A simple two-page proof of local realism

[gill1109]

The real issue for me is the discrepancy you are seeing between your S^2 simulation in R and the prediction of the cosine function by my analytical model (as well as by the recipe called quantum mechanics).

So the real question for me is: Is the minute discrepancy between the S^2 simulation in R and the cosine correlation a real discrepancy? I very much doubt it.

I wonder what difference does it make if \theta_0 is taken in the full range of [0, 2\pi] instead in the range of [0, \pi/2]. I do not believe it will make any difference (because of the codomain [0, 1] of the sine function), but just in case there is something in R that is seeing the difference.

The discrepancy is real.

Changing [0, pi/2] to [0, 2 pi] makes no difference at all.

Notice that 1/2 sin^2 theta_0 lies between 0 and 1/2. If we compare the cosine of another angle with this, it means we are varying the radii of Caroline's circular caps between 0 and 60 degrees. Why this particular range? Why that particular distribution?

[Joy Christian]

Notice that 1/2 sin^2 theta_0 lies between 0 and 1/2. If we compare the cosine of another angle with this, it means we are varying the radii of Caroline's circular caps between 0 and 60 degrees. Why this particular range? Why that particular distribution?

I have answered this question above (see my reply of Sat Feb 15, 2014 11:39 am). The short answer is that the factor of 1/2 and the distribution of sin^2 follow from the rotational invariance of the state (e, t), as explained in the long footnote on the pages 242 and 243 of my longer paper (the book chapter).

In fact, if you increase the factor gradually from 1/2 to 1, then you will approach the PR-box correlation (the bound of 4 instead of 2\/2 on CHSH). Michel has checked this in his simulation. If you fix the factor to 0 instead of 1/2, then you will recover the original local model of Bell (linear correlation, or bound of 2 on CHSH). So the numerical factor ranging from 0 to 1/2 to 1 takes you from Bell-1964 to QM to PR-box.

Why not try to introduce a tiny phase shift between A and B to see what happens?

[gill1109]

In fact, if you increase the factor gradually from 1/2 to 1, then you will approach the PR-box correlation (the bound of 4 instead of 2\/2 on CHSH). Michel has checked this in his simulation. If you fix the factor to 0 instead of 1/2, then you will recover the original local model of Bell (linear correlation, or bound of 2 on CHSH). So the numerical factor ranging from 0 to 1/2 to 1 takes you from Bell-1964 to QM to PR-box.

Why not try to introduce a tiny phase shift between A and B to see what happens?

Anyone who likes can copy-paste from my script and try out such modifications themselves. I'll also prepare a "cleaner" version which only simulates the S^2 model, so the code is less cluttered, more transparent, more easy to experiment with.

[Ben6993]

Anyone who likes can copy-paste from my script and try out such modifications themselves. I'll also prepare a "cleaner" version which only simulates the S^2 model, so the code is less cluttered, more transparent, more easy to experiment with.

Yes, please. I installed R language a few days ago but have been busy on something else since so have not yet played with the language. I could use your S^2 code to try to learn R. And if I find that I can manipulate the phase shifting, I will report back (eventually).

[gill1109]

Yes, please. I installed R language a few days ago but have been busy on something else since so have not yet played with the language. I could use your S^2 code to try to learn R. And if I find that I can manipulate the phase shifting, I will report back (eventually).

Your wish is my command.

To warm up (how to generate a uniform random sample of points on the sphere) see http://rpubs.com/gill1109/13340

Then go to the simulation of the S^2 model:

http://rpubs.com/gill1109/13270

The code could be made more efficient at the cost of a little transparency.

On my computer I can increase the sample size to 100 million (at present it is set at 1 million) and not run into time or memory limits. At that sample size, the Monte Carlo error is ten times smaller than the difference between the cosine and the (effective) model correlation. See http://rpubs.com/gill1109/13366

[Joy Christian]

Anyone who likes can copy-paste from my script and try out such modifications themselves. I'll also prepare a "cleaner" version which only simulates the S^2 model, so the code is less cluttered, more transparent, more easy to experiment with.

Yes, please. I installed R language a few days ago but have been busy on something else since so have not yet played with the language. I could use your S^2 code to try to learn R. And if I find that I can manipulate the phase shifting, I will report back (eventually).

Ben,

What is interesting to check from the perspective of my 3-sphere model is the effect of introducing two tiny phase-shifts and normalization on Bob’s side as follows:

B(b; e, c, d) = sign{+cos (b – e + c)} / sqrt{cos^2 (b – e + c) + sin^2 (b – e + d)},

where c and d are two adjustable phase-shifts (to be kept constant through all trials). The curve may not fit the cosine curve exactly for c = 0 and d = 0, but it may improve with fine-tuning of the two non-vanishing phase-shifts. Note that the normalization in the denominator of B is a nontrivial variable (which reduces to 1 for
c = d). The normalization on Alice’s side is 1, because no phase-shifts are necessary on her side. Note that these are not necessarily fudge factors. As we discussed elsewhere in the context of Chantal's simulation, two phase-shifts like these may be essential for Alice and Bob to not end up detecting some of the same particles.

[Ben6993]

For Richard:

Your wish is my command.

You are too kind! Thank you.

Thank you also for your link re vectors on a sphere. Chantal had already given me references for the maths techniques of that and I must have tried ten or more different ways in Excel VB, including
some of my own, but none of them worked well, which is probably why my simulation was only approximate.

At that sample size, the Monte Carlo error is ten times smaller than the difference between the cosine and the (effective) model correlation.

OK. I take that as your evidence that there is a genuine bias in the simulation of the cosine curve.


For Christian:

I note your suggestions for phase shifts.

You know that I do not have a good grasp of the 3-sphere, but I have never thought of the phase shifts as fudge factors. Pi, I think, gives a phase shift from a plane mirror, and that is not a fudge. My analogy, probably completely way off beam, is that to reflect light back to you from a concave mirror requires two reflections and therefore two angles. And the two angles will be different depending on where the light strikes the mirror. The two angles, at least in my analogy, might have more stable in effects if it was a off a paraboloid rather than a sphere.

[gill1109]

I've revised several of my R scripts (and added a lot of comments, hopefully making them more understandable).

At http://rpubs.com/gill1109 you can find the whole collection.

Chantal Roth has similarly been busy at http://rpubs.com/chenopodium/

[Joy Christian]

I've revised several of my R scripts (and added a lot of comments, hopefully making them more understandable).

At http://rpubs.com/gill1109 you can find the whole collection.

Chantal Roth has similarly been busy at http://rpubs.com/chenopodium/

Your extra comments are quite misleading. It is simply wrong to neglect the fact that the initial state of the system both in my model and in your version of Michel's simulation is the pair (e, t), not just a vector e. You are misleading the physics community in thinking that my model has something to do with the detection loophole.

Also, why are you afraid of trying out my suggestion of introducing two tiny phase shifts in Bob's measurement result as described below?

B(b; e, c, d) = sign{+cos (b – e + c)} / sqrt{cos^2 (b – e + c) + sin^2 (b – e + d)}

To see where these phase shifts and the normalization factor are coming from, please see eqs. (A28) to (A31) of this paper: http://arxiv.org/abs/1301.1653.

[Ben6993]

Hi Richard

I have copied and pasted and run the code from: "To warm up (how to generate a uniform random sample of points on the sphere) see http://rpubs.com/gill1109/13340"
OK
[Presumably one can only rotate the output diagram if one has a touch sensitive screen?]

"Then go to the simulation of the S^2 model: http://rpubs.com/gill1109/13270"
OK

"I've revised several of my R scripts (and added a lot of comments, hopefully making them more understandable). At http://rpubs.com/gill1109 you can find the whole collection. "
[I ran the first page of code] OK


But I have not looked in detail at the code yet ....

-------
The following two links gave '404' messages.

"At that sample size, the Monte Carlo error is ten times smaller than the difference between the cosine and the (effective) model correlation. See http://rpubs.com/gill1109/13366gill1109"

"Chantal Roth has similarly been busy at http://rpubs.com/chenopodium/gill1109"

[gill1109]

The editor seems to have garbled the text.
Try again:

At that sample size, the Monte Carlo error is ten times smaller than the difference between the cosine and the (effective) model correlation. See http://rpubs.com/gill1109/13366

"Chantal Roth has similarly been busy at http://rpubs.com/chenopodium/

[Joy Christian]

Chantal Roth has similarly been busy at http://rpubs.com/chenopodium/

Contrary to what Chantal asserts on this website, neither your nor her simulations have anything to do with the detection loophole. Both of you are using a pair of hidden variables like (e, t) as initial states, where e is a vector and t is a scalar. It is simply incorrect to attribute any relation between e and t as exploiting the detection loophole. The initial state of the system is a pair (e, t), not just e, and every initial state (e, t) uniquely leads to a detection. If there are no states, then there can be no outcomes. There is one-to-one correspondence between the states (e, t) and the outcomes (A, B). If there are no clouds, then there can be no rain; and if there is rain, then there must have been clouds. To claim otherwise is to make a false claim. These comments have nothing to do with R^3 versus S^3 per se.

[gill1109]

Chantal Roth has similarly been busy at http://rpubs.com/chenopodium/

Contrary to what Chantal asserts on this website, neither your nor her simulations have anything to do with the detection loophole. Both of you are using a pair of hidden variables like (e, t) as initial states, where e is a vector and t is a scalar. It is simply incorrect to attribute any relation between e and t as exploiting the detection loophole. The initial state of the system is a pair (e, t), not just e, and every initial state (e, t) uniquely leads to a detection. If there are no states, then there can be no outcomes. There is one-to-one correspondence between the states (e, t) and the outcomes (A, B). If there are no clouds, then there can be no rain; and if there is rain, then there must have been clouds. To claim otherwise is to make a false claim. These comments have nothing to do with R^3 versus S^3 per se.

There are two ways in which I interpret my S^2 version of the algorithm which Minkwe described and implemented:

(1) The hidden state is (e,t) where e is drawn uniformly at random from S^2 and independently of that, t is drawn from the distribution of 1/2 sin^2 theta, where theta in its turn is drawn uniformly at random from [0, pi/2]. The measurement functions A and B take values in {-1, 0, 1}.

(2) The hidden state is (e,t) where (e,t) are drawn from the joint conditional distribution described above but conditional on A(a; e,t) != 0 and B(b; e,t) != 0.

According to common understanding and terminology, model (1) is a model where particles sometimes fail to be detected (coded by "outcome" 0). Model (2) is a conspiracy loophole model: the probability distribution of the hidden variables depends on the measurement settings. The difference between (1) and (2) is whether we keep or we discard the occasions when A or B takes the value 0.

Anyone is welcome to come up with new interpretations, for instance

(3) The simulation is an imperfect implementation of Joy Christian's local hidden variable model based on S^3. A possible source of "imperfection" is through the way the condition "for all x" has been implemented. I only check x=a and x=b. So I check two particular values of x but not all values of x. From the geometric picture of what is going on it is clear that if we truly had checked all possible values of x, no state would have survived at all.

[Joy Christian]

There are two ways in which I interpret my S^2 version of the algorithm which Minkwe described and implemented:

(1) The hidden state is (e,t) where e is drawn uniformly at random from S^2 and independently of that, t is drawn from the distribution of 1/2 sin^2 theta, where theta in its turn is drawn uniformly at random from [0, pi/2]. The measurement functions A and B take values in {-1, 0, 1}.

According to common understanding and terminology, model (1) is a model where particles sometimes fail to be detected (coded by "outcome" 0).

This is very misleading, if not plain wrong. For the state (e, t) we can certainly say mathematically that the measurement functions A and B take values in {-1, 0, 1}.

But physically 0 does not correspond to failure of detecting a particle. We cannot detect what is not there in the first place. If there is no particle to begin with, then there is nothing to detect in the first palce.

(2) The hidden state is (e,t) where (e,t) are drawn from the joint conditional distribution described above but conditional on A(a; e,t) != 0 and B(b; e,t) != 0.

Model (2) is a conspiracy loophole model: the probability distribution of the hidden variables depends on the measurement settings.

In my model it does not. In my model the probability distribution of the hidden variable does not depend on the measurement settings.

The difference between (1) and (2) is whether we keep or we discard the occasions when A or B takes the value 0.

For the state (e, t) there are no "occasions" when A or B takes the "value" 0, because physically no such value could "occasion" to begin with.

Anyone is welcome to come up with new interpretations, for instance

(3) The simulation is an imperfect implementation of Joy Christian's local hidden variable model based on S^3.

Sure.

A possible source of "imperfection" is through the way the condition "for all x" has been implemented. I only check x=a and x=b. So I check two particular values of x but not all values of x. From the geometric picture of what is going on it is clear that if we truly had checked all possible values of x, no state would have survived at all.

True in R^3, but not true in S^3. The set of complete states Lambda is not a null set in S^3, as I have proven in this one-page document.

[gill1109]

I said there are two ways in which *I* interpret the algorithm which I coded in the R language.

I said that anyone else was welcome to any other interpretation.

The algorithm itself is clear. This is not a matter of interpretation. It does what it does:

repeatedly, do the following:

* pick e uniformly at random from S^2

* independently thereof, pick t = 1/2 sin^2 theta, where theta is uniform at random from [0, pi/2]

* calculate A(a; e, t) and B(b; e, t) taking values in {-1, 0, 1} according to the formulas in Joy's one page document.

* whenever neither is zero, take the product

Average the so obtained products (only over the occasions when it is non-zero)

My R code implements this algorithm. It is as good as numerical precision allows (double precision reals, state of the art pseudo random generators). Anyone can believe whatever they like about it, in particular, what it represents, whether or not it has any connection (and if so, what connection) to real or imaginary EPR-B experiments, or to Joy's 3-sphere based theory, ...

[FrediFizzx]

The one page proof Joy linked to above proves that that algorithm is related to 3-sphere geometry. Did you find a fault in the proof? Or is there something you don't understand in the proof and have a question about?

[gill1109]

The one page proof Joy linked to above proves that that algorithm is related to 3-sphere geometry. Did you find a fault in the proof? Or is there something you don't understand in the proof and have a question about?

I made these comments a long time ago, and did not get any kind of resolution.

No, I don't understand the proof.

I don't understand Joy's use of the quantifier "for all x": .

I don't know where comes from, what we are supposed to assume about it.

Joy states that the hidden variable should satisfy a certain condition for all , but in fact only imposes the condition for
and for .

Unless , there are no satisfying the condition .

Joy told us that "for all" could also be understood as "for any" but I think he has silently slipped into "for some", i.e., "there exists", .

[Joy Christian]

I don't understand Joy's use of the quantifier "for all x": .

There is nothing mysterious or ambiguous about this universal quantifier. You can look up the standard definition.

I don't know where comes from, what we are supposed to assume about it.

is an arbitrary unit vector in , just as is. There is nothing mysterious about either of them.

Joy states that the hidden variable should satisfy a certain condition for all , but in fact only imposes the condition for and for .

I don't impose anything. The universal condition on follows from the properties of the parallelized 3-sphere, as defined in the very first equation. What holds for all , holds for any . Therefore it holds for any and any . In practice all we can do is choose any two directions at once (there are only two detectors). All other directions are exclusive and counterfactual to the chosen ones. But in principle one could choose any pair .

Unless , there are no satisfying the condition .

This is false. Well, it is true in R^3, but not in S^3. Locally S^3 = S^1 x S^2, and thus contains infinitely many S^2's.

Joy told us that "for all" could also be understood as "for any" but I think he has silently slipped into "for some", i.e., "there exists", .

The confusion here stems from not being able to escape from the R^3 picture. Recall that S^3 is a surface in R^4, not in R^3. For all means for all, or for any. It is the standard universal quantifier. The property under discussion is an intrinsic property of the 3-sphere, as derived in the one-page document Fred has mentioned.

[gill1109]

Unless , there are no satisfying the condition .

This is false. Well, it is true in R^3, but not in S^3. Locally S^3 = S^1 x S^2, and thus contains infinitely many S^2's.

Hm. So each x in S^2 should really be thought of as a pair x,t with t in S^1. Does "for all x" mean "for all x,t"? Or does it mean, "for all x, there exists t, such that ..."?

Ambiguous notation and terminology. Dangerous.

[Joy Christian]

Unless , there are no satisfying the condition .

This is false. Well, it is true in R^3, but not in S^3. Locally S^3 = S^1 x S^2, and thus contains infinitely many S^2's.

Hm. So each x in S^2 should really be thought of as a pair x,t with t in S^1. Does "for all x" mean "for all x,t"? Or does it mean, "for all x, there exists t, such that ..."?

Ambiguous notation and terminology. Dangerous.

No! imperfect knowledge and imperfect understanding. I used the word "locally" in topological sense. Globally S^3 is not= S^1 x S^2. If it were, then life would be very simple indeed, because then there would be no EPR-Bohm correlation. It is the fact that S^3 is globally not a product of S^1 and S^2 that makes life "difficult" for some. More to the point, your whole line of questioning here is misguided. The correct way of looking at a (parallelized) S^3 is as a set of unit quaternions. It is then easy to prove, as I have done in the one page I keep citing, that the set of all complete states is not a null set. But you, and almost all other Bell-believers keep thinking of your x as vectors in R^3 rather than vectors that appear in the set of unit quaternions. Your vectors x, for example, do not satisfy geometric products like ab = a.b + a /\ b. In fact your vectors do not even form a proper algebra (there is no such thing as vector "algebra"), whereas my vectors do form a genuine algebra.