in parameter λ. The underlying assumption here is that the values M and N are affected by some common factors (represented by λ), local factors a and b, and some residual randomness, independent of either global or local influences. This residual randomness is the reason we have probabilities P1, P2 at all, without it we would have
deterministic functions M(a,λ) and N(b,λ).

Eq (10) is valid for any given a,b,λ. Say, we discovered that the number of cases in both Lyons and Lille is influenced by day of week, so we are only comparing the results on a given day (say on Friday). Then we discovered a correlation with the stock market, so we only compare the results from only those Fridays when stock market was bearish. etc. And we keep doing that that until the residual randomness is independent.

I repeat, P1 and P2 are independent by design. If they turn out not to be independent, it just means we didn't do a good enough job with λ and overlooked some common factor. There is no limit on what λ can contain, except for local factors a and b, in accordance with physical model of local realism.

Now, there is an easy way to get rid of the residual randomness, by lumping it into λ. We can introduce random variables χ and η representing residual randomness of M and N. In case of Lyons and Lille they would represent the health of the population, their susceptibility to heart attack, including random fluctuations. χ(a,λ) might be a random function which tells whether a person in Lyons is going to have a heart attack given local and global factors a and λ. M and M then become deterministic functions M=M(χ,a,λ), N=N(η,b,λ), probabilities P1(M|χ,a,λ) and P2(N|η,b,λ) become {0,1} and eq (10) is automatically satisfied. Then we just redefine λ to include χ,η: λ' = {λ,χ,η}. It does expand the meaning of λ, which now means not just common global factors but any factors at all whether local or global, but excluding a and b. This is basically what was done from the outset in eq (2) in Bell's EPR paper.

Jaynes says that the fundamentally correct equation is
P(AB|abλ) = P(A|Babλ) P(B|abλ) (15)
Well, where did that come from? It's just the axiom of conditional probability P(AB) = P(A|B) P(B) with abλ tucked in. It is of course trivially true, but the locality assumptions and the special role of λ have been thrown out with the bathwater. Basically, while (14) is a physical model of a particular EPR setup with added local realism assumption, (15) is a tautology in a form 2*2*x = 4*x which tells us absolutely nothing.

Now, let's talk about 1st of the two objections:
I assume Jaynes refers here to the following quite:
Note the subtle difference: Jaynes talks about causal dependence of one outcome random variable on another random variable, while Bell talks about dependence of random variable on free parameter. The difference is, with two random variables they may be dependent and you cannot say whether X causes Y, Y causes X or both X and Y are caused by some third factor. In Bell's case of random variable and free parameter, dependency is clearly one way: the outcome depends on the parameter but not the other way around. The parameter is a given, it does not depend on anything else. This is actually one assumption which is implied and not stated directly. Violation of this assumption represents superdeterminism loophole, which is currently being discussed in another wardthread.

As an illustration of his point, Jaynes gives Bernoulli Urn example. Let's start with eq (16):
P(R1|I)=M/N
I'd say I was introduced here to mimic Bell's λ. But what is the meaning of I exactly?
So I is not a random variable, nor a parameter. It does not have a set of values you can integrate over. It never changes. Basically it does absolutely nothing. Also note conspicuous absence of local parameter a or its equivalent. And without a, the whole thing misses the point.

Now if we are to re-introduce a and λ according to Bell's recipe, we would define a as a free local parameter which applies to the first measurement only. Say, a is a location of the ball to be picked during the first draw. Correspondingly b is the location of the ball to be picked on the second draw. λ is a random variable which by definition includes everything else which might possibly affect the outcomes. In this case λ would be exact arrangement of the balls in the urn. Clearly λ and a together completely determine which ball is drawn first: R1 = R1(a,λ). State of the urn after the first draw is γ=γ(a,λ) and second ball R2=R2(b,γ)=R2(a,b,λ). Note that expression for R2 violates Bell locality assumption and so the whole setup is clearly different from Bell's. Anyway, R1 and R2 are fully determined by a, b, and λ and therefore do not depend on anything else:
P(R1|R2abλ)=P(R1|aλ)={1: R1=R1(a,λ), 0: otherwise}
P(R2|R1abλ)=P(R2|abλ)={1: R2=R2(a,b,λ), 0: otherwise}.
Easy to see that factorization P(R1 R2|a,b,λ)= P(R1|aλ)P(R2|abλ) is in fact correct.

R1=R1(a,λ) and R2=R2(a,b,λ) above are deterministic functions, like in EPR paper. We could add some local residual randomness to them to get the equation similar to eq. (10) from Berltmann's Socks paper. For example, a and b would select x-coordinate of the ball to be drawn and y-coordinate would be picked at random. As long as random functions R1 and R2 are independent, the factorization will be valid. Again, this randomness can always be moved from R1 and R2 into λ.

So what is missing in Jaynes paper? Well, the elephant in the room of course, I mean the λ. λ is a key feature of Bell's paper and it is completely absent in Jaynes example. λ by definition encapsulates all randomness and all parameters in the system, except a and b. Once particular values of λ,a,b are fixed, everything else is predetermined. Without λ, the best posteriori estimate of conditional probability P(R1|...) would necessarily include dependency on R2 and vice versa. Once we nail down λ,a,b, all other dependencies disappear.

Delta Kilo

No it does not. The results A and B are already fully defined by a,b, and λ.

PeterDonis

I didn't say that knowledge of the results changes the results. I said that knowledge of one result changes the *posterior probability* that you would compute for the other result.

DrChinese

Here is my point. Start with ONE photon, not 2. Apply realism to that. That means that there is a well defined value for the result of a polarization measurement at 0, 120 and 240 degrees. So this means that p(0=H,120=H,240=H) or any permutation is >=0. Do you agree with this? If so, yours and mine and Bell's definitions are alike. The problem Bell found starts here. You can see that when you try to put down values for what they would be for any reasonable sample - it won't agree with Malus (and I do mean Malus here).

So what I am saying is that once you set up the realistic scenario you are looking to test, you add an entangled (essentially cloned) photon in to help accomplish that. When that photon is tested remotely to the first, you are also require the assumption of observational locality - a setting here does not affect an outcome there, and vice versa. How can a local realist object to this?????

So if Jaynes were to agree with this definition of realism, I really don't see what his objection would be to Bell. Again, I am not trying to derail the conversation so much as understand it. If Jaynes is picking on a detail of what Bell wrote, but which has since been readily clarified by hundreds of writers, I just miss the issue entirely.

DrChinese

This helps. Thanks.

Delta Kilo

It does it if you don't know λ: P(A|Bab) ≠ P(A|ab)
But for a given a,b,λ it doesn't. P(A|Babλ) = P(A|aλ) = { 1: A=A(a,λ), else 0 }

And that is the crux of the argument. λ is what makes Bell's factorization possible but Jaynes completely ignores it in his paper.

PeterDonis

Well of course, if you have a completely deterministic theory (which Bell's "local realistic" theory is), and you have complete knowledge of initial conditions, then you have complete knowledge of outcomes. I was talking about the case (which is the only case of real interest if we're trying to compare a "local realistic" theory in Bell's sense with QM) where we don't know λ, since that's the case Bell and Jaynes are discussing.

(And of course the actual QM probabilities do *not* factorize as above; that is, there is *no* "local realistic", in Bell's sense, set of hidden variables λ that allows perfect prediction of outcomes.)

I'm not disputing that "λ makes Bell's factorization possible", and I don't think Jaynes was either. As I've said in previous posts, I think Jaynes was saying that requiring there to be some such set of hidden variables λ might not be the correct definition of "local realism".