Bell's Theorem with Easy Math - Stuck

[Ilja]

EPR talks about predicting values for a particle based on somehow knowing the same value on its paired/entangled particle. However, Bell's theorem talks about predicting measurements of a value for a particle based on the result of a measurement of the same value on its paired/entangled particle.

It is this leap from talking about theoretical real values to just the results of measurements of those values, which is probably intuitive for you guys, which I have difficulty understanding.[emph. mine]

So we have the measurement result A which somehow depends on the state of reality λ and what is measured a. These are, without doubt, different things.

The aim of EPR was to prove that, given Einstein causality and realism, the λ essentially contains the A(a) for all a, thus, contains more than allowed if QM is complete. So they wanted to prove something about λ.

Instead, the aim of Bell was an impossibility theorem. Given Einstein causality and realism, we obtain a contradiction with the empirical prediction of QM. So, Bell's interest was not to prove something about λ. It can be whatever you like. The contradiction follows from the predictions about the observables A. We need realism only for a single purpose: A should have to form A(a,λ) for some λ, whatever it is.

If one is red then it is very much more likely that the other two are opposites than the same, so if I randomly pick two of the three my probability of getting red+red is about 1/4 not 1/3.

No. You have three choices to pick two of the three cards: 1+2, 1+3, 2+3. Whatever the colors of the cards, at least one choice gives equal color, or red+red or black+black, because it is impossible that all three pairs have different color. (Think about picking immediately a pair, not of picking them separately, it is easier.)

 

[Badvok]

The aim of EPR was to prove that, given Einstein causality and realism, the λ essentially contains the A(a) for all a, thus, contains more than allowed if QM is complete. So they wanted to prove something about λ.

Yes, I see that. However, is it not true that in a case like testing photon polarisation A(λ,a) would depend in part on something in the measurement device, i.e. λ would be likely to include a LHV from the measurement apparatus?

 

[DrChinese]

This is a step that I have trouble understanding. Why are we reducing it to + and -? Why is there a preset limit for saying it is + or -? I don't see how that comes from EPR. The + and - are simply limits imposed by the experimental apparatus and I can't see how they are in themselves elements of reality.
My current understanding of the LHV concept is that whether an experiment registers + or - will be dependent on a LHV λ, and as Bell says that can be one or any number of locally hidden values. For a photon polarisation experiment is it not reasonable to assume that part of λ is in the detectors and not just in the photon? Or in other words, we can't reduce it to an exact either/or situation, we can only get a probability for + and a probability for - for any given LHV that is defined only for the photon.

The element of reality is ASSOCIATED with the observable. We don't claim to understand WHAT is the REAL component or components. So suppose that the true hidden variables are something I represent as follows: {13, -6, -18}. The sum of these (completely made up) hidden variables is -11. Let's say if the sum is negative, you see a - at the detector and if it is positive you see a +.

Now, all we know is the + or - and cannot see the {13,-6, -18} set. All we know is that Alice and Bob see perfect correlations. For all we know, when Alice's photon is {13,-6, -18}, Bob's photon might actually be {-4,-3, 5}. They both give the - result so that would work.

But you are not correct when you say "we can't reduce it to an exact either/or situation, we can only get a probability". Perfect correlations say that both will be same every time! And notice that the state of the separated measuring devices makes no difference! You can completely ignore that, because it obviously won't be a determining factor - otherwise sometimes one measuring device would influence in one direction, the other would influence in the opposite direction.

 

[Badvok]

You can completely ignore that, because it obviously won't be a determining factor - otherwise sometimes one measuring device would influence in one direction, the other would influence in the opposite direction.

But that is exactly what I don't understand, how can I completely ignore that? Haven't we known for ages that a photon polarised at θ will only pass a polarisor set at angle α with a probability proportional to cos²(α - θ)? So it is possible that one device would influence in one direction and the other could influence in the opposite direction? I don't see how you can not include the possibility of something in the measuring devices influencing the measurements. Isn't it this that makes it impossible to experimentally achieve perfect correlation?

 

[DrChinese]

DrC constantly suggesting I do some exercises on paper to show how it works is totally disingenuous, I can see how the examples work it is simply the assumptions the examples make that I don't understand. The examples all assume that the three options are all equally likely to be +/-, 1/0, red/black, i.e. the three selections are in no way related to each other. However, in reality (CFD?) this is not the case is it? If one is red then it is very much more likely that the other two are opposites than the same, so if I randomly pick two of the three my probability of getting red+red is about 1/4 not 1/3.

And why do I suggest this? It is precisely because you think 1/4 is a reasonable result. Because it is not reasonable in a local realistic world! If you run your example you will actually get 1/3, not 1/4 as you imagine! I never say that the results of one have nothing to do with the results of another, because in fact they do. And in preparing your examples, you can keep that in mind so you can come as close to the quantum predictions as possible. But you won't get 1/4. If you think I am wrong, prepare a set and present it.

If you ran the exercise you would see that it is necessary for Alice to know in advance what Bob is doing to get these results. If Alice and Bob separately and independently select which angle they measure (of 0/120/240 degrees), those results can NEVER be made to match experimental observations, where these 3 requirements are to be met:

a) perfect correlations/anti-correlations at 0/90/180/270 degrees.
b) cos^2(theta) rule everywhere else.
c) Alice does not not Bob's choice of setting, and vice versa.

If Alice and Bob must know what each is going to measure in order to get the proper outcome, then you are saying that we live in an observer dependent reality (and there is contextuality). That is what EPR (wrongly it turns out) rejected by assumption. And thus when you say that only 2 angles need to be considered, you are rejecting the EPR criterion that all elements of reality do NOT need to be simultaneously observable. They felt that ascribing reality to only 2 at a time (the 2 you can actually measure) was unreasonable. That was, by analogy, equivalent to saying the moon exists only when you are looking at it.

The entire point of Bell was in fact to dissuade you from glossing over the 1/3 versus 1/4 situation (although again Bell never used my specific example). If you make the EPR local realistic assumptions, you cannot get the QM result. As long as you hand wave around this point, you will go in circles. You MUST give up something to avoid a contradiction.

 

[DevilsAvocado]

EPR talks about predicting values for a particle based on somehow knowing the same value on its paired/entangled particle. However, Bell's theorem talks about predicting measurements of a value for a particle based on the result of a measurement of the same value on its paired/entangled particle.

Thanks Badvok, this makes it much easier. Let’s talk electrons and positrons instead, with spin along x, y, z axis. According to HUP we cannot with absolute certainty know the non-commuting operators x & y spin at the same moment in time.

Entangled electrons and positrons are anti-correlated, so if Alice measures her electron as y$\uparrow$ Bob will measure his positron as y$\downarrow$.

Now, what happens if Alice first measures her electron as y$\uparrow$ and then it’s Bobs turn; Bob will now know with 100% certainty that IF he measures the y-axis it will be down$\downarrow$, right? So, what happens if Bob instead chose to measure the x axis? Will he violate HUP and get precise knowledge about the counterfactual properties of spin x & y??

Well, it turns out that if Bob chose to measure the x-axis the result will be completely decoupled from Alice and the result will always be 100% random, and Bob will measure 50% x$\leftarrow$ and 50% x$\rightarrow$.

This leads naturally to Bell’s inequalities, where it is explicitly assumed that every possible measurement – even if not performed – must be included in the statistics. Okay?

IF you believe that non-commuting operators actually has a value [though not yet accessible to us], then you have to include these values in your statistics – EVEN if it is never measured, right?

Bell's theorem proves that every type of quantum theory must necessarily violate either locality or CFD (/Realism).

You can check out this video where DrPhysicsA takes you thru the EPR example with electron/positron, note however: He gets it wrong @10:34 where he says “he [Bob] can’t measure the x coordinate” which is not correct. Bob can measure x but it will be 100% random.

https://www.youtube.com/watch?v=0x9AgZASQ4k

It is this leap from talking about theoretical real values to just the results of measurements of those values, which is probably intuitive for you guys, which I have difficulty understanding.

This is as close we ever get to x and p, hope it helps.

Likewise, with the triple value examples, it is not the examples themselves that I have an issue with it is how they could possibly relate to reality. DrC constantly suggesting I do some exercises on paper to show how it works is totally disingenuous, I can see how the examples work it is simply the assumptions the examples make that I don't understand.

Bell's theorem is a logical/mathematical theorem. It does not provide a complete description for Local Realism, it only makes the minimal assumptions that there has to be “something there” – a definite state – and that casual influences cannot propagate faster than light. That’s all.

Now it’s up to you to provide a Local Realism that violates Bell’s inequalities!

 

[DrChinese]

But that is exactly what I don't understand, how can I completely ignore that? Haven't we known for ages that a photon polarised at θ will only pass a polarisor set at angle α with a probability proportional to cos²(α - θ)? So it is possible that one device would influence in one direction and the other could influence in the opposite direction? I don't see how you can not include the possibility of something in the measuring devices influencing the measurements. Isn't it this that makes it impossible to experimentally achieve perfect correlation?

No. Bell tests are able to exclude LR models by 30+ standard deviations. Yet they are compatible with the QM prediction. They is exactly opposite of what you expect.

If you model that the observation device is part of the hidden variables, that is fine as far as modeling goes. It is true that such cannot be excluded out of hand, in a limited sense. But the problem you end up with is that it doesn't allow a pathway to get the QM results. Instead, your model will simply flop because it doesn't get you even a hair closer!

----------------------------

In fact, using your idea: the predicted result at the perfect correlation angles would actually start getting closer to 75% rather than 100%. That is what you get when you measure *unentangled* Type I PDC photon pairs at random (and identical) angles. The key is that the same pairs give different statistics according to whether they are polarization entangled or not. Only the entangled pairs violate the Bell Inequality. The unentangled ones do not. Yet the physical apparatus is exactly the same either way. That is a difficult one to model around, because there is no classical way to explain why one set gives one set of statistics, and the other gives different ones.

 

[DrChinese]

To add to the point in my post#77:

A single Type I PDC will generate HH pairs from a V pump laser - call this the H case. Or alternately, a single Type I PDC will generate VV pairs from a H pump laser, call this the V case.

In either case (H or V), you get a 75% actual correlation rate when you set the measuring angle to 45 degrees for both. This is obviously NOT perfect correlation by a long shot. Instead, this is a typical classical regime.

Here is the problem: you can combine the 2 streams (that of the H case with that of the V case) to get a new case, we will call this H+V. Classically, this must always give 75% too. That is the average of .75 and .75, right? But in a quantum world, the H+V case is entangled and the actual result now jumps to 100% - perfect correlations.

This defies classical modeling.

 

[DevilsAvocado]

So it is possible that one device would influence in one direction and the other could influence in the opposite direction? I don't see how you can not include the possibility of something in the measuring devices influencing the measurements.

That won’t save your rear part, only if one device could influence the other device, you could make it work, but that’s a violation of locality...

Isn't it this that makes it impossible to experimentally achieve perfect correlation?

[B]Measurement 1[/B]
[B]A[/B] = 10101 01010
[B]B[/B] = 10101 01010

[B]Measurement 2[/B]
[B]A[/B] = 11001 10011
[B]B[/B] = 11001 10011
 
[B]Measurement 3[/B]
[B]A[/B] = 01000 10111
[B]B[/B] = 01000 10111

All these 3 measurements show prefect correlations (for Bell state Type I), and it works every time in EPR-Bell experiments.

 

[DevilsAvocado]

Here is the problem: you can combine the 2 streams (that of the H case with that of the V case) to get a new case, we will call this H+V. Classically, this must always give 75% too. That is the average of .75 and .75, right? But in a quantum world, the H+V case is entangled and the actual result now jumps to 100% - perfect correlations.

Nice DrC!

 

[DrChinese]

In either case (H or V), you get a 75% actual correlation rate when you set the measuring angle to 45 degrees for both. This is obviously NOT perfect correlation by a long shot. Instead, this is a typical classical regime.

OOPS! It should be 50%, not 75% as I indicated. My bad.

Same conclusions though, actually just emphasizes the point. That being: when you combine 2 independent streams, each with a lot of random correlations, they suddenly become 100% correlated if entangled. But if the streams consist of independently created photon pairs (ie in separate PDC crystals), how did this happen?

How would you explain this from a hidden variable perspective? Where are those hidden variables residing? And, once you speculate on their location, how can you zero in on them via experiment? Once you go through these steps, the issues become very difficult for the local realist. Just ask Marshall or Santos, who have attempted to construct a number of stochastic (classical) models. None of these have had any traction.

 

[Badvok]

How would you explain this from a hidden variable perspective? Where are those hidden variables residing?

I haven't a clue! What I don't get though is how Bell proves that it would be impossible to do so. My understanding of his formulae seems to indicate that they don't take account of possible LHVs that are part of the measuring devices, i.e. why is there just one λ and not a λ_a and λ_b?

Telling me to roll a dice myself and see that I get a 6 roughly 1 in 6 times is pointless, but helping me understand me why I am using a six sided dice instead of a ball might be better - or in other words it is not the probability bit that is an issue, it is the assumptions that determine the possibilities that is an issue for my understanding.

 

[Nugatory]

My understanding of his formulae seems to indicate that they don't take account of possible LHVs that are part of the measuring devices, i.e. why is there just one λ and not a λ_a and λ_b?

You're misreading the formulae then - there's nothing in Bell's argument that stops you from including the state of the measuring devices (that is, LHVs associated with the measuring devices) in λ. Indeed the detector angle itself, which obviously is part of the computation, is a (not especially well hidden) LHV. What you can't do is use the variables associated with detector A to calculate the result at detector B and vice versa - if you did that you wouldn't be using local hidden variables.

You might also want to look at the text immediately under equation 3 in the paper.

 

[Nugatory]

helping me understand me why I am using a six sided die instead of a ball might be better - or in other words it is not the probability bit that is an issue, it is the assumptions that determine the possibilities that is an issue for my understanding.

Here the difference between a six-sided die and a ball is just the difference between a discrete eigenvalue spectrum and a continuous one. It is easier to construct examples and experiments around observables that have discrete spectra, but there's nothing in Bell's argument that limits it to such hidden variables.

However, this is the second time you've raised this concern, so clearly I'm not understanding what you're asking well enough to give you a useful answer...

 

[Badvok]

You're misreading the formulae then - there's nothing in Bell's argument that stops you from including the state of the measuring devices (that is, LHVs associated with the measuring devices) in λ. Indeed the detector angle itself, which obviously is part of the computation, is a (not especially well hidden) LHV. What you can't do is use the variables associated with detector A to calculate the result at detector B and vice versa - if you did that you wouldn't be using local hidden variables.

You might also want to look at the text immediately under equation 3 in the paper.

So there isn't just one set of variables λ then? i.e. in the two functions A(a,λ) and B(b,λ) the λ isn't actually the same thing?

 

[Nugatory]

So there isn't just one set of variables λ then? i.e. in the two functions A(a,λ) and B(b,λ) the λ isn't actually the same thing?

It's the same λ, a complete specification of the whole shebang. The text under equation 3 ("some may prefer...") explains why we don't need a separate λ_a and λ_b; and the text under equation 1 explains the locality constraint which λ and the functions A and B of λ are assumed to obey.

 

[Badvok]

It's the same λ, a complete specification of the whole shebang. The text under equation 3 ("some may prefer...") explains why we don't need a separate λ_a and λ_b; and the text under equation 1 explains the locality constraint which λ and the functions A and B of λ are assumed to obey.

I don't understand that text, if λ can include factors that are 'local' to each of the measurement devices how can you get a function A(a,λ) that doesn't depend in some way on b?
He also says that "our λ can then be thought of as initial values of those variables at some suitable instant", but which instant is that? Is there really a suitable instant? Is it when A(a,λ) is measured, when B(b,λ) is measured, or some other time? And how can we assume that λ doesn't change wrt time?

 

[DrChinese]

I haven't a clue! What I don't get though is how Bell proves that it would be impossible to do so. My understanding of his formulae seems to indicate that they don't take account of possible LHVs that are part of the measuring devices, i.e. why is there just one λ and not a λ_a and λ_b?

As Nugatory says, no problem with there being more sets of hidden variables living alongside the measuring devices. But Alice can't communicate her setting (ie her hidden variables) to Bob because, as mentioned, that would violate locality QED.

But if the local measurement setting HVs are not communicated to the other spot, how are you going to get perfect correlations unless the effects exact cancel each other out at ANY similar setting for Alice and Bob? And if they exactly cancel out, they then didn't need to be considered in the first place QED.

So either way, we are back to the same point. Where are the HVs? Clearly not a part of the measuring devices; but if they are, they cancel out. Please note that you could use completely DIFFERENT (in the physical sense) methods of measuring photon polarization and the results will be the same. For example: beam-splitters versus polarizing filters. You could use a variety of methods (such as wave plates) to first rotate the photon's polarization by various amounts (presumably introducing yet more devices - and therefore more HVs - to consider). All of this makes no difference, it's *theta* (A-B) that rules. And theta is a quantum non-local variable that does not consider anything from the measuring devices OTHER than the net angle setting.

 

[DrChinese]

I don't understand that text, if λ can include factors that are 'local' to each of the measurement devices how can you get a function A(a,λ) that doesn't depend in some way on b?
He also says that "our λ can then be thought of as initial values of those variables at some suitable instant", but which instant is that? Is there really a suitable instant? Is it when A(a,λ) is measured, when B(b,λ) is measured, or some other time? And how can we assume that λ doesn't change wrt time?

There is a function A() and a function B(), and if time t is to be a factor in the function: sure, it could vary over time.

The issue, as we keep saying, is that doesn't give you perfect correlations if it does. Because you would, according to your concept, ONLY observe perfect correlations when you measured A and B at the SAME time (or some periodic interval). In fact, you get perfect correlations regardless of the relative time of observation.

So your hypothetical HV function, and the initial conditions, must be such that you get perfect correlations. That requirement causes virtually everything to exactly cancel out (if it was ever a factor in the first place). So you are, again, left only with theta.

 

[Nugatory]

I don't understand that text, if λ can include factors that are 'local' to each of the measurement devices how can you get a function A(a,λ) that doesn't depend in some way on b?

Here's a trivial example. Suppose λ is {Q=23, R=T_a, S=T_b} where T_a and T_b are the temperatures of the two detectors. If A(a,λ)=a+Q+R and B(b,λ)=b+Q+S, then λ includes factors that are local to both measurement devices, yet A(a,λ) is unaffected by anything that happens at device b and B(b,λ) is unaffected by anything that happens at device a.

Of course in this case we could just as easily have written λ_a={Q=23,R=T_a} and λ_b={Q=23,S=T_b}, but as Bell pointed out in the text below equation 3 this is just a notational preference.

He also says that "our λ can then be thought of as initial values of those variables at some suitable instant", but which instant is that? Is there really a suitable instant? Is it when A(a,λ) is measured, when B(b,λ) is measured, or some other time? And how can we assume that λ doesn't change wrt time?

He says that specifically to allow for the possibility that λ does change with time. In the example above you can easily imagine that the detectors gradually cool off so that T_a and T_b are functions of time - and then you'd need to know their temperature at some specific time (any time before the experiment when it's convenient to measure the temperature) and the rate of change of temperature with time to know the value of T_a and T_b at the time that we measure A(a,λ) and B(b,λ).

 

[Badvok]

If A(a,λ)=a+Q+R and B(b,λ)=b+Q+S, then λ includes factors that are local to both measurement devices, yet A(a,λ) is unaffected by anything that happens at device b and B(b,λ) is unaffected by anything that happens at device a.

Sorry, I guess this is maths that I don't understand, how can A(a,λ)=a+Q+R not include S when S is part of λ.

 

[Nugatory]

Sorry, I guess this is maths that I don't understand, how can A(a,λ)=a+Q+R not include S when S is part of λ.

Continuing with the trivial examples (and using trivial examples because I'm pretty sure that you're just getting hung up on Bell's notation here):

A(a,λ)=a+Q+R+(S-S) includes S but the value of A still doesn't depend on S.

More generally, λ is a set of conditions, and nothing requires that you use every member of that set in the definition of every function of that set. If a theory says that A(a,λ) uses the B-local conditions (except in the trivial self-cancelling sort of way that I just did), then that theory is non-local. Bell's theorem is a statement about the behavior of theories that are not non-local in this sense,

 

[DrChinese]

Sorry, I guess this is maths that I don't understand, how can A(a,λ)=a+Q+R not include S when S is part of λ.

It could, IF you wanted to switch to a NON-LOCAL version of hidden variables.

But otherwise, the shared variables do not include information about the measuring devices. The measuring devices can include any number of variables though, as long as a doesn't depend on b and vice versa.

 

[Badvok]

Continuing with the trivial examples (and using trivial examples because I'm pretty sure that you're just getting hung up on Bell's notation here):

A(a,λ)=a+Q+R+(S-S) includes S but the value of A still doesn't depend on S.

More generally, λ is a set of conditions, and nothing requires that you use every member of that set in the definition of every function of that set. If a theory says that A(a,λ) uses the B-local conditions (except in the trivial self-cancelling sort of way that I just did), then that theory is non-local. Bell's theorem is a statement about the behavior of theories that are not non-local in this sense,

Thanks, but Bell then goes onto express an expectation value as the integral with respect to λ of the product of A, B, and the probability distribution of λ. Again, I'm unsure how that can work when there are different λs.

 

[DrChinese]

Thanks, but Bell then goes onto express an expectation value as the integral with respect to λ of the product of A, B, and the probability distribution of λ. Again, I'm unsure how that can work when there are different λs.

Shared set λ (since λ are those local variables present when entanglement begins); while sets a and b are not shared. So there are 3 total sets of variables. The only restriction is that a is not shared with b, and vice versa.

 

[DrChinese]

And again, I would suggest trying to provide a specific example to work through so you can see the difficulties with your ideas. For example, suppose there is some formula, the answer to which is +/- or 1/0 or similar. Make the components of that formula such that we can get an answer with different inputs. Try to fix it so that the result is a perfect correlation when a and b are the same on one parameter (which we will associate with angle setting).

For example: suppose we get 0 if the result of our function is even, 1 if the result is odd. Our function is simply a sum of the inputs (this is not supposed to be a serious example in any physical sense.

The EntangledSourceHV1 (shared) is 6.
The AliceHV1 (not shared) is 9.
The BobHV1 (not shared) is 13.
The AliceMeasementAngle is 2.
The BobMeasementAngle is 2.

Alice's result = A(EntangledSourceHV1, AliceHV1, AliceMeasementAngle) = A(6+9+2)=1 (since sum is odd)
Bob's result = A(EntangledSourceHV1, BobHV1, BobMeasementAngle ) = A(6+13+2)=1 (since sum is odd)

So this works out for the perfect correlation at angle=2, so that is good. And you can add as many HVs as you like using this idea.

Now, try varying the measurement hidden variables with each side. You will see that as long as they change in tandem, everything is fine - but not otherwise. But if they change in tandem, then they are not observer independent, are they?

 

[Badvok]

OK, it doesn't matter what factors affect the measurements so long as they are the same for both and not linked to the observer setting. If they were slightly different, e.g. magnetic field strength, then that would simply affect how close to perfect correlation the experiment could get but it would still be able to achieve better than classical physics would predict.

Still a bit confused about why A(a,λ) and B(b,λ) need to be restricted to ±1 though. Is this just to make the maths easier or is this a fundamental part of the proof itself?

I see the following image (or variations of it) on a lot pages that discuss Bell's Inequalities.
http://upload.wikimedia.org/wikipedia/commons/7/77/StraightLines.svg
This is used to illustrate the difference between the QM prediction (and experimental results) and a 'local realist' prediction. The straight lines of the LR prediction obviously arise simply from constraining A and B to ±1 but I do wonder what the graph would look like without this constraint.

 

[Ilja]

The restriction follows from the particular experiment considered here. Once what is observed are only two possible results, up and down, this particular experiment cannot be explained by theories which allow for three, four or more possible results.

There may be other experiments, with other possible results and, therefore, other mathematical proofs and other resulting inequalities. But this is not quite relevant. If our world is local (better Einstein-causal) and realistic, this particular experiment needs an explanation in terms of such an Einstein-causal realistic theory. Once this is impossible for the particular experiment, Einstein-causal realism is dead.

 

[Badvok]

Thanks again for all your help with this.

If anyone is interested I've knocked up a little JavaScript model to play with the various assumptions so you can see the effects they have (though of course nothing allows you to get closer to the quantum predictions/real test results). You can even tweak the rate at which entangled photons are generated so you can get closer to a realistic simulation. It is on my home server here. There's no advertising or anything nasty there, just a very simple page with some script. Feel free to take it, and reuse it or change it if you wish. If I get the time I may later expand it to include some pretty graphical animations.