Bell's Theorem with Easy Math - Stuck

[DevilsAvocado]

Thanks all for your assistance with this but it looks like I'm falling between two stools. On the one hand I see simplified explanations that make assumptions that I can't clearly see the validity of, on the other hand my mathematical ability is not up to the required level to read and fully understand the actual papers. So I have a lot more learning to do before I can get my head around these concepts.

Why do I get the feeling a non-local bullet just penetrated my little green heart...?? ()

Could this be a remedy?

Everything should be made as simple as possible, but not simpler.

If it worked for J.S. Bell, why shouldn’t it work for us? In lectures he used exactly the same example as I gave you:

N(+30°, -30°) ≤ N(+30°, 0°) + N(0°, -30°)​

Sorry, I know that, that's not what I meant. I meant that if I choose to measure a particle's position I can do that fairly accurately, I don't limit the position measurement to a yes/no result (is it in one region or another?) And likewise, I can measure momentum fairly accurately without limiting it to a yes/no result. I know I can't measure both on the same particle.

If this is what stopping you from proceeding, I don’t really understand why... on a normal macroscopic scale we can pin down objects to a precise position in continuous space, but can we really do this in the QM world?

Well, from the Stern–Gerlach we know that angular momentum takes only certain quantized values:

https://www.youtube.com/watch?v=rg4Fnag4V-E


Then the question arises - Is space[time] itself quantized?

According to Loop quantum gravity it is, consisting of an extremely fine fabric of finite loops. The size of this structure is the Planck length which is approximately 10⁻³⁵ meters.

If spacetime is quantized – and you want to measure position – you will get a 'quantized' Yes/No answer...

(Besides, all particles in QM are ‘wobbling around’ due to energy and virtual particles bumping in and out of the QM soup. And if you try to remove the heat/energy by freezing them near absolute zero to a Bose–Einstein condensate and lowest accessible quantum state, they become indistinguishable!)

If it helps, you can think of the entangled photon as having a superposition of all angles between 0-360°, and you can set the polarizer to any degree + arcminute + arcsecond and so on, for any ‘continues resolution’ you want. The answer will however be Yes/No for the measurement...


... I wish I could understand what the problem is ...

 

[DevilsAvocado]

If I understand you properly, you're describing the difference between a continuous eigenvalue spectrum and a discrete one. Bell's proof is formulated in a way that covers both cases, but to see this you have to have to read the actual Bell paper and that takes us beyond "easy math".

Please have patience with my ignorance – but are there continuous eigenvalues in QM? Where and what is it?


(... getting prepared for embarrassment ... )

 

[Nugatory]

Please have patience with my ignorance – but are there continuous eigenvalues in QM? Where and what is it?

x and p for unbound particles, for example. Eigenfunctions are delta functions, eigenvalues form a continuous spectrum, is what I'm thinking about.

 

[DevilsAvocado]

x and p for unbound particles, for example. Eigenfunctions are delta functions, eigenvalues form a continuous spectrum, is what I'm thinking about.

Well... I did warn for the embarrassment...

Still try to wrap my head around this, but could a continuous spectrum, from let’s say a free electron becoming bound to an hydrogen ion, be used as entanglement in EPR-Bell experiments?

I.e. two free electron meet and get entangle and are then sent to respective hydrogen ion. Could we somehow measure correlations from the spectrum?

(sounds hard)


EDIT:
This can’t work, it’s the electrons that are entangled not the spectrum, right?

 

[lugita15]

DevilsAvocado, any two objects can be entangled, with respect to any observable, whether the observable has discrete eigenvalues, like spin or polarization, or continuous eigenvalues, like momentum.

 

[DevilsAvocado]

I know lugita, the thing that has gotten my nut temporarily dizzy is the continuous spectrum. Could we measure correlations in the spectrum? What are we looking for? A smeared out signal?

 

[lugita15]

I know lugita, the thing that has gotten my nut temporally dizzy is the continuous spectrum. Could we measure correlations in the spectrum? What are we looking for? A smeared out signal?

What spectrum are you talking about? Spectrum refers to the set of eigenvalues of some observable. What observable are you talking about?

 

[DevilsAvocado]

What spectrum are you talking about?

In my example above that would be the light emitted by the two free electrons becoming bound to an hydrogen ion.

Could we see the correlations in that continuous spectrum?

(sounds impossible to me)

 

[DrChinese]

In my example above that would be the light emitted by the two free electrons becoming bound to an hydrogen ion.

Could we see the correlations in that continuous spectrum?

(sounds impossible to me)

When PDC creates 2 photons from 1, the pairs are entangled on the basis of their frequency/wavelength among other things. The frequency is a continuous spectrum of values, they are not limited to discrete values. On the other hand, their polarization is only either 1 or 0 (or +/- or whatever).

As to seeing correlations... absolutely! A lot of observables are available for correlation.

 

[DevilsAvocado]

The frequency is a continuous spectrum of values, they are not limited to discrete values. On the other hand, their polarization is only either 1 or 0 (or +/- or whatever).

As to seeing correlations... absolutely! A lot of observables are available for correlation.

Gosh, there must be some short circuit in the guacamole... I can’t think straight... it’s been a long day... (must blame something :shy:)

Let’s break it down (to the avocado level):

• [Light] frequency is a continuous spectrum of values [of course].
• Higher frequency = higher energy.
• Entangled photons can have any frequency.
• Polarization of photons is not coupled to frequency, or?
• In QM, photon polarization is calculated with the Jones vector and applied to the Poincaré sphere.

Question: Is the Jones vector continues or discrete?

[my guess is continues...]


EDIT:
Of course when we measure the polarization is either 1 or 0, or thru/stopped, etc.

 

[Badvok]

The answer will however be Yes/No for the measurement...

... I wish I could understand what the problem is ...

You actually hit the nail on the head with that statement about the measurement. In all the examples the possible outcomes of a measurement are taken to be the 'elements of reality', this is the same assumption I think Bell makes?

In the EPR paper it says: "If, without in any way disturbing a system, we can predict with certainty (i.e. with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity." To me that doesn't imply that we can actually measure that physical quantity or that we should be able to predict the result of a measurement of that quantity with equal certainty.

From Nugatory's posts I see that EPR refers to correlations between continuous spectrum eigenvalues, i.e. x and p - is that right?

However, Bell refers to measurements of those values that result in Boolean results. Now I guess this is because spin and polarisation are considered discrete spectrum eigenvalues? However, since we can only ever measure spin and polarisation by apparatus that gives us a Boolean result, I don't see how Bell and EPR are actually talking about the same thing. If we could measure the actual spin/polarisation is it possible that we might find that there was an 'element of reality' that was a continuous spectrum eigenvalue? And therefore I don't see how Bell rules this type of LHV out. My guess is that the maths would still work and rule it out but I don't have the ability to do that sort of maths myself.

Next we have the concept of 'perfect correlation' used in yours and Nick Herbert's examples. Now I understand that in a ideal system 'perfect correlation' would exist and that it could be shown to exist in terms of conservation of momentum/energy and such but I don't get how this concept can be extrapolated to apply to the later local interaction of a particle with a local measurement device.

Lastly, on to the logic in DrC's and Ilja's examples and looking at the Scholarpedia article (these are all ones that talk about triple values). In the Scolarpedia article it appears to make the assumption that all three values can't be the same in one bit and then forgets this later (?). If we actually look at this assumption in the context of Ilja's cards then the set of cards from which the selection is made is reduced to just 4 cards so we can at most get two the same (instead of the infinite set used to get 50:50 probability for each selection). With just that limited set the probabilities change and I get a bit lost trying to get my head around them, i.e. the probability of selecting a red card and then another red card from a set of three cards that have in turn been selected from a set of four cards (= 0.25?). Now I don't know whether the assumption that the three values can't all be the same is correct or not, in DrC's example with photon polarisation and 0,120,240 test angles, there is a small but definitely non-zero probability that a photon polarised at angle θ will pass all three polarisers.

I hope I don't raise anyones ire with my language here, I'm not suggesting that I think anything or anyone is wrong, just that I don't understand it.

 

[stevendaryl]

However, Bell refers to measurements of those values that result in Boolean results. Now I guess this is because spin and polarisation are considered discrete spectrum eigenvalues? However, since we can only ever measure spin and polarisation by apparatus that gives us a Boolean result, I don't see how Bell and EPR are actually talking about the same thing. If we could measure the actual spin/polarisation is it possible that we might find that there was an 'element of reality' that was a continuous spectrum eigenvalue?

Bell's analysis does not assume that the hidden variables are discrete, but he assumes that the discrete results that one gets from a spin measurement must be a function of those hidden variables. He gives as a "toy" example of such a hidden variable:

Suppose that associated with every electron is a vector \vec{S}. When you measure the spin in direction \vec{P}, then you get +1 if \vec{S} \cdot \vec{P} \geq 0, and you get -1 if \vec{S} \cdot \vec{P} \leq 0.

So this model has a continuous "element of reality", since the vector \vec{S} can point in any direction. It doesn't agree with the predictions of QM, though.

 

[harrylin]

Thanks all, I think I've got it now.

The bit I was stumbling over in Nick Herbert's proof is this: "Starting with two completely identical binary messages". Where do these messages come from? So far as I could see the system has a stream of randomly polarized photons, no actual binary message. Thus the only mismatch that could be measured is by comparing the results obtained at A & B, which obviously links the two detectors and makes the mismatch 75% based on the mutual misalignment angle.[..]

Herbert's proof was also discussed on this forum:

https://www.physicsforums.com/showthread.php?t=90770
and
https://www.physicsforums.com/showthread.php?t=589134

 

[Ilja]

In all the examples the possible outcomes of a measurement are taken to be the 'elements of reality', this is the same assumption I think Bell makes?

No. Bell assumes that there is some reality λ, and this reality, together with the decisions of the experimenters a,b what to measure, defines the outcomes A and B of the experiment: A=A(λ,a,b) and B=B(λ,a,b).

In the EPR paper it says: "If, without in any way disturbing a system, we can predict with certainty (i.e. with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity." To me that doesn't imply that we can actually measure that physical quantity or that we should be able to predict the result of a measurement of that quantity with equal certainty.

In this case, we have a fortunate situation where we can measure it, and where we can predict it, and whenever we really measure it, we find that the prediction is correct.

What can be predicted is the result of A given a=b if we measure B. The EPR criterion can be applied if we assume that measuring B in direction b does not disturb the system at A, so A in direction a=b can be predicted without disturbing this system by measuring B in direction b.

From Nugatory's posts I see that EPR refers to correlations between continuous spectrum eigenvalues, i.e. x and p - is that right?

It is right but quite irrelevant. The discrete version is easier to understand, so there is no reason to consider this earlier variant.

However, Bell refers to measurements of those values that result in Boolean results. Now I guess this is because spin and polarisation are considered discrete spectrum eigenvalues? However, since we can only ever measure spin and polarisation by apparatus that gives us a Boolean result, I don't see how Bell and EPR are actually talking about the same thing.

Very simple: We consider the situation considered by Bell, with discrete results, but apply the argument (the criterion of reality) proposed by EPR for a different, otherwise irrelevant, continuous situation.

If we could measure the actual spin/polarisation is it possible that we might find that there was an 'element of reality' that was a continuous spectrum eigenvalue?

It doesn't matter. The 'elements of reality' themself, as used in the assumption, may be arbitrary - Bell's inequality follows for discrete as well as continuous hidden variables λ.

Next we have the concept of 'perfect correlation' used in yours and Nick Herbert's examples. Now I understand that in a ideal system 'perfect correlation' would exist and that it could be shown to exist in terms of conservation of momentum/energy and such but I don't get how this concept can be extrapolated to apply to the later local interaction of a particle with a local measurement device.

First, there is no need for any extrapolation. Once Bell's inequality is violated in one experiment, it is violated. Point.

Then, there is a minor problem with an argument based on an ideal assumption. The perfect correlation happens only in the ideal case if the direction is ideally the same. So in any real experiment the correlation will not be ideal.

But there are variants of the inequalities, slightly more complicated, which do not depend on this, so that one needs only approximate accuracy in the whole reasoning.

Lastly, on to the logic in DrC's and Ilja's examples and looking at the Scholarpedia article (these are all ones that talk about triple values). In the Scolarpedia article it appears to make the assumption that all three values can't be the same in one bit and then forgets this later (?). If we actually look at this assumption in the context of Ilja's cards then the set of cards from which the selection is made is reduced to just 4 cards so we can at most get two the same (instead of the infinite set used to get 50:50 probability for each selection). With just that limited set the probabilities change and I get a bit lost trying to get my head around them, i.e. the probability of selecting a red card and then another red card from a set of three cards that have in turn been selected from a set of four cards (= 0.25?). Now I don't know whether the assumption that the three values can't all be the same is correct or not,

I don't assume it. If I choose three times the same card, you win with certainty, this would be stupid for me. But I can do this. Therefore, one can prove only an inequality, >=1/3.

If I choose only different cards, and you choose the two cards by accident, without any possibility for me to predict your choice, you have a chance 1/3. If I use three cards of the same color, your chance is 1. So it is always >= 1/3.

 

[DrChinese]

Gosh, there must be some short circuit in the guacamole... I can’t think straight... it’s been a long day... (must blame something :shy:)

Let’s break it down (to the avocado level):

• [Light] frequency is a continuous spectrum of values [of course].
• Higher frequency = higher energy.
• Entangled photons can have any frequency.
• Polarization of photons is not coupled to frequency, or?
• In QM, photon polarization is calculated with the Jones vector and applied to the Poincaré sphere.

Question: Is the Jones vector continues or discrete?

[my guess is continues...]EDIT:
Of course when we measure the polarization is either 1 or 0, or thru/stopped, etc.

Debil,

Polarization of photons is not tied to frequency or wavelength, as you suppose.

Imagine input pump of 400nm wavelength, typically would get out a pair of photons both around 800nm each. But you could also get out one at 802nm, another at 798nm (values are approx.). Or a pair at 804.3nm and 795.7nm. There are no specific values that are prohibited as long as conservation is preserved.

 

[DrChinese]

1. You actually hit the nail on the head with that statement about the measurement. In all the examples the possible outcomes of a measurement are taken to be the 'elements of reality', this is the same assumption I think Bell makes? ... If we could measure the actual spin/polarisation is it possible that we might find that there was an 'element of reality' that was a continuous spectrum eigenvalue?

2. Now I don't know whether the assumption that the three values can't all be the same is correct or not, in DrC's example with photon polarisation and 0,120,240 test angles, there is a small but definitely non-zero probability that a photon polarised at angle θ will pass all three polarisers.

1. Yes, p and q can be predicted with certainty for entangled particles just as spin can.

2. A photon has a definite probability of passing three such aligned filters. All things being equal, that would be 1/2 * 1/4 * 1/4 or about 3%.

You will see the problem if you think of it this way:

a. If I can predict the result of any measurement on Bob by first performing the same measurement on Alice, then you might at first glance that Bob is essentially a clone of Alice. How else to explain the fact that you can predict one by measuring the other? This is the position of the local realist, and it is the position of EPR.

b. In fact, there are literally hundreds if not an infinite number of different measurements that can be performed on Bob and predicted in advance (by measuring Alice). You can do at 1 degree, 2 degrees, 3 degrees, etc, or 1.1 degrees, 1.2 degrees, 1.3 degrees. Gosh, Bob must be carrying around a LOT of hidden variables! Alice too! And that is just the spin degrees of freedom. There are many more.

c. Now try to map values to those. For example, make 1 degree be +, 2 degrees be +, etc until finally you find one where you decide to say it is -. Maybe that is at 115 degrees. Whatever you say it is. Do this for all 360 degrees. (Or for simplicity, every 10 degrees or something.) Keep in mind that these values are preset if you follow the EPR program. You don't know what they are, but they must be something!

d. Here is the Bell stumbling block: they not only have to be "something", but across a series of successive measurements on different entangled pairs, they must match the quantum (QM) predictions when the angles are NOT the same! And what is that predictions? It is cos^2(theta) where theta is the difference in the measurements on Alice and Bob. It doesn't really matter how he figured it out, but Bell found that this requirement "broke the bank" on the EPR argument, so to speak.

e. It turns out there are NO sets of values that BOTH support the EPR outcomes - the perfect correlations of b.) - AND the QM requirement of d. If you actually try to come up with such a set, you will see in short order what the problem is. Just do a handful of examples and you will quickly see that you can't make it work out. Try it! Really! Get out a piece of scratch paper for 15 minutes and you will see what is wrong with the EPR program. (Keep in mind that Bell was the first person in 30 years to discover this. So don't beat yourself up that you need to invest 15 more minutes!)

f. Bell's Inequality is simply a generalized proof of this same fact.

 

[DevilsAvocado]

Thanks DrC

 

[Badvok]

c. Now try to map values to those. For example, make 1 degree be +, 2 degrees be +, etc until finally you find one where you decide to say it is -. Maybe that is at 115 degrees. Whatever you say it is. Do this for all 360 degrees. (Or for simplicity, every 10 degrees or something.) Keep in mind that these values are preset if you follow the EPR program. You don't know what they are, but they must be something!

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.

 

[DevilsAvocado]

You actually hit the nail on the head with that statement about the measurement. In all the examples the possible outcomes of a measurement are taken to be the 'elements of reality', this is the same assumption I think Bell makes?

In the EPR paper it says: "If, without in any way disturbing a system, we can predict with certainty (i.e. with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity." To me that doesn't imply that we can actually measure that physical quantity or that we should be able to predict the result of a measurement of that quantity with equal certainty.

Now you’re thinking about Counterfactual Definiteness (CFD), right? It’s quite interesting how hard it is to pin down the most fundamental property of human life – reality!

Einstein and Bohr was very gifted and bright, still they argued over these matters for nearly 30 years without consensus, with famous arguments like:

Einstein: Do you really think the moon isn't there if you aren't looking at it?
Bohr: How do you know? How can you prove the moon exists, if you don't observe it?

My guess is that it would be a mistake trying to repeat this debate once again...

Einstein certainly had his flaws, but so did Bohr, and the true champ in this debate is Bell who settled the matter with an experiment. One can have any ‘philosophical’ objections one like, but if you question rigorous experiments then you’re on thin ice...

From Nugatory's posts I see that EPR refers to correlations between continuous spectrum eigenvalues, i.e. x and p - is that right?

It looks like it’s possible, though it must be harder to execute in a real experiment.

However, Bell refers to measurements of those values that result in Boolean results. Now I guess this is because spin and polarisation are considered discrete spectrum eigenvalues? However, since we can only ever measure spin and polarisation by apparatus that gives us a Boolean result, I don't see how Bell and EPR are actually talking about the same thing.

You mean that if EPR used ‘continuous values’ for defining local realism, Bell will be faulty testing this assumption with Boolean ‘discrete values’, right?

This is where you lost me because there’s no explanation how/why continuous values ever could violate Bell inequalities with local realism preserved. In fact, I think you would run into even greater difficulties...

AND EPR-Bell experiments is never performed with only one pair, but the correlations are measured over an ensemble of photons and using the CHSH inequality the upper bound for QM is 2.828427...

A very ‘continuous value’ indeed!

But space is free, if you could construct an EPR toy model with ‘continuous values’ that violates Bell inequalities, my guess is the whole physics community would read the paper (if proven correct).

If we could measure the actual spin/polarisation is it possible that we might find that there was an 'element of reality' that was a continuous spectrum eigenvalue? And therefore I don't see how Bell rules this type of LHV out.

I do not get this. The photon spin is continuous and when we measure it thru a polarizer we get 0/1 or thru/stopped. Exactly the same phenomena happen in your classical polarized sunglasses. There is no “QM cheating” here... you can’t be saying that polarization is not real because is not continuous??

Next we have the concept of 'perfect correlation' used in yours and Nick Herbert's examples. Now I understand that in a ideal system 'perfect correlation' would exist and that it could be shown to exist in terms of conservation of momentum/energy and such but I don't get how this concept can be extrapolated to apply to the later local interaction of a particle with a local measurement device.

Maybe it’s my ignorance, but I have never heard that conservation of energy has anything to do with Bell’s inequalities... in SPDC yes, but in Bell??

Lastly, on to the logic in DrC's and Ilja's examples and looking at the Scholarpedia article (these are all ones that talk about triple values).

I’ve seen many go nuts over “the triple values”, thinking “Aha! A mathematical trick!”. It’s not; it’s the simplest way demonstrating Bell’s inequalities. You can’t do it with only two values – then we’re back to “tossing coins/gloves in a box” -type of correlations, which could take another 30 years to sort out. Of course you can try 4, 5, 6, 7... values but that doesn’t change anything.

I hope I don't raise anyones ire with my language here,

Absolutely not (and you can ignore my silly jokes = genetic disease ;).

 

[Badvok]

OK, I still seem to be having a problem expressing the points that I'm stuck on, here's another attempt. (Please note that I have no doubts whatsoever that QM predictions tally with experimental results - that sort of thing is unquestionable for someone at my level.)

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.

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. 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.

 

[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.