Various QRC simulations in one Mathematica file

[Mikko]

Baseless speculation,

No speculation at all. That there is a non-local model that the simulation code simulates is easy easy to see when one thinks about it. Perhaps I should have said that there may be still other models, but the point is that assuming there is only one is incorrect; and an apparent disagreement may reflect that participants are talking about different models, each believing to be talking about the only one.

[Joy Christian]

Baseless speculation,

No speculation at all. That there is a non-local model that the simulation code simulates is easy easy to see when one thinks about it. Perhaps I should have said that there may be still other models, but the point is that assuming there is only one is incorrect; and an apparent disagreement may reflect that participants are talking about different models, each believing to be talking about the only one.

You are seeing things.

[FrediFizzx]

It's clear you don't understand what the program is doing. To help you here is a short review of what is happening:

M is the number of experiments analyzed, in my example this is 10.

No. It is 6. The other 4 don't exist in the first place so how can they be analyzed? Joy knows exactly what the program is doing. That is not the issue here. The issue is looking at the whole scenario from an S^3 perspective. Granted, that doesn't seem to be very easy to do for some people.

I would like to understand how the experiment looks from the enlightened viewpoint of the S^3 perspective. From my viewpoint, that of an ignorant flatlander, a source generates two electrons which travel in opposite directions (to conserve momentum), and with opposite spins (to conserve angular momentum). They go into two detectors and are analyzed by their Stern Gerlach apparatuses which are set at different angles. We can assume all electrons enter one of the detectors. Now, if the electron's spin is oriented parallel to the detector, the detector outputs a 1, if antiparallel to the detector a -1 is output. However if the spin is oriented close to 90 degrees, the detector response will be below some threshold, and there will be no output. In this case, one detector has an output, and the other will not have an output. If this happens, we will get a zero, and that experiment is still recorded as having a zero. A flatlander sees this as one of the experiments that contributes to the detector loophole since it will be rejected later in the analysis.

How is this experiment explained from the viewpoint of S^3? Please don't just say go read a paper. Kindly describe it in detail in terms that an ignorant flatlander can understand.

[Joy Christian]

How is this experiment explained from the viewpoint of S^3? Please don't just say go read a paper. Kindly describe it in detail in terms that an ignorant flatlander can understand.

Thank you, John. Neither Fred nor I are trying to skirt the issues you are raising. We just have a different perspective. Unfortunately this perspective is not easy to understand without first understanding the geometry and topology of the 3-sphere. But one can use simplified metaphors to understand what is going on. One such metaphor is to define an effective metric on S^3 such that orthogonal directions are simply cut off from the corresponding embedding space. This is what topologist call "surgery." It is a technical term, but it is not difficult to understand. Here is a very transparent definition of a metric on S^3 that accomplishes such a "surgery":





Here the generalized orthogonality of the vectors and is defined, evidently, by the condition , and it depends on the parameter .

Note that there are no strictly orthogonal directions in the space defined by the above metric. The orthogonal directions have been simply cut off and thrown away.

In probabilistic terms one can say that the probability of two directions being strictly orthogonal is zero. As Bell notes in his famous 1964 paper, "[a]ctually this leaves the result undetermined when = 0, but as the probability of this is zero we will not make special prescriptions for it." So John Bell agrees with me on this point.

If you still think that what I am doing is wrong, then your first quarrel is with John Bell, not me. In fact my model is far superior than John Bell's original local model: http://arxiv.org/abs/1405.2355.

[jreed]

How is this experiment explained from the viewpoint of S^3? Please don't just say go read a paper. Kindly describe it in detail in terms that an ignorant flatlander can understand.

Thank you, John. Neither Fred nor I are trying to skirt the issues you are raising. We just have a different perspective. Unfortunately this perspective is not easy to understand without first understanding the geometry and topology of the 3-sphere. But one can use simplified metaphors to understand what is going on. One such metaphor is to define an effective metric on S^3 such that orthogonal directions are simply cut off from the corresponding embedding space. This is what topologist call "surgery." It is a technical term, but it is not difficult to understand. Here is a very transparent definition of a metric on S^3 that accomplishes such a "surgery":





Here the generalized orthogonality of the vectors and is defined, evidently, by the condition , and it depends on the parameter .

Note that there are no strictly orthogonal directions in the space defined by the above metric. The orthogonal directions have been simply cut off and thrown away.

In probabilistic terms one can say that the probability of two directions being strictly orthogonal is zero. As Bell notes in his famous 1964 paper, "[a]ctually this leaves the result undetermined when = 0, but as the probability of this is zero we will not make special prescriptions for it." So John Bell agrees with me on this point.

If you still think that what I am doing is wrong, then your first quarrel is with John Bell, not me. In fact my model is far superior than John Bell's original local model: http://arxiv.org/abs/1405.2355.

In Bell's paper, he isn't considering any lost data. His detectors are perfect and no data is recorded with zeros. Our situation is different.

When you say topological surgery, this implies something has been removed or cut out from a starting collection of objects. What is left is then analyzed. What is the starting collection? This sounds exactly like what I'm saying: You start with the vectors {x, y, z} and using the R function "length", you cut out some of the values where the observations on this set have zeros, ending up with the new collection {o, p, q}. {o, p, q} are then analyzed. Isn't this correct? If not please explain what I'm missing here.

[Joy Christian]

In Bell's paper, he isn't considering any lost data.

Neither am I.

Both in my analytical model as well as in its simulation there is one-to-one correspondence between the initial states "e" and the measurement functions A and B.

Thus no data is lost.

His detectors are perfect and no data is recorded with zeros.

My detectors are also perfect and no data is recorded with zeros.

Our situation is different.

Our situation is not different.

When you say topological surgery, this implies something has been removed or cut out from a starting collection of objects.

No. What has been cut out has nothing to do with the starting collection of objects. It has to do with the procedure of identifying the correct collection of objects.

What is left is then analyzed.

No. What is analyzed is what happens in the topological space that is constructed by the surgery. That is the starting space. Everything happens within that space.

What is the starting collection?

The starting collection in this simulation is the set of vectors "e". There are N such vectors. In your example N = 6. You start with M = 10, but that number has nothing to do with either physics or the actual content of the simulation.

This sounds exactly like what I'm saying: You start with the vectors {x, y, z} and using the R function "length", you cut out some of the values where the observations on this set have zeros, ending up with the new collection {o, p, q}. {o, p, q} are then analyzed. Isn't this correct?

No. That is not correct. Your mistake is in this phrase: "observations on this set have zeros." What observations? No one is making any observations on the set {x, y, z}. The vector w = {x, y, z} in the simulation is not the initial state, and the functions C and D are not what is observed by Alice and Bob. So why do you keep referring to the functions C and D as "observations"? These functions have nothing to do with the observations made by Alice and Bob.

The functions used by Allice and Bob for observations are the functions A and B. The initial states of the spins they receive are the vectors "e". And as I noted, there is one-to-one correspondence between the initial states "e" and the measurement functions A and B. Therefore the question of detection loophole does not even arise.

[jreed]

This sounds exactly like what I'm saying: You start with the vectors {x, y, z} and using the R function "length", you cut out some of the values where the observations on this set have zeros, ending up with the new collection {o, p, q}. {o, p, q} are then analyzed. Isn't this correct?

No. That is not correct. Your mistake is in this phrase: "observations on this set have zeros." What observations? No one is making any observations on the set {x, y, z}. The vector w = {x, y, z} in the simulation is not the initial state, and the functions C and D are not what is observed by Alice and Bob. So why do you keep referring to the functions C and D as "observations"? These functions have nothing to do with the observations made by Alice and Bob.

I ran the small simulation and again started with 10 samples. 6 of these were non-zero. Here are the starting values of x, y, z, the points on the unit sphere:

x = {0.443175, -0.964391, -0.653491, 0.349304, 0.825805, -0.815974,
-0.0922178, 0.365188, -0.812601, 0.891069}

y = {0.89448, -0.145578, 0.692179, 0.870631, 0.474944, 0.362727,
-0.988365, 0.0164345, -0.185999, 0.412685}

z = {-0.05918, 0.220811, -0.306329, 0.346393, 0.304096, 0.450128,
-0.120955, 0.930789, 0.552344, -0.188908}

Now I found the samples o, p, q, obtained by eliminating zero values from x, y, z:

o = {0.443175, -0.964391, 0.825805, -0.815974, -0.812601, 0.891069}

p = {0.89448, -0.145578, 0.474944, 0.362727, -0.185999, 0.412685}

q = {-0.05918, 0.220811, 0.304096, 0.450128, 0.552344, -0.188908}

If you examine these, you'll find the values in o, p, and q are all in the x, y, z arrays. It is incorrect to say that x, y, and z were not observed. The program observed these to find the vectors o, p, and q, which are the values of the x, y, and z vectors where the detector got a value larger than the function f.

[Joy Christian]

The program observed these to find the vectors o, p, and q, which are the values of the x, y, and z vectors where the detector got a value larger than the function f.

This is wrong. What on earth do you mean by ""program observed these"? What on earth do you mean by "detector" when you say "detector got a value larger than the function f." There is no detector as yet. Alice and Bob are not even in their labs as yet, as I have pointed out several times by now. You are horribly stuck and unlikely to understand what is going on. So it is best that we agree to disagree.

To anyone reading this, John is simply unable to understand the difference between a preparation of the initial states "e" and the actual detection of the events A and B, as so clearly separated out in this simulation: http://rpubs.com/jjc/105450.

[FrediFizzx]

To anyone reading this, John is simply unable to understand the difference between a preparation of the initial states "e" and the actual detection of the events A and B, as so clearly separated out in this simulation: http://rpubs.com/jjc/105450.

Also, it is like that he and others forgot what "simulation" means.

[jreed]

The program observed these to find the vectors o, p, and q, which are the values of the x, y, and z vectors where the detector got a value larger than the function f.

This is wrong. What on earth do you mean by ""program observed these"? What on earth do you mean by "detector" when you say "detector got a value larger than the function f." There is no detector as yet. Alice and Bob are not even in their labs as yet, as I have pointed out several times by now. You are horribly stuck and unlikely to understand what is going on. So it is best that we agree to disagree.

To anyone reading this, John is simply unable to understand the difference between a preparation of the initial states "e" and the actual detection of the events A and B, as so clearly separated out in this simulation: http://rpubs.com/jjc/105450.

I have written a Mathematica version of your program so that I could get a feeling for what it's doing.
I'll try to explain what the program is doing, as I understand it:

The detector in your program is the function g in the R code:

s = runif(M, 0, pi)
f = -1 + (2/sqrt(1 + ((3 * s)/pi)))
g = function(u,v,s){ifelse(abs(colSums(u*v)) > f, colSums(u*v), 0)}

f is a function you got from the paper by Pearle about the detector loophole, and it is used in g, so that if the absolute value of the dot product of the vectors u and v is greater than f, the dot product u*v is output, otherwise a zero is output. That's what the ifelse statement above says. These statements form a detector that outputs a value if the components of two vectors are above a certain value. I call that a detector. I hope you can understand this. That's what on earth I'm talking about. Is this clear to you?

Now, in the program this detector is applied to the vector set w = {x, y, z} which are the initial set of random unit vectors on a unit sphere in the statements:

C = sign(g(a,w,s))
D = sign(g(b,w,s))

Here, a and b are vectors representing the directions of Alice and Bob's detectors. These statements generate C and D, the outputs of their detectors, either a 1, representing alignment of spin and detector, -1 representing anti-alignment, or zero, which indicates the response was below the threshold of detection. C and D have a length of M, which is ten for the small simulations I have used to demonstrate this. C and D have values of 1, -1 and zero. These are the detected outputs for all the {x, y, z} random vectors that were generated at the start of the program. The program used (observed) these values to generate the detector response C and D. That's what on earth I mean by observed. I hope this is understandable to you.

Now you use C and D in the statements:

o = x[C & D]
p = y[C & D]
q = z[C & D]

to select the {x, y, z} vectors where their detected responses (C & D) are both non-zero and combine them into e, which you call the initial states. Why on earth do you call o, p, and q the initial states when x, y and z are clearly the initial states? o, p and q are the states remaining after eliminating those states in x, y and z which have a zero in their detected outputs. The length of o, p and q is less than x, y and z, since some values have been eliminated (the detector loophole).

I don't think you understand the concept of what you're doing in this simulation. It's clearly using the detection loophole as discussed by Pearle. This is guaranteed to work since Pearl's paper is a demonstration of this. You have just repeated his ideas with a numerical simulation. As I mentioned in a previous post, the CHSH inequality won't work in this case, and your simulation doesn't violate Bell's theorem. If you need more help on understanding this, please ask.

[Joy Christian]

The detector in your program is the function g in the R code:

I disagree with this statement, so there is no point in going any further. Please read the correct meaning of g in the original simulation: http://rpubs.com/jjc/84238.

[Heinera]

And let me just add to jreed's post that o, p, q cannot be the initial states, since C and D depend on the detector settings (obvious from the above code), and thus will o, p, q depend on the settings as well.

[Joy Christian]

And let me just add to jreed's post that o, p, q cannot be the initial states, since C and D depend on the settings (obvious from the above code), and thus will o, p, q depend on the settings as well.

Nonsense, as I have already pointed out to you:

What jreed realizes, based on the code, is that the exact manner of how the 10 experiments are reduced to 6 depends on the detector settings.

There are no 10 experiments. There are only 6 experiments in John Reed's example. Once again, you are seeing things. You have no clue what you are talking about.

Ok, let me rephrase:

What jreed realizes, based on the code, is that the exact manner of how the 10 pairs are reduced to 6 experiments depends on the detector settings.

No, it does not:

In the code I have the following line which is not used in the calculation of the main correlation:

Code: Select all

# corrs[i,j] = length(A*B)/N   # Verifies (# of A*B) = (# of e)


Run the code with this line and see what you get for the correlation. You should now be able to understand what that means. It mean what it says in the comments.

[Heinera]

From the code:

C = sign(g(a,w,s))
D = sign(g(b,w,s))

What on earth are a and b doing there?

[Joy Christian]

The detector in your program is the function g in the R code:

I disagree with this statement, so there is no point in going any further. Please read the correct meaning of g in the original simulation: http://rpubs.com/jjc/84238.

Also, I have made an entire post explaining the correct concepts: viewtopic.php?f=6&t=188#p5129. This has gone totally over the heads of both Heinera and jreed.

In particular, neither has bothered to read or understand the original analytical model: http://arxiv.org/pdf/1405.2355.pdf.

PS: For anyone reading this, g is not a detector. That would be like calling a "detector." The detectors in this simulation are represented by the vectors a and b, as is always done in any Bell-type local model. No one in their right mind would call a "detector." In other words, jreed's statement is a gobbledygook.

[FrediFizzx]

Ok, this thread has gone somewhat off topic and is going in circles. It is being locked. I think there already is a thread about Joy's simulation that this discussion should probably be in. Or feel free to start a new topic.