## Entanglement

 Quote by vanesch Well, because of some view that there should be an underlying unity to physics. You're not required to subscribe to that view, but I'd say that physics then looses a lot of interest - that's of course just my opinion. The idea is that there ARE universal laws of nature. Maybe that's simply not true. Maybe nature follows totally different laws from case to case. But then physics reduces to a catalog of experiments, without any guidance. A bit like biology before the advent of its molecular understanding. I think that the working hypothesis that there ARE universal laws has not yet been falsified. Within that frame, you'd think that ONE AND THE SAME theory must account for all experimental observations concerning optics. We have such a theory, and it is called QED. Of course we had older theories, like Maxwell's theory and even the corpuscular theory ; and QED shows us IN WHAT CIRCUMSTANCES these older theories are good approximations ; and in what circumstances we will get deviations from their predictions. It just turns out that in EPR type experiments you are in fact NOT in a regime where you can use Maxwell's theory because it is exactly the same regime in which you have the anti-coincidence counts. In one case however, Maxwell gives you (I'd say, by accident) an answer which corresponds to the QED prediction, in the other case, it is completely off.
I don't think that adopting a more easily visualizable 'classical'
explanation when possible for some experiments destroys the idea
that there are universal organizing principles. I believe that
a coherent 'big picture' that is close to the 'deep' or 'true'
nature of the universe can eventually be developed. (I think that
it will be some sort of wave mechanics that will account
for both the orderly and the chaotic/turbulent aspects of
reality, and that it will provide a communicable 'picture'
in a way that current quantum theory doesn't.) But that
belief isn't why I study physics.

Yes, QED can account for the instrumentally produced
data. But that isn't a picture of the sub-microscopic,
sub-atomic reality. It's a picture of the experimental data.
There is no picture of what light actually is, just
sometimes paradoxical experimental results. Using the
same single-photon light source you can make light behave
as if it is composed of indivisible 'particles' or
divisible waves. (The same setups that produce
anti-coincidence counts can be modified to produce
interference effects.) This could be due to the
interference-producing setups analyzing indivisible units
in aggregate (via combined streams using interferometer
in the beamsplitter setups or long time exposure using
detection location data in the double-slit setups), or it
could be due to instrumental insensitivity to sub-threshold
(divisible) wave activity. The answer isn't clear yet, afaik.

In any case, I don't think the fact that the cos^2 theta
formula works in the standard two-detector optical EPR/Bell
setup, and the fact that it's a 200 year old optics formula is
just a coincidence. (Remember all that stuff about
an "underlying unity to physics" above? :) )

There do seem to be organizing principles that are peculiar
to certain scales and contexts. The phenomenology
of, say, human social interactions is certainly different
than the phenomenology of quantum interactions.

It seems unlikely to me that there will ever be anything
like a quantum gravity. Gravitational behavior (in accordance
with the equivalence principle by the way) can be thought of
as emerging via complex wave interactions many orders of
magnitude greater in complexity than the simpler interactions
that are characterized as quantum. This isn't to say that
there aren't quantum interactions happening in and between
gravitating bodies -- they just aren't important in that
context, they don't *determine* gravitational behavior.

String theory, on the other hand, by positing the existence
of an underlying universal particulate medium, seems very well
motivated, though obviously a contrivance. I think it's
sort of the wrong approach, and even if they get it to
work mathematically for everything that won't necessarily
mean that it's a 'true' description of reality.

 Quote by vanesch ... for me the essence of physics is the identification of an objective world with the Platonic world (the mathematical objects), in such a way that the subjectively observed world corresponds to what you can deduce from those mathematical objects. MWI, CI and Bohmian mechanics are different mappings between an objective world and the Platonic world ; only they lead to finally the same subjectively observed phenomena. Now if physics would be "finished" then it is a matter of taste which one you pick out. But somehow you have to choose I think. However, physics is not finished yet. So this choice of mapping can be more or less inspiring for new ideas.
For me, the essence of physics is the recognition of associations
or connections wrt natural and experimentally observed phenomena
and the ability to quantify those (intuitive?) associations.
(For example, I'll bet you've wondered why there is any motion
at all. Most people just take it as a given. There's motion,
now proceed to Newton's Laws and so on. But, there are
observations that indicate that the universe is expanding
omnidirectionally. Could these observations be the basis
for a new fundamental, universal law?)

I agree that physics is not only not finished, it's pretty
much just getting started. I also think that MWI, CI and Bohmian
mechanics *are* a matter of taste, and not very inspiring. :)

 Quote by vanesch I think that the perfect understanding is a fully coherent mapping between a postulated objective world and the platonic world of mathematical objects, in such a way that all of our subjective observations are in agreement with that mapping. There may be more than one way of doing this. I am still of the opinion that there exists at least one way. Apart from basing the meaning of "explanation" on intuition (and we should know by now that that is not a reliable thing to do), I don't know what else can it mean, to "explain" something.
If there's more than one way of doing it (and using the
method that you advocate almost assures that there will
always be more than one way) then why would you consider
any one of those ways to be the 'perfect' understanding?

One's 'intuition' changes as one learns and observes.

My intuition tells me that, for example, MWI, CI and
Bohmian mechanics are *not* providing us with a true
picture of the real world -- regardless of how
'coherently' they 'map'. I think that most scientists'
intuitions would tell them this, and I think that
scientists intuitive judgements about things should
be taken seriously.

 Quote by vanesch If you have a theory which makes unambiguous, correct predictions of experiments, then in what way is there still something not "understood" ? I can understand the opposite argument: discrepancies between a theory's prediction and an experimental result can point to a more complex underlying "reality". But if the theory makes the right predictions ? I would then be inclined to think that the theory already possesses ALL the ingredients describing the phenomenon under study, no?
Well, yes and no. :) For example, quantum theory makes
correct predictions. But, the *phenomena* under study are
experimental results, not an 'underlying reality' that
the results are, as presumed by some, about. So, you
sometimes get incomprehensible results. From this, the
CI view is that the 'quantum world' is simply
incomprehensible, and that analogies from the world of
our sensory experience are simply inapplicable. And, I
consider that to be a very wrongheaded view.

As for my statement regarding GR as simplistic:
if gravitational behavior is complex wave
interactions, then GR is an oversimplification.
Lots of people think that GR, and even the
Standard Model, won't be up to the task of
handling recent astronomical observations.

And, regarding MWI, I don't consider it to be a
physical theory -- even though it might be
a very clean mapping. :)

 Quote by vanesch The whole "mystery" resides then in 2 things: 1) what about this non-locality ? Clearly it is contained in the quantum formalism (a la Copenhagen) and clearly also it doesn't correspond to any specific dynamics. One cannot say that it is "due to a force yet to be discovered", because it *is* already present in the formalism, and it is NOT some dynamics of unknown sort. 2) How does it come that Bob changes the states in such a way at Alice's that a) this directly influences the probabilities of outcomes Alice will observe, but b) that the mixture of influences that Bob prepares for Alice (when repeating the experiment) is exactly such, that, when weighting with Bob's mixture of outcomes, Alice finds finally a 50/50 probability AS IF Bob didn't influence her stuff. That, to me, sounds like a serious conspiracy :-) It is here that I see a certain superiority of the MWI view: we know why this has to remain 50/50 because after the unitary evolution at Bob, the length of the vectors at Alice weren't influenced.
I don't think this clearly states the essence of the real physical
mystery, which I view as concerning whether all of the light
incident on a polarizer during a certain coincidence interval
associated with a photon detection is being transmitted by the
polarizer or not (there are similar considerations for the two-slit
and beamsplitter setups -- is the emitted light associated with a
photon detection going through both slits when they are both open,
and is the emitted light associated with a photon detection being
both reflected and transmitted after interacting with a beamsplitter?).
That is, it's known what photons *are* theoretically and to a certain
extent instrumentally, but the actual physical nature of photons
isn't known. Hence, there are some interpretational problems.

As for the projection, it's based on the idea that Alice and Bob
are analyzing in the joint context the same value of some physical
property during a certain interval associated with the production of
that value. The projected axis is taken as the axis of
maximum probability of detection because it produced a
detection. This in itself doesn't imply a nonlocal
physical connection between Alice and Bob. The nonlocal
stuff comes from people thinking that Bell proved
that the light incident on the polarizers couldn't
have a common motional property.

But, this is the essence of what Schroedinger called entanglement --
that two objects which have interacted, or have been produced
by the same process (like being emitted via one and the same
atomic transition), carry with them in their subsequent motion
information of the motion imparted via the interaction or
the process that created them. This shared property of
motion will stay with the objects no matter how far apart
they travel, as long as no external torques are introduced
which might modify the value of the shared property.

Probabilities are not explanations. They're descriptions of
behavior at the level of instrumental detection, which to
a certain extent can't be controlled.

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 Quote by Sherlock In any case, I don't think the fact that the cos^2 theta formula works in the standard two-detector optical EPR/Bell setup, and the fact that it's a 200 year old optics formula is just a coincidence. (Remember all that stuff about an "underlying unity to physics" above? :) )
Well, it doesn't really work. It works for ONE specific correlation, under the assumption (which is semi-classically correct) that you have a probability of clicking proportional to incident intensity, namely A+ and B-. Now if you use ABSORBING polarizers, that's all you get, so there it is ok. But if you use *polarizing beam splitters*, it DOESN'T work for some of the other correlations, as I tried to point out in post number 42 in this thread.

Now, if, within the same experiment, a certain way of reasoning explains SOME results, and is in contradiction with OTHERS, then that way of reasoning IS WRONG.

Like my old physics teacher used to say: we know that many solids have a dilatation as a function of temperature. Now, in summer, days are longer, and they are hotter too... (but it doesn't work for the summer nights...)

cheers,
Patrick.

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 Quote by Sherlock That is, it's known what photons *are* theoretically and to a certain extent instrumentally, but the actual physical nature of photons isn't known. Hence, there are some interpretational problems.
Que veut le peuple ?

If you know what they are "theoretically" and you know what they mean instrumentally, what else is there to know ???
A "mechanical" picture (like the discussions people had in the 19th century about *in what matter* the E and B fields had to propagate) ?

cheers,
Patrick.

Regarding cos^2 theta correlation curve in EPR/Bell experiments
you wrote:

 Quote by vanesch Well, it doesn't really work. It works for ONE specific correlation ...
It describes the data curves for a class of setups. Which have
some things in common with the setup from which it was originally
gotten.

You disappoint me if you don't see at least the possibility
of some connection between the two.

Regarding photons, you wrote:

 Quote by vanesch Que veut le peuple ? If you know what they are "theoretically" and you know what they mean instrumentally, what else is there to know ??? A "mechanical" picture (like the discussions people had in the 19th century about *in what matter* the E and B fields had to propagate) ?
I know what 'gods' are 'theoretically'. And I know how people
react to the word. But I have no idea what gods *are*.
That is, I have no way of knowing how (in what form) or if
they exist outside those contexts.

It's sort of the same with photons, except that photons
are a much more interesting subject -- especially entangled
ones.

So, yes, I'd say that there's a lot more to be known
Some sort of mechanical picture of the deep reality
would be nice. Do you think that's impossible?

I think that not being curious in this way would
make physics a lot less interesting.

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 Quote by Sherlock Regarding cos^2 theta correlation curve in EPR/Bell experiments It describes the data curves for a class of setups. Which have some things in common with the setup from which it was originally gotten. You disappoint me if you don't see at least the possibility of some connection between the two.
Sorry to disappoint you :-)

The link is however, rather clear. In the QED picture, the AVERAGE photon count rate is of course equal to the classical intensity, and we know that the classical intensities are related with a cos^2 theta curve.
So if you consider that the light beams are made up of classical *pulses* with random orientation, and you look at the intensities per pulse that get through the polarizers, then you get the cos^2 theta relationship. On average, then, the photon counting rates must also be related by a cos^2 theta relationship.
So *A* way to respect this constraint is just to have a correlation PER EVENT which is given by cos^2 theta. But that doesn't NEED to be so. For A+ and B-, it is so, agreed. But for A+ and A-, they have, in the same classical picture, intensities which vary from 50-50 to 0-100 (namely 50-50 when the incoming classical pulse is under 45 degrees with the polarizing BS orientation, and 0-100 when the classical pulse is parallel (or perpendicular) to the BS orientation). So you would expect a certain correlation rate (about 50%: you have EQUAL intensities in the 50-50 -> full correlation and you have anti-correlation in the 0-100 case).
Well, this IS NOT THE CASE. You find perfect anticorrelation. So this illustrates that the picture of a classical pulse with a random polarization, and a probability of triggering PER CLASSICAL PULSE of the photodetector, proportional to the classical intensity of the individual pulse, DOES NOT WORK IN THIS SETUP. If it doesn't work for certain aspects of the set-up, it doesn't work AT ALL.
The proportionality of detections and classical intensitis only works ON AVERAGE, not nessesarily PULSE PER PULSE.

The ONLY picture which gives you a consistent view on all the data is the photon picture, with a SINGLE DETECTABLE ENTITY PER "PULSE" in each arm. And if you accept THAT, you appreciate the EPR "riddle", and you do not explain it with the old cos^2 theta law, because that SAME cos^2 theta law would also give us SIMULTANEOUS HITS in A+ and A-, which we don't have. The EPR problem is only valid in the case where you do not have simultaneous
YES/NO answers, of course, otherwise you have, apart from a +z and a -z answer, also a (+z AND -z) answer, which changes Bell's ansatz.

But I repeat my question: people do experiments with light because of 2 reasons: it is feasable, and they *assume* already that we accept the photon picture. If you do not do so, then doing the EPR experiment with light is probably not very illuminating (-:.
However, (at least on paper), you can do the same thing WITH ELECTRONS. Now, I take it that you accept that a single electron going onto two detectors will only be detected ONCE, right ? Well, according to quantum theory, you get exactly the same situation (the cos^2 theta correlation) there. So how is this now explained "classically" ?

(ok, the angle is now defined differently because of the difference between spin-1 and spin-1/2 particles).

Do you:
a) think that QM just makes a wrong prediction there ?
b) do not accept that a single electron can only be detected in 1 detector ?
c) other ?

cheers,
Patrick.

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 Quote by Sherlock So, yes, I'd say that there's a lot more to be known about photons, about light, than is currently known. Some sort of mechanical picture of the deep reality would be nice. Do you think that's impossible?
No, it is not impossible, Bohm's theory does exactly that.
The main objection I have against the view that we need a mechanical picture as an explanation, is: what MORE does a mechanical picture explain ? Isn't it simply because we grew up with Newton's mechanics, and the associated mathematics (calculus) and we develloped more "gut feeling" for it ? What is so special about some mechanical view of things ? I have nothing *against* a mechanical view, but I don't think a mechanical view is worth sacrifying OTHER ideas. And that's what, for instance, Bohm's theory does: it sacrifices locality (and so does the projection postulate).

I will agree with you that quantum theory, or general relativity, or whatever, doesn't give us a "final view" on how nature "really" works ; for the moment however, it is the best we have. 300 years from now, I'm pretty sure that our paradigms will have changed completely, and people will look back on our discussions with a smile in the same way we could look back on people develloping a "world view" based upon a newtonian picture. And they are being naive, because 600 years from now, their descendants will again have changed their views :-)

So for short I think it is a meaningless exercise to try to say what nature "really" looks like. But what you can try to do is to build a mental picture that gives you the clearest possible view on how nature is seen using things that we KNOW right now. It is in that context that I see MWI. I do not know/think/hope that the MWI view is the "real" view on the world (which, I outlined, I don't think we'll ever have). I think that MWI is about the purest mental picture of quantum theory, because *it respects most of all its basic postulates*. That's all. If you do formally ugly things, such as the projection postulate, to get "closer to your gutfeeling about nature" I think you miss the essential content of quantum theory, and as such I think you're in a bad shape to see where it could be extended, because you already mutilated it !

cheers,
Patrick.

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 Quote by vanesch That's the point. There are no hidden variables, and everything is local. So what gives, in Bell ? What gives is that, from Alice's point of view, Bob simply didn't have a definite result, and so you cannot talk about a joint probability, until SHE "decided" which branch to take. But when she did, information was present from both sides, so the Bell factorisation hypothesis is not justified anymore. ... As I said, it is much less spectacular this way, because you only have Alice having a "superposition" of states of her photon. It's more spectacular to have her have a superposition of states of Bob. cheers, Patrick.
Thanks, that helps me to understand this perspective better!

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 Quote by Sherlock In any case, I don't think the fact that the cos^2 theta formula works in the standard two-detector optical EPR/Bell setup, and the fact that it's a 200 year old optics formula is just a coincidence. (Remember all that stuff about an "underlying unity to physics" above? :) )
There are definitely TWO ways to look at that statement. Some of the vocal local realists argue that the cos^2 law isn't correct! They do that so the Bell Inequality can be respected; and then explain that experimental loopholes account for the difference between observation and their theory.

Clearly, classical results sometimes match QM and sometimes don't; and when they don't, you really must side with the predictions of QM. Even Einstein saw that this was a steamroller he had to ride, and the best he could muster was that QM was incomplete.

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I said the following:

 Quote by vanesch And if you accept THAT, you appreciate the EPR "riddle", and you do not explain it with the old cos^2 theta law, because that SAME cos^2 theta law would also give us SIMULTANEOUS HITS in A+ and A-, which we don't have. The EPR problem is only valid in the case where you do not have simultaneous YES/NO answers, of course, otherwise you have, apart from a +z and a -z answer, also a (+z AND -z) answer, which changes Bell's ansatz.
and I would like to illustrate WHERE it changes Bell's ansatz.

Consider again 3 directions, a, b and c, for Alice and Bob.

Alice has an A+ and an A- detector, and Bob has a B+ and a B- detector.
Usually people talk only about the A+ hit or the "no-A+ hit" (where it is understood that the no-A+ hit is an A- hit).

We then take as hidden variable a bit for each a, b and c:

If we have a+ this means that Alice will have A+ and bob will have no B+ in the a direction, if we have a b+ that means that Alice will have an A+ and bob will have no B+ in the b direction, and ...

So we can have: a(+/-) b(+/-) c(+/-) as hidden state. But that description already includes the anti-correlation: if A+ triggers, then A- does NOT trigger, and if A- triggers, then A+ does not trigger. When A+ and A- do not trigger, that is then assumed to be due to the finite quantum efficiencies of the detector, which lead to the "fair sampling hypothesis".

But if we accept the possibility that A+ AND A- trigger together, then each direction has, besides the + and - possibility, a THIRD possibility namely X: double trigger. So from here on, we have 27 different possible states. This changes completely the "probability bookkeeping" and Bell's inequalities are bound to change. The local realist cloud even introduces a fourth possibility: A+ and A- do not trigger, and this is not due to some inefficiency, with symbol 0.

So we have a(+/-/X/0), b(+/-/X/0), c(+/-/X/0) which gives us 64 possibilities.
You can then easily show that Bell's inequalities are different and that experiments don't violate them.

The blow to this view is that whenever you make up a detector law as a function of intensity which allows you to consider the 0 case, you also have to consider the X case. The X case is never observed, so there are reasons to think that the 0 case doesn't exist either, especially because QED tells us so, and that you do get out the right results (including the observed number of 0 cases) when applying the quantum efficiency under the fair sampling hypothesis.

cheers,
Patrick.

 Quote by DrChinese That's sorta funny, you know. Application of classical optics' formula $$cos^2\theta$$ is incompatible with hidden variables but consistent with experiment.
The cos^2 theta formula isn't incompatible with hidden
variables.

For the context of individual results you can write,

P = cos^2 |a - lambda|,

where P is the probability of detection, a is the
polarizer setting and lambda is the variable
angle of emission polarization.

This doesn't conflict with qm. If you knew
how it was varying (other than just that
it's varying randomly), then you could more
accurately predict individual results (by
individual results I mean the data streams
at one end or the other).

How do we know that there *is* a hidden
variable operating in the individual measurement
context? Because, if you keep the polarizer
setting constant the data stream varies
randomly.

Now, this hidden variable doesn't just
stop existing because we decide to
combine the individual data streams wrt
joint polarizer settings.

However, the *variability* of lambda
isn't a factor wrt determining coincidental
detection.

 Quote by DrChinese a b and c are the hypothetical settings you could have IF local hidden variables existed. This is what Bell's Theorem is all about. The difference between any two is a theta. If there WERE a hidden variable function independent of the observations (called lambda collectively), then the third (unobserved) setting existed independently BY DEFINITION and has a non-negative probability. Bell has nothing to do with explaining coincidences, timing intervals, etc. This is always a red herring with Bell. ALL theories predict coincidences, and most "contender" theories yield predictions quite close to Malus' Law anyway. The fact that there is perfect correlation at a particular theta is NOT evidence of non-local effects and never was. The fact that detections are triggered a certain way is likewise meaningless. It is the idea that Malus' Law leads to negative probabilities for certain cases is what Bell is about and that is where his selection of those cases and his inequality comes in. Suppose we set polarizers at a=0 and b=67.5 degrees. For the a+b+ and a-b- cases, we call that correlation. The question is, was there a determinate value IF we could have measured at c=45 degrees? Because IF there was such a determinate value, THEN a+b+c- and a-b-c+ cases should have a non-negative likelihood (>=0). Instead, Malus' Law yields a prediction of about -10%. Therefore our assumption of the hypothetical c is wrong if Malus' Law (cos^2) is right.
Bell demonstrated that using the variability of lambda
to augment the qm formulation for coincidental
detection gives a result that is incompatible
with qm predictions for all values of theta
except 0, 45 and 90 degrees.

Now, there's at least two ways to interpret Bell's
analysis. Either (1) lambda suddenly stops existing when we
decide to combine individual results, or (2) the variability
of lambda isn't relevant wrt joint detection.

I think the latter makes more sense, and in fact
it's part of the basis for the qm account which
assumes that photons emitted by the same atom
are entangled in polarization via the emission
process. This is why you have an entangled
quantum state prior to detection. So, all you
need to know to accurately predict the
*coincidental* detection curve is the angular
difference between the polarizer settings. And,
as in all such situations where you're analyzing,
in effect, the same light with crossed linear
polarizers the cos^2 theta formula holds.

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 Quote by Sherlock The cos^2 theta formula isn't incompatible with hidden variables. For the context of individual results you can write, P = cos^2 |a - lambda|, where P is the probability of detection, a is the polarizer setting and lambda is the variable angle of emission polarization.
Ok, that's the probability for the A+ detector to trigger. And what is the probability for the A- detector to trigger, then ? P = sin^2 |a - lambda| I'd say...

cheers,
Patrick.

EDIT:

I played around a bit with this, and in fact, it is not so easy to arrive at a CORRELATION function which is cos^2(a-b). Indeed, let's take your probability which is p(a+) = cos^2(lambda-a).
Assuming independent probabilities, we have then that the correlation, which is given by p(a+) p(b+) = cos^2(lambda-a) sin^2(lambda-b) for an individual event. (the b+ on the other side is the b- on "this" side)

Now, by the rotation symmetry of the problem, lambda has to be uniformly distributed between 0 and 2 Pi, so we have to weight this p(a+) p(b+) with this uniform distribution in lambda:

P(a+)P(b-) = 1/ (2 Pi) Integral (lambda=0 -> 2 Pi) cos^2(lambda-a) sin^2(lambda-b) d lambda.

If you do that, you find:

1/8 (2 - Cos(2 (a-b)) ) = 1/8 (3-2 Cos^2[a-b])

And NOT 1/2 sin^2(a-b) !!!

I checked this with a small Monte Carlo simulation in Mathematica and this comes out the same. Ok, in the MC I compared a+ with b+ (not with b-), and then the result is 1/8 (2+cos(2(a-b)))

So this specific model doesn't give us the correct, measured correlations...

cheers,
Patrick.

I attach the small Mathematica notebook with calculation...
Attached Files
 coslaw.zip (10.6 KB, 4 views)

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 Quote by Sherlock The cos^2 theta formula isn't incompatible with hidden variables. For the context of individual results you can write, P = cos^2 |a - lambda|, where P is the probability of detection, a is the polarizer setting and lambda is the variable angle of emission polarization. This doesn't conflict with qm. If you knew the value of lambda, or had any info about how it was varying (other than just that it's varying randomly), then you could more accurately predict individual results (by individual results I mean the data streams at one end or the other). How do we know that there *is* a hidden variable operating in the individual measurement context? Because, if you keep the polarizer setting constant the data stream varies randomly. Now, this hidden variable doesn't just stop existing because we decide to combine the individual data streams wrt joint polarizer settings. Now, there's at least two ways to interpret Bell's analysis. Either (1) lambda suddenly stops existing when we decide to combine individual results, or (2) the variability of lambda isn't relevant wrt joint detection. I think the latter makes more sense, and in fact it's part of the basis for the qm account which assumes that photons emitted by the same atom are entangled in polarization via the emission process. This is why you have an entangled quantum state prior to detection. So, all you need to know to accurately predict the *coincidental* detection curve is the angular difference between the polarizer settings. And, as in all such situations where you're analyzing, in effect, the same light with crossed linear polarizers the cos^2 theta formula holds.
Or Lambda=LHV does not exist, a possibility you consistently pass over. It is a simple matter to show that with a table of 8 permutations on A/B/C, there are no values that can be inserted that add to 100% without having negative values at certain angle settings.

A=___ (try 0 degrees)
B=___ (try 67.5 degrees)
C=___ (try 45 degrees)

Hypothetical hidden variable function: __________ (should be cos^2 or at least close)

1. A+ B+ C+: ___ %
2. A+ B+ C-: ___ %
3. A+ B- C+: ___ %
4. A+ B- C-: ___ %
5. A- B+ C+: ___ %
6. A- B+ C-: ___ %
7. A- B- C+: ___ %
8. A- B- C-: ___ %

It is the existence of C that relates to the hidden variable function. What you describe is just fine as long as we are talking about A and B only. (Well, there are still some problems but there is wiggle room for those determined to keep the hidden variables.) But with C added, everything falls apart as you can see.

You can talk all day long about joint probabilities and lambda, but that continues to ignore the fact that you cannot make the above table work out. If you are testing something else, you are ignoring Bell. After you account for the above table, then your explanation might make sense. Meanwhile, the Copenhagen Interpretation (and MWI) accounts for the facts that LHV cannot.
 I would like to point out, in a previous round against Vanesh about EPR and many worlds, the following point (1) : Usual "orthodox Copenhagen QM" contains 1) a local hidden variable that corresponds to the specification of the PRECISE endstate when the latter is degenerate. The "standard" Copenhagen QM is a special configuration of the endstate that corresponds to it's maximum. However, there is more : 2) a NON-LOCAL hidden variable. Let see the latter : a non-local measurement is obtained by the operator : $$\sigma_z\otimes\(\sigma_z\cdot\vec{n}_b)$$...hence Both side are measured, and there is no 1 operator on the other (non disturbing operator). Let consider $$\theta_b=0$$ Hence : both directions of measurement are the same. The clearly the only 2 possible endstates are : |+-> or |-+>, with $$p(+-)=|<+-|\Psi>|^2=\frac{1}{2}=p(-+)$$ This sounds very like more than intuitive and easy to understand. However, one can see the things in an other way, by looking that : $$M=\sigma_z\otimes\sigma_z=\left(\begin{array}{cccc} 1 &&&\\&-1&&\\&&1&\\&&&-1\end{array}\right)$$ Hence, then eigenvalues of M are 1,-1 and are both degenerate. 1 corresponds to |A=B> and -1 to |A<>B> (same or different results in A and B). Here again, the eigenSPACE can be parametrized : $$|same>=\left(\begin{array}{c}\cos(\chi)\\0\\\sin(\chi)\\0\end{array}\ri ght)$$ $$|different>=\left(\begin{array}{c}0\\cos(\delta)\\0\\\sin(\delta)\end{a rray}\right)$$ $$|\Psi>=\frac{1}{\sqrt{2}}\left(\begin{array}{c}0\\1\\-1\\0\end{array}\right)$$ So that : $$p(different)=||^2=\frac{1}{2}\cos(\delta)^2$$ $$p(same)=||^2=\frac{1}{2}\sin(\chi)^2$$ Where $$\chi,\delta$$ are GLOBAL HIDDEN VARIABLES... So that in fact 2p(same)=1 at MAX.......what is the interpretation of this, if there is no mistake of course....??

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 Quote by kleinwolf So that in fact 2p(same)=1 at MAX.......what is the interpretation of this, if there is no mistake of course....??
To me the interpretation is that your chi and delta are just variables that parametrize the eigenspaces of the operator sigma_z x sigma_z.

However, I don't understand your calculation. When you write out sigma-z x sigma-z, I presume in the basis (++, -+,+-,--), then I'd arrive at a diagonal matrix which is (1,-1,-1,1)... You seem to have taken the DIRECT SUM, no ?

cheers,
Patrick.
 Yes, you're entirely right...my mistake is unforgivable, since this will change all the afterwards calculation and interpretation of $$\delta$$. Then the result is $$p(same)=0\quad p(diff)=\frac{1}{2}(1-\sin(2\chi))$$ However, you admit there are 2 visions of computing the probabilities with your correct M : locally : p(+-)=p(-+)=1/2 globally, the endstate |->_g=(0,cos(a),sin(a),0), gives the prob : p(+-)=cos(a)^2, p(-+)=sin(a)^2...hence on average or special values of a, the same as locally....but a infinite of possibilities more are allowed. Can this be measured on the statistical results in an experiement, and how to find how to change the value of a experimentally ??