Bell's theorem: actual experiment event by event

[DrChinese]

Data stream 1
A: 0 1 0 0 1 0 1 1 0 1
B: 0 1 0 0 1 0 1 1 0 1

Data stream 2
A: 0 1 0 0 1 0 1 1 0 1
B: 1 0 1 1 0 1 0 0 1 0


So if Type I experiment counts only matching values, it concludes the first sequence is 100% correlated and the second sequence 0% correlated, thus failing to recognize the second sequence is actually 100% correlated as well? It seems they should really be sampling and analyzing data over larger sequences, not just making one to one comparison.

At this point you are so lost...

You can measure any values you like in an experiment of your own making. I really can't help you on that. The rest of us have standard setups to discuss and I have given you any number of those. Everyone knows the difference between matches and correlation. And anti-correlation too, which is what you are mentioning. None of this has anything to do with a Bell test beyond what I have explained so far.

The issue is the average relationship between pairs of entangled particles. In quantum theory, that relationship remains in place as long as the pair is described by a single wave function. In classical theory, that relationship ends when they are no longer in causal contact because classical theories follow local realism.

I will again ask you to read the references I have provided. When you understand those, many of your questions will be resolved. Your idea that the scientists working this area do not understand the basics of their experiments (and are comparing wrong sequences) is... well, what do you really expect?

Learn the background, THEN try to tear it apart. Not the other way around.

 

[DrChinese]

I think I see a new law of physics, new to me at least.

Humble's 1st Law of Binary Sequences
- two binary sequences can never be more than 50% uncorrelatedEDIT:
The moment where "matching pairs correlation" drops below 50%, is the same moment where "opposite pairs correlation" raises above 50%.

When measuring and reporting "correlations", the usual standard is:

1 means fully correlated.
-1 means fully anti-correlated.
0 means neither, or essentially random.

 

[humbleteleskop]

When measuring correlations:

1 means fully correlated.
-1 means fully anti-correlated.
0 means neither, or essentially random.

How detector knows whether it is +1 or -1? , what does it measure?

How detector knows it's 0, what does it measure, or how it calculates?

The issue is the average relationship between pairs of entangled particles. In quantum theory, that relationship remains in place as long as the pair is described by a single wave function. In classical theory, that relationship ends when they are no longer in causal contact because classical theories follow local realism.

Type I experiment counts and records both matching pairs and opposite pairs?

How is that average correlation calculated, do you know the formula?

 

[humbleteleskop]

Yikes, you arrived at 33% a COMPLETELY different way than I did. Sorry, I didn't look at the detail of your code. I only count trials in which theta is 60 or 120 degrees different. The QM prediction for that is 25%.

If I don't count correlations when P1 and P2 angles are the same, then I get this:

100 measurements 18% - 27%
1000 measurements 20% - 24%
10000 measurements 21% - 22%


New algorithm, changes in bold:

Event T0:
L1= 1
[B]L2= -1[/B]

Event T1:
P1= RNG1(A, B, C)
P2= RNG2(A, B, C)

Event T2:
L1= L1 x P1
L2= L2 x P2

Event T3:
[B]if P1 == P2 goto SKIP:[/B]
if L2 == L1 then CORRELATED++
[B]SKIP:[/B]
N_MEASURE++
RESULT= CORRELATED/(N_MEASURE/100)
if N_MEASURE < N_REPEAT goto Event T0

 

[DrChinese]

How detector knows whether it is +1 or -1? , what does it measure?

How detector knows it's 0, what does it measure, or how it calculates?

Type I experiment counts and records both matching pairs and opposite pairs?

How is that average correlation calculated, do you know the formula?

Type I photons have the same polarization. Type II pairs have orthogonal (perpendicular) polarization. Type has no bearing on whether what you decide to record, that is part of the rest of the setup.

A Polarizing Beam Splitter (PBS) allows horizonally (H) polarized photons straight through, and deflects vertical (V) ones through some angle specific to the PBS. Keep in mind the H and V designations are relative. The PBS can itself be rotated. The important thing is that you can orient the PBS so you can split the input stream in such a way as photons emerge with a known polarization. You can label it as makes sense.

The PBS will have photodetectors at each of the 2 output channels. When it fires, the detector and time is recorded. Obviously, you can consider that an H, V, +, -, 0, 1, -1 or whatever as long as it is consistent. Different people tend to use different notation.

I usually talk about match % because it is easier to discuss and relate to the cos^2 formula. Then the formula is:

Matches/(Matches+NonMatches) where hits that cannot be paired are ignored. In real experiments, unpaired hits are usually noted and discussed.

 

[humbleteleskop]

Type I photons have the same polarization.

Ok, at the beginning photon pairs in Type I experiment always have the same polarization. I'm now only interested in the final part, measurements and data, what is recorded, what is not recorded, what is counted, and what is not counted in Type I experiment.

In Type I experiment we have these two data streams:

Data stream 1
A: -1 +1 -1 -1 +1
B: -1 +1 -1 -1 +1

Data stream 2
A: -1 +1 -1 -1 +1
B: +1 -1 +1 +1 -1

Stream 1:
matches= 5
measurements= 5
result= 5/(5/100)= 100% correlated ?

Stream 2:
matches= 0
measurements= 5
result= 0/(5/100)= 0% correlated ?


Is that how/what is recorded, counted, and calculated in Type I experiment?

 

[humbleteleskop]

The PBS will have photodetectors at each of the 2 output channels. When it fires, the detector and time is recorded. Obviously, you can consider that an H, V, +, -, 0, 1, -1 or whatever as long as it is consistent.

How does photo-detector know when to record -1, and when to record +1? What is it measuring, how can it tell the difference?

 

[Nugatory]

How does photo-detector know when to record -1, and when to record +1? What is it measuring, how can it tell the difference?

The photodetector is measuring the polarization of the photon relative to the photodetector. This will be one of two values, either vertical or horizontal.

When the two photodetectors measure the same value, we count a match.

You have to keep separate counts for the cases in which the detectors are 0, 30, and 60 degrees apart. Bell's inequality is a relationship between the probabilities of a match in each of these three cases.

 

[Nugatory]

Data stream 1
A: -1 +1 -1 -1 +1
B: -1 +1 -1 -1 +1

Data stream 2
A: -1 +1 -1 -1 +1
B: +1 -1 +1 +1 -1

Stream 1:
matches= 5
measurements= 5
result= 5/(5/100)= 100% correlated ?

yes.

Stream 2:
matches= 0
measurements= 5
result= 0/(5/100)= 0% correlated ?

No. That is 100% anti-correlated (or -100% correlated). The correlation is defined as the number of matches minus the number of mismatches, then divided by the total; it ranges from -1 to 1.

A 0% correlation happens when the number of matches is the same as the number of mismatches, which happens when the two streams are random and unrelated to one another.

 

[humbleteleskop]

The photodetector is measuring the polarization of the photon relative to the photodetector.

I know photo-detectors can measure intensity and frequency, but how can it measure polarization of a single photon?

This will be one of two values, either vertical or horizontal.

Are you sure for each recorded data both photons actually arrive at both detectors? Or is maybe non-arrival also recorded and counted in some way?

When the two photodetectors measure the same value, we count a match.

Measure the same value, we count a match, ok.

What do you count if they measure different value, something or nothing?

What do you count if one gets blocked by polarizer, something or nothing?

When does counter of "performed measurements" increase and when not?

You have to keep separate counts for the cases in which the detectors are 0, 30, and 60 degrees apart. Bell's inequality is a relationship between the probabilities of a match in each of these three cases.

Why then not just measure each angle combination separately? What's the point of RNG and shuffling angles, why not make measurements while polarizers have fixed angles until we're done measuring for that particular combination? What's the results when we do that?

 

[humbleteleskop]

No. That is 100% anti-correlated (or -100% correlated).

I understand correlation of mismatches is equally strong and valid as correlation of matching pairs. I am only to sure whether DrChinese said Type I experiments do not actually count mismatches at all, but only matches.

The correlation is defined as the number of matches minus the number of mismatches, then divided by the total; it ranges from -1 to 1.

A: -1 +1 -1 -1 +1 -1 +1 -1 -1 +1
B: -1 +1 -1 -1 +1 +1 -1 +1 +1 -1

match= 5
mismatch= 5
num_data= 10
result= (5-5)/(10/100)= 0% correlated?

Aren't those two sequences A and B actually 100%, or 50% correlated at least?

A 0% correlation happens when the number of matches is the same as the number of mismatches, which happens when the two streams are random and unrelated to one another.

So we have data stream 1 and 2, like this:

Data stream 1
A: -1 +1 -1 -1 +1
B: -1 +1 -1 -1 +1

Data stream 2
A: -1 +1 -1 -1 +1
B: +1 -1 +1 +1 -1

Both data streams are 100% correlated. One is correlated in matching pairs and the other is correlated in mismatching pairs. But when we put them together, we conclude the resulting data stream is 0% correlated. What sense does that make? Shouldn't resulting correlation be 50%, at least?

 

[Nugatory]

I know photo-detectors can measure intensity and frequency, but how can it measure polarization of a single photon?

The paper that DrChinese pointed you at in post 23 of this thread describes one of the ways in which this is done.

Are you sure for each recorded data both photons actually arrive at both detectors? Or is maybe non-arrival also recorded and counted in some way?

Again, there are many ways of handling this. One easy approach that is often used is to connect the electronic outputs of both photodetectors to a computer which stores both streams and also keeps a running total of the number of matches, mismatches, and detections at one detector without a near-simultaneous detection at the other.

Because you're doing a simulation, you get to decide exactly which experimental setup you're going to simulate, so...

Measure the same value, we count a match, ok.
What do you count if they measure different value, something or nothing?
What do you count if one gets blocked by polarizer, something or nothing?
When does counter of "performed measurements" increase and when not?

... until you have the basic principles working right, you should keep it simple. Simulate as if every time an entangled pair is generated, both members of the pair make it to the detectors and are detected, and these are the only photons that are detected. Once you have this working right, you can start adding the complications.

Do this and you only need to count, for each run, the total number of pairs, the number of matches, and the number of mismatches (and actually you only need to count two of these three because the the total is the sum of the matches and mismatches so if you have any two you can calculate the third).

Why then not just measure each angle combination separately? What's the point of RNG and shuffling angles, why not make measurements while polarizers have fixed angles until we're done measuring for that particular combination? What's the results when we do that?

Most often the experiments do use fixed angles, changing them only when we're done measuring for that combination. Selecting the angles at random on the fly is needed only in particularly sophisticated variations that are trying to exclude some particularly arcane possibilities. Either way, the results have agreed with the quantum mechanical predictions.

You will still need a random number generator in your simulation of course... You need it to generate the simulated stream of entangled photons.

 

[humbleteleskop]

I think I found the answers in this article:
http://www.askamathematician.com/20...-god-really-does-play-dice-with-the-universe/


Photo-detectors do not measure any polarization. They either "click" or don't, depending on if their photon went through, or it got blocked by their polarizer. For example, if we take 10 photon pairs, and set angles to 0 and 30 degrees:

P1= 0 -> Malus's law -> 100% ~ 10 out of 10
P2= 30 -> Malus's law -> 75% ~ 7 out of 10

S1: 1 1 1 1 1 1 1 1 1 1
S2: 0 1 0 1 1 1 0 1 1 1

match= 7
mismatch= 3
num_data= 10
result= (7-3)/(10/100)= 40% correlation


Is that "correct" result predicted by QM for delta 30 degrees angle?

 

[Nugatory]

I understand correlation of mismatches is equally strong and valid as correlation of matching pairs. I am only to sure whether DrChinese said Type I experiments do not actually count mismatches at all, but only matches.

The number of matching pairs plus the number of mismatching pairs is equal to the total number of pairs, so you only need to count any two of these three and then you can calculate the third on demand.

A: -1 +1 -1 -1 +1 -1 +1 -1 -1 +1
B: -1 +1 -1 -1 +1 +1 -1 +1 +1 -1

match= 5
mismatch= 5
num_data= 10
result= (5-5)/(10/100)= 0% correlated?

Aren't those two sequences A and B actually 100%, or 50% correlated at least?

No, they are 0% correlated. They are also (as you know, because you carefully constructed them by hand) not random; if something like that showed up in an experimenter's raw data everyone would be looking for a piece of equipment that started malfunctioning after the fifth sample.

So we have data stream 1 and 2, like this:
...
What sense does that make? Shouldn't resulting correlation be 50%, at least?

No, because that's not how the mathematical quantity called "correlation" is defined. (You are free to define some other mathematical quantity that obeys different rules, but if you do, please don't call it "correlation" or we'll be using the same word to mean two different things, which is confusing to all concerned).

And please do remember that the smallest correlation value possible is -1, not zero, so 50% is not halfway in between.

 

[humbleteleskop]

Again, there are many ways of handling this.

Not many. I'm attempting to talk about only one and very specific way, only one specific experiment I refer to as "Type I" since that's how DrChinese seems to call it.

Because you're doing a simulation, you get to decide exactly which experimental setup you're going to simulate, so...
... until you have the basic principles working right, you should keep it simple.

I don't want to decide anything, I want to exactly replicate this "Type I" experiment. Rather than basics, I need specifics.

Most often the experiments do use fixed angles, changing them only when we're done measuring for that combination. Selecting the angles at random on the fly is needed only in particularly sophisticated variations that are trying to exclude some particularly arcane possibilities. Either way, the results have agreed with the quantum mechanical predictions.

Doesn't Malus's law predict the same thing?

 

[Nugatory]

Doesn't Malus's law predict the same thing?

Malus's Law gives the probability that light with a given polarization angle will make it through a polarizer. It says nothing about how the polarization angles of two entangled photons are related.

 

[humbleteleskop]

No, because that's not how the mathematical quantity called "correlation" is defined.

There are many different kinds of "correlation". The term belongs to statistics rather than mathematics. What particular definition are you talking about? Can you give me a link?

And please do remember that the smallest correlation value possible is -1, not zero, so 50% is not halfway in between.

Those are two different things. One is "sequence correlation" expressed as a percentage, the other is "binary correlation" expressed as true or false (match/mismatch).

Malus's Law gives the probability that light with a given polarization angle will make it through a polarizer. It says nothing about how the polarization angles of two entangled photons are related.

You didn't really answer if Malus's law alone would predict the same thing as QM, or not. The only thing we measure, the only thing we count, is just how many photons manage to pass through, the difference in intensity, and that's exactly what Malus's law is for. Is it not?

 

[humbleteleskop]

Malus's law based simulation, sample size 10,000 measurements:

P1= 0
P2= 0
Result: 100%

P1= 0
P2= 180
Result: 100%

P1= 0
P2= 30
Result: 51%

P1= -30
P2= 30
Result: 25%

P1= 30
P2= 60
Result: 24%

P1= 0
P2= 60
Result: 50%

#include <math.h>
#include <time.h>

#define A -30
#define B  0
#define C +30

void main()
{
Init_Setup:;
srand (time(NULL));
int         N_MEASURE= 0;
int         N_REPEAT= 10000;
int         MATCH= 0;
int         MISMATCH= 0;
float       P1= 0.0174533 * -30;
float       P2= 0.0174533 * 30;BEGIN:;
        //Event_T0
        float L1= 0;
        float L2= 0;

        //Event_T2
        L1= (cos(P1) * cos(P1)) * 100;
        L2= (cos(P2) * cos(P2)) * 100;

        printf("\nP1=%.1f L1=%.2f   P2=%.1f L2=%.2f\n", P1, L1, P2, L2);

        if(rand()%100 < L1)
            L1= 1; else L1= 0;

        if(rand()%100 < L2)
            L2= 1; else L2= 0;        //Event_T3
        printf("\n%d: ", N_MEASURE);
//        if (P1 == P2)
            //same angle - don't count correlations

            if(L1 == L2)
            {
                MATCH++;
                printf("SAME");
            }
            else
            {
                MISMATCH++;
                printf("DIFF");
            }

        N_MEASURE++;
        if (N_MEASURE < N_REPEAT) goto BEGIN;

printf("\nRESULT: %d%%", abs(MATCH - MISMATCH)/(N_MEASURE/100));
printf("\n\nPress a key to repeat.");
getch(); goto Init_Setup;
}

/*
P1= 0
P2= 0
Result: 100%

P1= 0
P2= 30
Result: 51%

P1= -30
P2= 30
Result: 25%

P1= 30
P2= 60
Result: 24%

P1= 0
P2= 60
Result: 50%

P1= 0
P2= 120
Result: 49%

P1= 0
P2= 120
Result: 49%

P1= 0
P2= 240
Result: 50%

P1= 120
P2= 240
Result: 23%

P1= 120
P2= 240
Result: 23%

P1= 0
P2= 180
Result: 100%

P1= 45
P2= 45
Result: 0%
*/

 

[DrChinese]

You didn't really answer if Malus's law alone would predict the same thing as QM, or not. The only thing we measure, the only thing we count, is just how many photons manage to pass through, the difference in intensity, and that's exactly what Malus's law is for. Is it not?

No, although there are obvious similarities. For example, in a Type I experiment where the MATCHES are consider, the formula happens to be the same as Malus: cos^2(theta). However, if you measured CORRELATION the formula is different: cos^2(theta)-sin^2(theta). Either way, there is a connection but it is wrong to think of it as the same case. It isn't. One involves a single photon, the other involves a pair of entangled photons.

And an FYI: if you are expecting me to help you make sense of your code model or one of your other incorrect models, I do not plan to assist. You need to understand the standard model first. So far, you efforts in that regard seem minimal. Have you read and understood the references provided?

 

[DrChinese]

I don't want to decide anything, I want to exactly replicate this "Type I" experiment. Rather than basics, I need specifics.

Earlier you said you wanted to keep things simple. Well, the details are complex as in most in modern experiments. You can model more or less, and there are no shortage of specifics in the references. So changing the scope every few post isn't going to help you. Until you understand Bell's conclusion that no classical dataset can match experimental results, the rest is going to be tough slogging. Einstein, Bohr, Heisenberg and all the other greats had all the tools to discover Bell's Theorem too. But they missed it. So my point is: now that we know about Bell, you must use it before moving forward. You may as well be hypothesizing that photons are little turtles (and its "turtles all the way down"). No one can really help you with that.

BTW that article you mentioned is OK, but I would ignore every one of the comments completely. And even the article should be taken with a grain of salt, because it is what is called "interpretation dependent" in the vernacular.

 

[humbleteleskop]

No, although there are obvious similarities. For example, in a Type I experiment where the MATCHES are consider, the formula happens to be the same as Malus: cos^2(theta). However, if you measured CORRELATION the formula is different: cos^2(theta)-sin^2(theta). Either way, there is a connection but it is wrong to think of it as the same case. It isn't. One involves a single photon, the other involves a pair of entangled photons.

I need the formula that relates to the experiment I am doing, I believe we call it "Type I experiment". I now need to calculate QM's predictions, so I can compare that with those results I got. What formula should I use? Is 'theta' angle difference? Do I add absolute values of angles to get 'theta'?

P1= 0
P2= 30
My result= 51%
QM prediction= ??

P1= -30
P2= 30
My result= 25%
QM prediction= ??

P1= -45
P2= 45
My result= 0%
QM prediction= ??

 

[DrChinese]

I need the formula that relates to the experiment I am doing, I believe we call it "Type I experiment". I now need to calculate QM's predictions, so I can compare that with those results I got. What formula should I use? Is 'theta' angle difference? Do I add absolute values of angles to get 'theta'?

P1= 0
P2= 30
My result= 51%
QM prediction= ??

P1= -30
P2= 30
My result= 25%
QM prediction= ??

P1= -45
P2= 45
My result= 0%
QM prediction= ??

The QM match prediction is cos^2(theta) for Type I PDC assuming perfect efficiency. So:

Theta=30, QM=.75
Theta=60, QM=.25
Theta=90, QM=0

 

[humbleteleskop]

The QM match prediction is cos^2(theta) for Type I PDC assuming perfect efficiency. So:

Theta=30, QM=.75
Theta=60, QM=.25
Theta=90, QM=0

Two out of three. And I get 75% for theta=30 if P1=-15 and P2= 15. Isn't that something?

My setup seem to be centered around zero degrees, somehow. It gives correct results but only if angles are spread from 0 to opposite signs. Should all photon pairs in Type I have only vertical polarization?

 

[DrChinese]

My setup seem to be centered around zero degrees, somehow. It gives correct results but only if angles are spread from 0 to opposite signs. Should all photon pairs in Type I have only vertical polarization?

Type I entangled photons are not vertically polarized. There is nothing special about 0 degrees or any other orientation. They do not have a specific predetermined polarization until measured. They only have a relationship with each other. This state does not really map to any classical idea.

If you do not plan to read the references, please let me know. I won't be able to continue if you are not going to do your homework.

 

[humbleteleskop]

Type I entangled photons are not vertically polarized. There is nothing special about 0 degrees or any other orientation. They do not have a specific predetermined polarization until measured. They only have a relationship with each other. This state does not really map to any classical idea.

Do you think you can explain why the algorithm produces correct results? Isn't that supposed to be impossible?

If you do not plan to read the references, please let me know. I won't be able to continue if you are not going to do your homework.

Haven't we already discussed everything? Is that paper you posted about Type I experiment? What "type" is experiment in this article: http://www.askamathematician.com/20...-god-really-does-play-dice-with-the-universe/

 

[humbleteleskop]

Here is what algorithm does, with an example...


P1= -25
P2= 25

Polarizer P1 is set to -25, and P2 to 25 degrees.


L1= (rand()%100 < ((cos(P1) * cos(P1)) * 100)) ? 1:0
L2= (rand()%100 < ((cos(P2) * cos(P2)) * 100)) ? 1:0

Using Malus's law calculate probability of photon L1 passing through polarizer P1, and L2 through P2. If random number between 0 and 100 is less than photon's probability percentage the photon goes through (= 1), otherwise it gets blocked (= 0).


if (L1 == L2) MATCH++ else MISMATCH++
RESULT= (MATCH - MISMATCH)/(N_MEASURE/100))

If both L1 and L2 passed through (1 = 1) or if both got stopped (0 = 0) add one to matching pairs counter, otherwise increase opposite pairs counter. That's all, just like in the experiment. Here is roughly what's happening with 100 photon pairs sample sequences:P1= -25 -> Malus's law -> 82% ~ 82 out of 100
P2= 25 -> Malus's law -> 82% ~ 82 out of 100

0111110011 1111111011 1111100111 1111111111 1111111110 1101111011 0111101110 1111111111 1111111111 1111101111
1111111111 1110111011 1111001111 1111111101 0111111101 0011011111 1011111011 1111110001 1110001111 1101110111

match= 71
mismatch= 29
num_data= 100
Result: (71-29)/(100/100) = 42%

QM: cos^2(50) * 100 = 41.32%Isn't that something? Spooky action at distance, pocket size. Those two are actual random sequences generated by the computer, relative to Malus probability. It is interesting to note "bunching" of those zeros between all the ones. It means that luck comes in streaks, whatever that means. But if there is anything spooky here, that would be it.

 

[humbleteleskop]

Spooky lucky streaks: what is "random" binary sequence?


a.) 1 1 1 1 1 1 1 1 1 1

b.) 0 0 0 0 0 0 0 0 0 0

c.) 1 0 1 0 1 0 1 0 1 0


Maybe total chaos is just another form of perfect order, or whatever it is, in a binary sequences at least, there are not two, but three "perfect" states. Thus random can be anything in between, but not anyone of the three.

Random will try to move away from being like a., and it will try to move away from being like b., but more it tries the closer it gets to c. Random doesn't want to be like c. either, so what does it do, where can it possibly go? A binary sequence can never be more than 50% random.

 

[DrChinese]

Do you think you can explain why the algorithm produces correct results?

Correct? You see a tree here and there and do not realize where you are in the forest.

First, the correct result is: cos^2(theta) for MATCHES. Not matches-mismatches.

Second, theta cannot depend on bunching around some specific angle such as 0, as previously mentioned. These photons are not polarized.

Third, when theta is zero, you should get 1 (also as previously mentioned). Your model produces something more like 75%.

Lastly, you need 3 angles to address Bell and your model doesn't do that. If you run it for the angles I have mentioned, it will fail. But no need to even consider this yet, you have too much prep work to complete first. READ ALL THE REFERENCES.

 

[Nugatory]

Spooky lucky streaks: what is "random" binary sequence?

Google for "random number tests".
Don't expect to gain any insight by thinking about strings of ten or twenty bits; it takes many more than that to observe randomness.

 

[humbleteleskop]

Correct? You see a tree here and there and do not realize where you are in the forest.

First, the correct result is: cos^2(theta) for MATCHES. Not matches-mismatches.

That's vague, what are you trying to say? It does not matter what do you believe how it should be calculated. The only thing that matters is whether simulation result matches actual experiment and QM prediction. And if it matches the experiments, then naturally that must be the correct way.
.

Second, theta cannot depend on bunching around some specific angle such as 0, as previously mentioned. These photons are not polarized.

I didn't say it does, just that it is interesting, kind of spooky.

Third, when theta is zero, you should get 1 (also as previously mentioned). Your model produces something more like 75%.

No, it produces 1, times 100 gives us the percentage of 100%.

P1= 0, P2= 0
Malus_Sim: 1 *100 = 100%

P1= 0, P2= 180
Malus_Sim: 1 *100 = 100%

P1= -90, P2= +90
Malus_Sim: 1 *100 = 100%

Lastly, you need 3 angles to address Bell and your model doesn't do that. If you run it for the angles I have mentioned, it will fail. But no need to even consider this yet, you have too much prep work to complete first.

It works for any angle. Try it and see for yourself. You just have to be careful to input relative, not absolute, angles in the equation.