A real EPR-Bohm scenario

[minkwe]

As I expected, so for each iteration, the q function is returning the exact same thing when p > f. It is returning 51 numbers. So g= p*q will also be 51 numbers in those cases and also sign(q) will be 51 numbers some -1 other +1 and so will sign(p)*sign(q) etc. Do you see now that this cannot be correct?

Not really. But what is more confusing is: why does the code work at all if q is wrong? If I remove q by setting f = 0, then I get the linear correlations. So somehow the q-factor produces the right correlation. Don't you find that curious? If q is wrong (and I am sure you are right about that), then why does p x q work in place of original g, with linear background restored as expected when q is set to 1 (by setting f =0)?

***

That is why I keep asking for clarity on what the model is because that expresses what the intention is. It is the wrong question to ask why it works. The code is clearly not right (to me at least). There is definitely a reason why the code "works" but that reason has to do with side-effects of the R-language that might take you a long time to figure out. You don't want to waste your time on that.

The real question should be -- does the model say that the q function should produce 51 numbers for a single particle? If the answer is no, then the code is wrong.

[minkwe]

Perhaps what you are trying to do is

Code: Select all

q = function(u,v,x){ifelse(abs(p(u,v)) > f, cos(x), 0)}

Then in the code you will always pass the angle along and have

Code: Select all

A = +sign(p(a,e))*sign(q(a,e,alpha))  # Alice's results A(a, e, s) = +/-1
B = -sign(p(b,e))*sign(q(b,e,beta))   # Bob's results B(b, e, s) = -/+1
 

Is that what you are trying to do?

[ajw]

u and v are unit vectors that change from one iteration to the next. s is a scalar, f is also a scalar. p is the dot product which is also a scalar, Angles is a vector of dimention 51 as defined earlier on the first line. What does cos(Angles) mean then??

Just to be sure you realize this: u and v are not unit vectors: they are arrays of unit verctors, s is an array of scalars (all dimension M).

[ajw]

Speaking about a real EPR-Bohm scenario, does any of you have ideas of how to use quaternions or other multivectors (constrained by S3) for an event by event simulation? I know at least in Python there are libraries for this. The currently discussed simulations are the R3 flatlanders versions of Joy's model, and therefore have the if/else then 0 at the polarizer stage, which limits the proving value of the model.
(maybe this post belongs in a new thread)

ajw, my code is using geometric algebra with Cl(0,3) through out. It is a simple tweak to change the algebra to some other layout. Like I mentioned, I can easily test any model by simply defining the two functions so I welcome any suggestions. All I need are:

1) Source: How are the particle pairs represented, and what information should they contain
2) Station: What are the steps involved to produce the detection events (+1, -1) and what other side-effects are caused by the detection (eg, time delays, memory, etc if relevant to the model)

The framework can handle all those.

This sounds great!
I can only guess that for the description of the spin a quaternion should be used and a random variable lambda as hidden variable for the sign of the base to simulate Joys model. I have no clue whether or not the polarizer settings should be described using a quaternion, and how to arrive at the detection events using these objects.

[FrediFizzx]

As I expected, so for each iteration, the q function is returning the exact same thing when p > f. It is returning 51 numbers. So g= p*q will also be 51 numbers in those cases and also sign(q) will be 51 numbers some -1 other +1 and so will sign(p)*sign(q) etc. Do you see now that this cannot be correct?

No it is not returning 51 numbers. It is 51 x 51 because the iterations are nested.

Each time the function is called it returns 51 numbers. The R code calls it 51x51 times. You can add a print statement inside the inner loop to print

length(q(a,e,s)) or length(sign(q(a,e,s)))

and you will see that it gives 51 each time for a total of 51x51 times

Well, there is actually more to it that just 51x51. The final "corrs" is a matrix that is 51x51 and those numbers in the matrix are averages depending on N, etc.
.

[minkwe]

u and v are unit vectors that change from one iteration to the next. s is a scalar, f is also a scalar. p is the dot product which is also a scalar, Angles is a vector of dimention 51 as defined earlier on the first line. What does cos(Angles) mean then??

Just to be sure you realize this: u and v are not unit vectors: they are arrays of unit vectors, s is an array of scalars (all dimension M).

That is right ajw. I agree that v is an array of vectors of size (M), but what about "a" and "b"?


Note that a and b are defined as

Code: Select all

alpha = Angles[i]
beta = Angles[j]
a = c(cos(alpha), sin(alpha), 0)
b = c(cos(beta), sin(beta), 0)

Thus a and b are 3-vectors.
Now note that every time q is called it is called as follows:

Code: Select all

A = +sign(p(a,e))*sign(q(a,e,s))  # Alice's results A(a, e, s) = +/-1
B = -sign(p(b,e))*sign(q(b,e,s))   #  Bob's results B(b, e, s) = -/+1


Thus in the call q(a,e,s), which u = a, v=e. The function p calculates the inner product of u*v, but what does that mean then when u is a 3-vector and v is an M-vector? You'll probably end up with a array of M scalars. Is this intentional? Let us call that number P, in the q function, we compare P with the M-vector f, and return 51-vector cos(Angles).

Perhaps it's my naivity with R showing. But something tells me the cos(Angles) shouldn't be there.

[ajw]

Is this intentional?

I'm no hero in R either, but I think it is. I analyzed some earlier and simpler version of the model here http://challengingbell.blogspot.nl/2014 ... ff-on.html
Not sure how the Angles 51-vector is behaving in R and what the purpose for the simulation is.

[minkwe]

Is this intentional?

I'm no hero in R either, but I think it is. I analyzed some earlier and simpler version of the model here http://challengingbell.blogspot.nl/2014 ... ff-on.html
Not sure how the Angles 51-vector is behaving in R and what the purpose for the simulation is.

The only thing that looks "off" to me is the cos(Angles). The rest seems normal, even including the M-vector aspects.

In any case, if the model is:

- setting "a" (or "b") is a random angle between (0, 2pi), e is also an angle between (0, 2pi)
- calculate |a - e| = c
- calculate polarizer = {+1 if c<pi else -1}
- calculate q = {cos(a*polarizer) if c > f else 0}
- detector output = sign(q)

I can confirm that it does not work according to my python code.

Fred, Joy,
Could you confirm if the intention for the q function is the same as:

Code: Select all

A = +sign(p(a,e))*sign(q(a,e,alpha))  # Alice's results A(a, e, s) = +/-1
B = -sign(p(b,e))*sign(q(b,e,beta))   # Bob's results B(b, e, s) = -/+1


I'm guessing that is the case because earlier Fred said:

Joy's q-function is returning cos(a) or 0 and cos(b) or 0 for each iteration. We think that is how it should be with the polarizer action separated out. And it works.

[FrediFizzx]

In any case, if the model is:

- setting "a" (or "b") is a random angle between (0, 2pi), e is also an angle between (0, 2pi)
- calculate |a - e| = c
- calculate polarizer = {+1 if c<pi else -1}
- calculate q = {cos(a*polarizer) if c > f else 0}
- detector output = sign(q)

I can confirm that it does not work according to my python code.

That is basically the model that works in both Mathematica and R for us. Mainly Mathematica. The R model is slightly different which I posted the complete code for already. I should note that "e" for the B particle needs to be "-e".

[FrediFizzx]

Fred, Joy,
Could you confirm if the intention for the q function is the same as:

Code: Select all

A = +sign(p(a,e))*sign(q(a,e,alpha))  # Alice's results A(a, e, s) = +/-1
B = -sign(p(b,e))*sign(q(b,e,beta))   # Bob's results B(b, e, s) = -/+1


I'm guessing that is the case because earlier Fred said:

Joy's q-function is returning cos(a) or 0 and cos(b) or 0 for each iteration. We think that is how it should be with the polarizer action separated out. And it works.

That doesn't seem right. It would be more like,

Code: Select all

A = +sign(p(a,e))*sign(q(a,e))  # Alice's results A(a, e, s) = +/-1
B = -sign(p(b,e))*sign(q(a,e))   # Bob's results B(b, e, s) = -/+1


with the q-function returning cos(alpha) or 0 and cos(beta) or 0. But I think I tried that and it didn't work. Let me try again here... it doesn't work. But yes, you would think that would be the right way of doing it.
.

[minkwe]

Why do you remove alpha/beta as a parameter for the q function but then say the q function returns "alpha"/"beta". Where does the function get those values to return?

[FrediFizzx]

Why do you remove alpha/beta as a parameter for the q function but then say the q function returns "alpha"/"beta". Where does the function get those values to return?

Oh, I see what you mean. OK, tried calling alpha and beta through the function and it still doesn't work. So we have a mystery of why Angles works and alpha and beta don't when it seems like they should.

[Joy Christian]

***
I am getting M = 100 numbers for both p and q. If q was wrong, then shouldn't I be getting an error message, or at least a different total number for q?

Code: Select all

(p(a,e))

##   [1]  0.67786550  0.64575118 -0.25426569  0.78422801 -0.89053366
##   [6]  0.49789059  0.07573709  0.53771959  0.82074388 -0.11736966
##  [11] -0.88538331 -0.77503251 -0.44758211 -0.15457592  0.78222645
##  [16]  0.54018283 -0.26501735  0.85073992 -0.40911859  0.90307235
##  [21]  0.40898502  0.84177919 -0.76453770 -0.26319791  0.93563109
##  [26]  0.30599767 -0.52642840 -0.66926828 -0.47240159 -0.80554639
##  [31]  0.68287312 -0.98037936 -0.86875765  0.12143532 -0.90840380
##  [36]  0.78656701  0.23015029 -0.39264213 -0.65777291 -0.44693819
##  [41] -0.24446008 -0.03685300 -0.25005645  0.43345175 -0.76212388
##  [46]  0.41154968  0.27593050 -0.77287365  0.45841924 -0.05939217
##  [51]  0.43375199 -0.97776246  0.63746385  0.51123556 -0.12323894
##  [56]  0.03761263 -0.46948100  0.49250628  0.77304947 -0.26475771
##  [61]  0.07379288  0.63319551  0.97488354 -0.29635812 -0.26413093
##  [66]  0.25511797  0.50163476  0.37540033  0.36596551  0.25750995
##  [71] -0.01478519 -0.35192678  0.96579040  0.55529209  0.42755873
##  [76] -0.07046557  0.28687501 -0.78136240 -0.43180049 -0.69471322
##  [81]  0.19307502  0.04680831  0.75985086 -0.86015136 -0.14600318
##  [86]  0.50782947 -0.41320527  0.27925798  0.32832982  0.81272125
##  [91] -0.42831146 -0.26824145  0.38873106 -0.37869934  0.73472381
##  [96] -0.16719547 -0.14914802 -0.05227766 -0.53342151  0.13650480


Code: Select all

(q(a,e,s))

##   [1]  0.00000000  0.99211470  0.00000000  0.92977649  0.87630668
##   [6]  0.80901699  0.72896863  0.63742399  0.53582679  0.00000000
##  [11]  0.30901699  0.18738131  0.06279052  0.00000000 -0.18738131
##  [16] -0.30901699 -0.42577929 -0.53582679  0.00000000 -0.72896863
##  [21] -0.80901699 -0.87630668 -0.92977649  0.00000000 -0.99211470
##  [26] -1.00000000  0.00000000 -0.96858316  0.00000000 -0.87630668
##  [31] -0.80901699 -0.72896863 -0.63742399  0.00000000  0.00000000
##  [36] -0.30901699  0.00000000 -0.06279052  0.06279052  0.00000000
##  [41]  0.00000000  0.42577929  0.00000000  0.63742399  0.72896863
##  [46]  0.00000000  0.87630668  0.92977649  0.00000000  0.00000000
##  [51]  1.00000000  1.00000000  0.00000000  0.96858316  0.00000000
##  [56]  0.00000000  0.80901699  0.72896863  0.63742399  0.53582679
##  [61]  0.00000000  0.30901699  0.18738131  0.06279052 -0.06279052
##  [66] -0.18738131 -0.30901699 -0.42577929 -0.53582679 -0.63742399
##  [71]  0.00000000 -0.80901699 -0.87630668 -0.92977649 -0.96858316
##  [76]  0.00000000  0.00000000 -0.99211470 -0.96858316 -0.92977649
##  [81] -0.87630668  0.00000000 -0.72896863 -0.63742399  0.00000000
##  [86] -0.42577929 -0.30901699  0.00000000  0.00000000  0.06279052
##  [91]  0.00000000  0.00000000  0.00000000  0.00000000  0.63742399
##  [96]  0.00000000  0.00000000  0.00000000  0.92977649  0.96858316


***

[minkwe]

So we have a mystery of why Angles works and alpha and beta don't when it seems like they should.

Isn't it time to go back to the original goal? Starting from the original model, introduce the polarizer action in a tractable manner?

Do you have any problem with this:

Code: Select all

def particle_detector(info):
    particle, setting = info
   
    z = particle[-1]    # represents which arm we are on, left or right
    f = particle[0]     
    particle[0] = 0; particle[-1] = 0 # zero out scalar and tri-vector parts leaving just vector
    a = ga.ang2Vec3d(theta=setting, phi=numpy.pi/2) # generate setting vector on a plane
    e = ga.MultiVector(ga.layout, value=particle) # e is a vector
    pol = (a*e).value[0] # dot product between setting vector, project particle vector on setting vector
    q = z*max(abs(pol) - f, 0) 
    out = numpy.sign(pol)*numpy.sign(q)
    return [0, setting,out]


This is equivalent to the original simulation, except there is a separate q function.

[minkwe]

***
I am getting M = 100 numbers for both p and q. If q was wrong, then shouldn't I be getting an error message, or at least a different total number for q?

Code: Select all

(p(a,e))

##   [1]  0.67786550  0.64575118 -0.25426569  0.78422801 -0.89053366
##   [6]  0.49789059  0.07573709  0.53771959  0.82074388 -0.11736966
##  [11] -0.88538331 -0.77503251 -0.44758211 -0.15457592  0.78222645
##  [16]  0.54018283 -0.26501735  0.85073992 -0.40911859  0.90307235
##  [21]  0.40898502  0.84177919 -0.76453770 -0.26319791  0.93563109
##  [26]  0.30599767 -0.52642840 -0.66926828 -0.47240159 -0.80554639
##  [31]  0.68287312 -0.98037936 -0.86875765  0.12143532 -0.90840380
##  [36]  0.78656701  0.23015029 -0.39264213 -0.65777291 -0.44693819
##  [41] -0.24446008 -0.03685300 -0.25005645  0.43345175 -0.76212388
##  [46]  0.41154968  0.27593050 -0.77287365  0.45841924 -0.05939217
##  [51]  0.43375199 -0.97776246  0.63746385  0.51123556 -0.12323894
##  [56]  0.03761263 -0.46948100  0.49250628  0.77304947 -0.26475771
##  [61]  0.07379288  0.63319551  0.97488354 -0.29635812 -0.26413093
##  [66]  0.25511797  0.50163476  0.37540033  0.36596551  0.25750995
##  [71] -0.01478519 -0.35192678  0.96579040  0.55529209  0.42755873
##  [76] -0.07046557  0.28687501 -0.78136240 -0.43180049 -0.69471322
##  [81]  0.19307502  0.04680831  0.75985086 -0.86015136 -0.14600318
##  [86]  0.50782947 -0.41320527  0.27925798  0.32832982  0.81272125
##  [91] -0.42831146 -0.26824145  0.38873106 -0.37869934  0.73472381
##  [96] -0.16719547 -0.14914802 -0.05227766 -0.53342151  0.13650480


Code: Select all

(q(a,e,s))

##   [1]  0.00000000  0.99211470  0.00000000  0.92977649  0.87630668
##   [6]  0.80901699  0.72896863  0.63742399  0.53582679  0.00000000
##  [11]  0.30901699  0.18738131  0.06279052  0.00000000 -0.18738131
##  [16] -0.30901699 -0.42577929 -0.53582679  0.00000000 -0.72896863
##  [21] -0.80901699 -0.87630668 -0.92977649  0.00000000 -0.99211470
##  [26] -1.00000000  0.00000000 -0.96858316  0.00000000 -0.87630668
##  [31] -0.80901699 -0.72896863 -0.63742399  0.00000000  0.00000000
##  [36] -0.30901699  0.00000000 -0.06279052  0.06279052  0.00000000
##  [41]  0.00000000  0.42577929  0.00000000  0.63742399  0.72896863
##  [46]  0.00000000  0.87630668  0.92977649  0.00000000  0.00000000
##  [51]  1.00000000  1.00000000  0.00000000  0.96858316  0.00000000
##  [56]  0.00000000  0.80901699  0.72896863  0.63742399  0.53582679
##  [61]  0.00000000  0.30901699  0.18738131  0.06279052 -0.06279052
##  [66] -0.18738131 -0.30901699 -0.42577929 -0.53582679 -0.63742399
##  [71]  0.00000000 -0.80901699 -0.87630668 -0.92977649 -0.96858316
##  [76]  0.00000000  0.00000000 -0.99211470 -0.96858316 -0.92977649
##  [81] -0.87630668  0.00000000 -0.72896863 -0.63742399  0.00000000
##  [86] -0.42577929 -0.30901699  0.00000000  0.00000000  0.06279052
##  [91]  0.00000000  0.00000000  0.00000000  0.00000000  0.63742399
##  [96]  0.00000000  0.00000000  0.00000000  0.92977649  0.96858316


***

Joy, even if the code is wrong, it could still be valid R and you won't get an error, even if it it does not correctly represent the model you are trying to simulate. Can you tell me what cos(Angles) is supposed to be doing in that expression? What is the corresponding mathematical expression you are trying to evaluate. Think about a single particle, what is the expression for it? In other words, what is the mathematical formula for producing just one of those numbers you printed above.

[Joy Christian]

Joy, even if the code is wrong, it could still be valid R and you won't get an error, even if it it does not correctly represent the model you are trying to simulate. Can you tell me what cos(Angles) is supposed to be doing in that expression? What is the corresponding mathematical expression you are trying to evaluate. Think about a single particle, what is the expression for it? In other words, what is the mathematical formula for producing just one of those numbers you printed above.

My understanding is that cos(Angles) in the q-function is supposed to represent cos(angle_x), where angle_x is the angle that the "vector a" makes with the x-axis.

***

[minkwe]

Joy, even if the code is wrong, it could still be valid R and you won't get an error, even if it it does not correctly represent the model you are trying to simulate. Can you tell me what cos(Angles) is supposed to be doing in that expression? What is the corresponding mathematical expression you are trying to evaluate. Think about a single particle, what is the expression for it? In other words, what is the mathematical formula for producing just one of those numbers you printed above.

My understanding is that cos(Angles) in the q-function is supposed to represent cos(angle_x), where angle_x is the angle that the "vector a" makes with the x-axis.

***

But that is not what the R-code is doing. What you are suggesting will be equivalent to

Code: Select all

A = +sign(p(a,e))*sign(q(a,e,alpha))  # Alice's results A(a, e, s) = +/-1
B = -sign(p(b,e))*sign(q(b,e,beta))   # Bob's results B(b, e, s) = -/+1

BUT, I did some more digging and now I can tell you what exactly the R-code with cos(Angles) does.

p(u,v) will produce an array of M dot products. ie M scalars.
f is also an array of M scalars. The comparison expression abs(p(u,v)) > f, produces an array of M booleans, the values will be TRUE when abs(p) > f and FALSE when abs(p) <= f. The operation ifelse(X, Y, 0) takes loops through the array X, when ever it finds a TRUE, it puts in the corresponding value from the Y array, otherwise it puts in 0. If it reaches the end of Y before it has reached the end of X, it loops back to the beginning of Y and continue until the end. if you replace cos(angles) in the expression with a sequence, you will find that it produces something like

Code: Select all

> abs(p(b,e)) > f
    [1] FALSE  TRUE FALSE  TRUE  TRUE  TRUE  TRUE FALSE  TRUE  TRUE  TRUE  TRUE
   [13]  TRUE FALSE  TRUE  TRUE FALSE  TRUE  TRUE FALSE FALSE FALSE  TRUE  TRUE
   [25]  TRUE  TRUE FALSE  TRUE  TRUE  TRUE  TRUE FALSE  TRUE  TRUE  TRUE  TRUE
   [37] FALSE FALSE FALSE FALSE FALSE FALSE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE
   [49]  TRUE FALSE FALSE  TRUE FALSE FALSE FALSE FALSE  TRUE FALSE  TRUE  TRUE

Code: Select all

> ifelse(abs(p(b,e)) > f, seq(from=1,to=10,by=1), 0)
    [1]  0  2  0  4  5  6  7  0  9 10  1  2  3  0  5  6  0  8  9  0  0  0  3  4
   [25]  5  6  0  8  9 10  1  0  3  4  5  6  0  0  0  0  0  0  3  4  5  6  7  8
   [49]  9  0  0  2  0  0  0  0  7  0  9 10  0  0  0  4  0  6  7  8  9 10  0  2



This is the kind of side-effect I was talking about. I doubt this was intentional.

[FrediFizzx]

So the Y array is cos(Angles). So when X is true, it is picking a value from the list of 51 Angles that matches up with the corresponding X position in its array. Is that correct? It seems like it is correct so that is how cos(Angles) works in the simulation.

Correct, it was not intentional. I found that it works by discovery after I couldn't get alpha and beta to work.

[minkwe]

Yes, There are M entries in X and 51 entries in Y, so it loops around Y. Try returning cos (random angle) instead of cos (Angles) and see if that still works.Because the corresponding angle whose cosine is being returned has nothing to do with Alpha, or Beta, that is why cos(alpha) did not work.

[FrediFizzx]

Yes, There are M entries in X and 51 entries in Y, so it loops around Y. Try returning cos (random angle) instead of cos (Angles) and see if that still works.Because the corresponding angle whose cosine is being returned has nothing to do with Alpha, or Beta, that is why cos(alpha) did not work.

Well, I have to figure out how to setup random angles in R but I don't expect it to work. But Mathematica does work with random angles with a slightly different process. Somehow the iteration loops in R are lining up properly using cos(Angles) in the q-function. And it still works if I set the fixed angles to 5 degree resolution instead of 7.2.

It might be more worthwhile to try to get alpha and beta working because it seems to me that they should work also now that I see the proper way to do them in the q-function. Thanks for that.