Bell's derivation; socks and Jaynes

lugita15

This may just be because I don't grasp Jaynes' argument, but it seems to me that there is no need to go deep in the weeds concerning the mathematics of conditional probabilities. As far as I know, proofs of Bell's theorem (except Bell's original) generally do not even depend on the notion of conditional probability. What is Jaynes' fundamental explanation for the experimental fact that there seem to be nonlocal correlations between measurements of entangled particles, of a kind that is different than the correlations that could arise just from the local sharing of hidden variables between the two particles? Phrased in this way, all the thorny issues of Bayesian probability inference and the like go out the window.

PeterDonis

I believe there are Bell-type results that do not involve probabilities. For example, suppose there were a scenario in which "local realism" would require, not just that probabilities obey certain inequalities, but that certain measurement results simply could *not* happen at all, whereas QM would predict that they could. I seem to remember reading about one such scenario constructed by Roger Penrose using spin-3/2 particles in The Emperor's New Mind, but I don't have my copy handy to check. As far as I can see, Jaynes' arguments wouldn't apply at all to such a scenario.

Delta Kilo

Well, the theory doesn't have to be deterministic. λ can include any number of random variables (in fact expected to include some). Since factorization works for any given value of λ, we don't need to know it, we just need to make sure λ exists. Specifically, we assume there exists probability distribution ρ(λ) independent from a and b: ρ(λ|ab) = ρ(λ) (of course A(a,λ) and B(b,λ) must exist as well, they only make sense together).

It is hard to tell where the breakdown occurs but we can guess. According to Bell, λ can be thought of as all relevant laws of physics and all relevant initial conditions with the exception of values a and b. If only we allow A to depend on b: A=A(a,b,λ), then everything clicks into place and we have a working model (QM). That suggests it's not a problem with general setup but specifically with factorizing a and b out of λ, that is local realism assumption.

harrylin

Yes, and that's why Bell didn't use electrons for his argument. But he dropped Bertlmann's socks and instead he gave an illustration with Booles's Lille and Lyon. However, some of us shortly discussed a Lille-Lyon counter example in a thread that I started a long time ago, but none of us appreciated it much; perhaps Lille-Lyon doesn't catch the detector setting aspect well. It would be more interesting to try an adapted variant of Bertlmann's socks.
So, here's the intro of an example that I had in mind. It's a shot in the dark as I don't know the outcome concerning Bell vs. Jaynes (likely it will support Bell which would "weaken" Jaynes, but I can imagine that it could by chance "invalidate" Bell):

A group of QM students get classes from Prof. Bertlmann. It's an intensive course with Morning class, Afternoon class and Evening class. The students wonder if Bell's story could actually be true and Bertlmann really wears different socks. However Bertlmann happens to wear long trousers and when he goes to sit behind his desk, his socks are out of sight.

Never mind, one student knows a little electronics and makes two devices with LED's to illuminate the socks and light detectors to determine if the sock is light or dark. He hides them on both sides under the desk, aiming at where Bertlmann's socks should appear. With a wireless control he can secretly do a measurement with the press of a button and the result is then indicated by two LED's that are visible for the students, but out of sight for Bertlmann. The next morning he fiddles a bit with the settings and then they wait for Bertlmann [to be continued]

Would such a scenario correspond to post #32 of IsometricPion? I intend to let Morning, Afternoon and Evening be selected by the students, as a and b.

PeterDonis

I'm not sure what you mean by this. He certainly used electrons to derive the *quantum* probabilities, which are what I was talking about in the passage you quoted. Bertlmann's socks, and the heart attacks in Lille and Lyon, are stipulated to be classical objects; there is nothing in their behavior corresponding to the behavior of electrons that undergo successive spin measurements in different directions. That's the point.

harrylin

That's my (and I think also your) point: it didn't make much sense for Bell to use electrons as example to defend the validity of his separation of terms; he had to use an example that we can understand - and he chose to use Lille-Lyon for that.
Now, his socks example is too simple, and none of us appreciated his Lille-Lyon example much when De Raedt presented a variant of it as counter example. And I think that we all agree that Jayne's example is also insufficient. Thus, it may be more instructive to improve Bertlmann's socks example into something like Lille-Lyon. My example keeps the physical separation and adds complexity as well as a certain "weirdness" of observed correlations at varying detector parameters. Only thing I was extremely busy until today so I have not yet worked out the probabilities. It's just a shot in the dark.

harrylin

I am now starting to study the outcomes of my little thought experiment in spreadsheet and it immediately gets interesting as I can now put much more meaning to the symbols and how they are used. Do you agree that the bold term should also apply on my example?

But then I encounter trouble! For what Bell next does (in his socks paper; it's instant in his first paper), is to multiply that term with dλ ρ(λ) [eq.11+12]. It looks to me that for every increment dλ there is a single λ, which appears to be a fixed set of variables because of Bell's "probability distribution" ρ(λ). That sounds pretty much fixed to me for the total experiment of many runs. If not, can someone please explain what the "probability distribution" ρ(λ) exactly means?

IsometricPion

Yes, assuming a local realistic theory for predicting the color of Bertlmann's socks (which I would say is the only intuitive kind in such ordinary situations).I think you're correct. In Bell's eq. 11 it is assumed that one knows the values of the variables that make up λ. His eq. 12 incorporates the fact that in actual experiments λ is not known so he multiplies the joint outcome probability by ρ(λ), the probability density for λ, and integrates with respect to λ to removing it from the equations. There is nothing that intrinsically prevents ρ(λ) from varying between runs in "real life". However, if one is numerically simulating the ensemble distribution of results of an experiment (which is what I assume you are doing), it should not be allowed to vary between runs.

harrylin

Yes, the measurements do not affect each other ("no action at a distance"). BTW, he is in reality wearing ordinary socks.
I'm about to start doing that (I need to add more columns and add a function etc.). So, I'm puzzled by your last remark; why should "real life" not be allowed in a Bell type calculation of reality?

PS. I guess that he wants to calculate the outcome for any (a, b) combination for all possible "real life" λ (thus all possible x), taking in account their frequency of occurrence. It seems plausible that λ (thus (x1,x2)) is different from one set of pair measurements to the next, and now it looks to me that Bell does account for that possibility (but can one treat anything as just a number?). And I suppose that according to Bell the total function of λ (thus X) cannot vary from one total experiment to the next, as the results are reproducible. Is that what you mean?

IsometricPion

I suppose real-life was a bad choice of words. At the time I was thinking of systematic effects that changed ρ(λ) from run to run but left P(AB|a,b) the same. Now that I have thought about it some more, I think a better way to put it is that if one were doing a time or space average (as would be necessary when simulating actual experimental runs, since they do not occur at the same points in space-time) ρ(λ) could vary from run to run and experiment to experiment (as long as P(AB|a,b) stays the same these would describe setting up indistinguishable experiments/runs). When producing different outcomes to obtain an ensemble distribution for a single run, ρ(λ) is fixed since it is part of the initial/boundry conditions of the run.

It is essentially the difference between a time-average and an ensemble average.

There is nothing preventing one from asserting from the start that ρ(λ) is the same for all experiments and experimental runs, it is merely a (reasonable) restriction on the set of hidden variable theories under consideration (which is almost certain to be necessary in order to make the analysis tractible).

harrylin

I'm not sure that I want to go there (at least, not yet); my problem is much more basic. It looks to me that for such an integration to be possibly valid, p(λ) - I mean P(x_A,x_B) - should be the same for different combinations of a and b. Isn't that a requirement?

IsometricPion

I am not entirely sure what you are asking (what do the x's stand for?). If you are asking about the dependence of ρ(λ) on the settings of the detectors a,b then the answer is yes, it should remain the same. This is because ρ(λ) can only depend on λ, which is required to be independent of a,b.

Delta Kilo

Well, ρ(λ) should not change from one run to another, otherwise you won't get repeatable results (I mean repeatable statistics for long runs of course, not repeatable single outcomes). If ρ(λ) does vary, it just means some random factor ζ has not been accounted for, it needs to be lumped into λ'={λ,ζ}, then ρ becomes joint distribution ρ(λ') = ρ(λ,ζ).

harrylin

@ Delta Kilo : Yes, that sounds reasonable, but I have a problem already with one full statistical experiment.
I'm not sure that your suggestion can actually be applied in reality to all types of λ. Even if each λ=(x_A,x_B) is independent of a and b, it seems to me that the probability distribution of the λ that play a role in the measurements could be affected by choices of a and b. Perhaps I'm just seeing problems that don't exist, or perhaps I'm arriving at the point that Jaynes and others actually were getting at, but didn't explain well enough.*

And certainly Bell didn't sufficiently defend properly that his integration is compatible with all possible types of λ. He simply writes in his socks paper: "We have to consider then some probability distribution ρ(λ)", but he doesn't prove the validity of that claim.

So, it may be best that I now give my example together with a small selection of results (later today I hope), and then try to work it out, perhaps with the help of some of you.

*PS: I'm now re-reading Jaynes and it does look as if his eq.15 exactly points at the problem that I now encounter.

harrylin

I don't know what test he talks about, but it appears to refer to slightly different predictions - and that's another way that this paradox could be solved perhaps. Take for example special relativity, would you say that general relativity is "contrary" to it? Moreover, many Bell type experiments do not exactly reproduce simplified QM predictions as often portrayed and which completely neglect correlation in time, selection of entangled pairs, etc.
Anyway, I'm here still at the start of Bell's derivation and which corresponds to Jaynes point 1.

Delta Kilo

Well, that's too bad, that was the whole point of the exercise :) Say, you have 2 photons flying from the source in opposite directions. The source generates λ and each photon carries this λ (or part of it, or some function of it, doesn't matter) with it. Once they fly apart, each photon is on its own as there is no way for any 'local realistic' (≤c) influence to reach one photon from another. Parameters a and b are chosen by experimenters and programmed into detectors while the photons are in mid-flight, again there is no way for the influence from parameter a to affect λ carried by photon B before it hits detector (and vice versa). When the photon hits detector, the outcome is determined by the λ carried by this photon and local parameter a or b.

BTW what are x_A and x_B exactly?

Well, if someone does come up with working Bell-type local realistic theory, they'd better have proper probability distribution ρ(λ), (and being proper includes ρ(λ)≥0, ∫ρ(λ)dλ=1), otherwise how are they going to calculate probabilities of the outcomes?

harrylin

That is a typical example of the kind of non-spooky models that Bell already knew not to work. He claimed to be completely general in his derivation, in order to prove that no non-spooky model is possible that reproduces QM - that is, also the ones that he couldn't think of. Else the exercise was of little use.
x is just my notation of the value of λ at an event (here at event A, and at event B), which I introduced early in this discussion for a clearer distinction with the total unknown X during the whole experiment.
That's not what I meant; I think that the probability distribution of the λ that correspond to a choice of A,B,a,b for the data analysis (and which thus are unwittingly selected along with that choice) could depend on that choice of data. It appears to me that Bell's integration doesn't allow that.