Exploring Bell's Assumptions and Quantum Theory

[JesseM]

With this setup, but negated result condition (RR or GG), I get:

What do you mean by this? Are you assuming that the wave sent to Alice has the same angle as the wave sent to Bob, rather than the assumption in your earlier experiment that the two waves were offset by 90 degrees? That's what you really should do if you want to satisfy my condition (1) above and ensure that whenever they choose the same angle, then regardless of what angle they both choose, they'll always get the same result (As you say below, I think it also works if the two waves are offset by 90 degrees and the detector threshold is cos^2(45), but it's a lot easier to understand conceptually if we just assume the waves have the same angle).

With cut-off cos(20):
-------- 1782/10000 (0.1782) (82.18% opposite)
-------- 100% Non-detections in subset (RR)
-------- 59.42% Total Non-detections (RR)
-------- Angles: A:0° B:120° C:240°

With cut-off cos(40):
-------- 336/10000 (0.0336) (96.64% opposite)
-------- 100% Non-detections in subset (RR)
-------- 29.49% Total Non-detections (RR)
-------- Angles: A:0° B:120° C:240°

With cut-off cos(45):
-------- 0/10000 (0) (100% opposite)
-------- NaN% Non-detections in subset (RR)
-------- 21.73% Total Non-detections (RR)
-------- Angles: A:0° B:120° C:240°

Note that, with cut-off > cos(45) all non-opposites are non-detections (RR). With cut-off < cos(45), all non-opposites turn into all-detections (GG). Apparently, cos(45) is the boundry where condition 1 is satisfied.

The reason this works is that if they both choose some angle theta, and the angle of Alice's wave is phi while the angle of Bob's wave is phi + 90 (in degrees rather than radians, of course), then at Alice's detector we have the intensity reduced by cos^2(theta - phi) and at Bob's we have it reduced by cos^2([theta - phi] - 90), which is just equal to sin^2(theta - phi). Since the relation sin^2(omega) + cos^2(omega) = 1 holds for arbitrary values of omega, it must hold for omega = (theta - phi), so it must be true that if cos^2(theta - phi) is larger than 0.5 then sin^2(theta - phi) is smaller than 0.5, and vice versa...and of course, cos^2(45) is exactly 0.5! The only problematic case would be if theta - phi = 45, but you could solve this by saying that Alice's detector goes off if the wave's intensity coming out of the polarizer is greater than or equal to 0.5 its intensity before hitting the polarizer, while Bob's detector goes off if the wave's intensity coming out of the polarizer is greater than 0.5 its intensity before hitting the polarizer. This would ensure that they always got opposite results on every trial where they picked the same detector angle regardless of the angle of the two waves (which must be offset by 90 degrees).

However, it really is a lot simpler conceptually to assume both waves have the same angle, and Alice and Bob always get the same result on every trial where they pick the same detector angle. Unless you really prefer the idea of a 90-degree offset between the two waves.

I added a count when you choose "Photon polarization:Same". When Alice!=Bob, subset(RR or GG) and cut-off cos(40), I get:

-------- 2109/10000 (0.2109)

2109 out of 10000 what, exactly?

-------- 65.58% Non-detections in subset (RR)
-------- 32.41% Total Non-detections (RR)

Your notation is still confusing to me, I thought RR meant that they both got a red light, i.e. neither detector went off...wouldn't that mean you have 100% non-detections in this subset? What's the difference between the percentage of "Non-detections" and the percentage of "Total Non-detections"? And why are you looking at subsets based on whether the detector went off, rather than subsets based on what angle was chosen by Alice and Bob? What I was asking for earlier was to look at the results both in the subset of trials where they chose the same angle, and the results in the subset of trials where they chose different angles.

-------- Angle distribution: 11:1114, 12:1109, 13:1155, 21:1158, 22:1099, 23:1075, 31:1071, 32:1086, 33:1133,

What do these numbers represent? What is the significance of the numbers 11, 12, 13, 21, 22, 23, 31, 32, and 33, and the significance of the numbers following each one?

edit: never mind, it just occurred to me that you are probably using 1,2,3 to represent the three possible angles rather than A,B,C, so you're just counting the number of trials with each possible combination of angles.

(A:R, B:R, C:G) (Alice:B, Bob:A) Result:(RR)Same Wave:55.33394317417163° <-
(A:R, B:G, C:R) (Alice:B, Bob:C) Result:(GR)Diff Wave:280.53612197198504°
(A:R, B:G, C:R) (Alice:A, Bob:C) Result:(RR)Same Wave:124.36775910448999° <-
(A:R, B:G, C:G) (Alice:B, Bob:C) Result:(GG)Same Wave:82.01282839063036° <-
(A:R, B:R, C:G) (Alice:B, Bob:A) Result:(RR)Same Wave:59.82230096148803° <-
(A:G, B:R, C:G) (Alice:B, Bob:A) Result:(RG)Diff Wave:21.064768816942056°
(A:G, B:R, C:R) (Alice:A, Bob:C) Result:(GR)Diff Wave:196.49632211000963°
(A:G, B:R, C:R) (Alice:C, Bob:A) Result:(RG)Diff Wave:353.717587748989°
(A:G, B:R, C:G) (Alice:C, Bob:B) Result:(GR)Diff Wave:204.43728181745195°
(A:G, B:R, C:G) (Alice:C, Bob:A) Result:(GG)Same Wave:20.983012712514256° <-
(A:G, B:G, C:R) (Alice:B, Bob:C) Result:(GR)Diff Wave:142.37439956338943°
(A:G, B:R, C:R) (Alice:B, Bob:C) Result:(RR)Same Wave:178.29335014198526° <-
(A:R, B:R, C:G) (Alice:C, Bob:B) Result:(GR)Diff Wave:259.7604226951664°
(A:R, B:G, C:R) (Alice:C, Bob:C) Result:(RR)Same Wave:118.22104001867596°
(A:R, B:R, C:G) (Alice:A, Bob:A) Result:(RR)Same Wave:243.49446988242934°
(A:R, B:G, C:R) (Alice:A, Bob:B) Result:(RG)Diff Wave:309.191668357614°
(A:G, B:G, C:R) (Alice:C, Bob:A) Result:(RG)Diff Wave:323.8647271988439°
(A:G, B:R, C:R) (Alice:A, Bob:A) Result:(GG)Same Wave:13.946472427685528°
(A:R, B:R, C:G) (Alice:C, Bob:C) Result:(GG)Same Wave:234.72520468634272°
(A:R, B:G, C:R) (Alice:C, Bob:C) Result:(RR)Same Wave:313.4941304009434°
(A:G, B:R, C:G) (Alice:C, Bob:A) Result:(GG)Same Wave:31.851310639507386° <-
(A:R, B:R, C:G) (Alice:B, Bob:A) Result:(RR)Same Wave:220.09222626242854° <-
(A:R, B:R, C:G) (Alice:B, Bob:A) Result:(RR)Same Wave:246.62362276103187° <-
(A:R, B:G, C:R) (Alice:C, Bob:B) Result:(RG)Diff Wave:105.67569489556982°
(A:G, B:R, C:R) (Alice:C, Bob:C) Result:(RR)Same Wave:344.86506464034517°
(A:R, B:G, C:G) (Alice:A, Bob:A) Result:(RR)Same Wave:93.73491259844621°
(A:G, B:R, C:R) (Alice:B, Bob:B) Result:(RR)Same Wave:17.372161662275264°
(A:G, B:G, C:R) (Alice:C, Bob:B) Result:(RG)Diff Wave:330.9473623306877°
(A:G, B:R, C:G) (Alice:B, Bob:B) Result:(RR)Same Wave:33.38294181576445°
(A:G, B:R, C:R) (Alice:B, Bob:B) Result:(RR)Same Wave:162.2677212635943°
(A:G, B:R, C:R) (Alice:C, Bob:B) Result:(RR)Same Wave:14.23186093208101° <-
(A:R, B:G, C:R) (Alice:C, Bob:C) Result:(RR)Same Wave:126.43868673509705°
(A:R, B:G, C:R) (Alice:B, Bob:A) Result:(GR)Diff Wave:104.45137908158726°
(A:R, B:G, C:R) (Alice:C, Bob:B) Result:(RG)Diff Wave:119.13226576630872°
(A:R, B:R, C:G) (Alice:A, Bob:A) Result:(RR)Same Wave:73.66093682638146°
(A:G, B:R, C:G) (Alice:A, Bob:B) Result:(GR)Diff Wave:203.67598891160878°
(A:R, B:R, C:G) (Alice:C, Bob:A) Result:(GR)Diff Wave:236.18228462020627°
(A:R, B:R, C:G) (Alice:A, Bob:C) Result:(RG)Diff Wave:254.44418076225924°
(A:R, B:G, C:R) (Alice:C, Bob:A) Result:(RR)Same Wave:317.2556548792309° <-
(A:G, B:G, C:R) (Alice:C, Bob:C) Result:(RR)Same Wave:159.75730971236067°
(A:R, B:G, C:R) (Alice:B, Bob:C) Result:(GR)Diff Wave:135.19957046552892°
(A:G, B:R, C:G) (Alice:A, Bob:C) Result:(GG)Same Wave:38.16741779834976° <-
(A:G, B:G, C:R) (Alice:B, Bob:C) Result:(GR)Diff Wave:147.42837822243075°
(A:G, B:R, C:R) (Alice:A, Bob:C) Result:(GR)Diff Wave:197.3206120463011°
(A:G, B:G, C:R) (Alice:A, Bob:C) Result:(GR)Diff Wave:143.86133740366955°
(A:G, B:G, C:R) (Alice:C, Bob:C) Result:(RR)Same Wave:335.5625220953304°
(A:R, B:R, C:G) (Alice:C, Bob:B) Result:(GR)Diff Wave:72.24531042336667°
(A:G, B:R, C:R) (Alice:A, Bob:B) Result:(GR)Diff Wave:174.84695084948635°
(A:R, B:G, C:R) (Alice:B, Bob:B) Result:(GG)Same Wave:313.9653741499807°
(A:R, B:R, C:G) (Alice:B, Bob:B) Result:(RR)Same Wave:225.45468768130047°
(A:G, B:G, C:R) (Alice:A, Bob:A) Result:(GG)Same Wave:153.10440542797002°
(A:R, B:R, C:G) (Alice:B, Bob:A) Result:(RR)Same Wave:79.03564053287022° <-
(A:G, B:G, C:R) (Alice:A, Bob:C) Result:(GR)Diff Wave:332.278541241043°
(A:R, B:R, C:G) (Alice:B, Bob:B) Result:(RR)Same Wave:51.437506275376464°
(A:R, B:G, C:R) (Alice:B, Bob:B) Result:(GG)Same Wave:131.94170120248234°
(A:R, B:R, C:G) (Alice:C, Bob:A) Result:(GR)Diff Wave:249.933870179784°
(A:G, B:R, C:R) (Alice:B, Bob:A) Result:(RG)Diff Wave:170.61355527100176°
(A:G, B:G, C:R) (Alice:A, Bob:C) Result:(GR)Diff Wave:324.4486209365025°
(A:G, B:R, C:R) (Alice:B, Bob:C) Result:(RR)Same Wave:176.41203368780063° <-
(A:G, B:R, C:G) (Alice:C, Bob:B) Result:(GR)Diff Wave:37.198388580297745°
(A:G, B:G, C:R) (Alice:C, Bob:A) Result:(RG)Diff Wave:339.3091989363497°
(A:R, B:R, C:G) (Alice:A, Bob:A) Result:(RR)Same Wave:252.22393108586778°
(A:G, B:R, C:R) (Alice:A, Bob:A) Result:(GG)Same Wave:1.9980738584536928°
(A:G, B:R, C:G) (Alice:B, Bob:B) Result:(RR)Same Wave:22.312254673238424°
(A:R, B:R, C:G) (Alice:A, Bob:A) Result:(RR)Same Wave:67.85310289343913°
(A:R, B:G, C:R) (Alice:A, Bob:B) Result:(RG)Diff Wave:316.95728140409244°
(A:R, B:R, C:G) (Alice:C, Bob:B) Result:(GR)Diff Wave:49.25931829314021°
(A:R, B:G, C:R) (Alice:A, Bob:C) Result:(RR)Same Wave:306.7749934488405° <-

OK, by my count you have 68 trials here, with 22 of these being ones where Alice and Bob chose the same setting (which is very close to 1/3 of the trials, so that's about what we should expect if they are choosing randomly). If we remove the trials where they chose the same setting and look only at the subset where they chose different settings, we have:

(A:R, B:R, C:G) (Alice:B, Bob:A) Result:(RR)Same Wave:55.33394317417163° <-
(A:R, B:G, C:R) (Alice:B, Bob:C) Result:(GR)Diff Wave:280.53612197198504°
(A:R, B:G, C:R) (Alice:A, Bob:C) Result:(RR)Same Wave:124.36775910448999° <-
(A:R, B:G, C:G) (Alice:B, Bob:C) Result:(GG)Same Wave:82.01282839063036° <-
(A:R, B:R, C:G) (Alice:B, Bob:A) Result:(RR)Same Wave:59.82230096148803° <-
(A:G, B:R, C:G) (Alice:B, Bob:A) Result:(RG)Diff Wave:21.064768816942056°
(A:G, B:R, C:R) (Alice:A, Bob:C) Result:(GR)Diff Wave:196.49632211000963°
(A:G, B:R, C:R) (Alice:C, Bob:A) Result:(RG)Diff Wave:353.717587748989°
(A:G, B:R, C:G) (Alice:C, Bob:B) Result:(GR)Diff Wave:204.43728181745195°
(A:G, B:R, C:G) (Alice:C, Bob:A) Result:(GG)Same Wave:20.983012712514256° <-
(A:G, B:G, C:R) (Alice:B, Bob:C) Result:(GR)Diff Wave:142.37439956338943°
(A:G, B:R, C:R) (Alice:B, Bob:C) Result:(RR)Same Wave:178.29335014198526° <-
(A:R, B:R, C:G) (Alice:C, Bob:B) Result:(GR)Diff Wave:259.7604226951664°
(A:R, B:G, C:R) (Alice:A, Bob:B) Result:(RG)Diff Wave:309.191668357614°
(A:G, B:G, C:R) (Alice:C, Bob:A) Result:(RG)Diff Wave:323.8647271988439°
(A:G, B:R, C:G) (Alice:C, Bob:A) Result:(GG)Same Wave:31.851310639507386° <-
(A:R, B:R, C:G) (Alice:B, Bob:A) Result:(RR)Same Wave:220.09222626242854° <-
(A:R, B:R, C:G) (Alice:B, Bob:A) Result:(RR)Same Wave:246.62362276103187° <-
(A:R, B:G, C:R) (Alice:C, Bob:B) Result:(RG)Diff Wave:105.67569489556982°
(A:G, B:G, C:R) (Alice:C, Bob:B) Result:(RG)Diff Wave:330.9473623306877°
(A:G, B:R, C:R) (Alice:C, Bob:B) Result:(RR)Same Wave:14.23186093208101° <-
(A:R, B:G, C:R) (Alice:B, Bob:A) Result:(GR)Diff Wave:104.45137908158726°
(A:R, B:G, C:R) (Alice:C, Bob:B) Result:(RG)Diff Wave:119.13226576630872°
(A:G, B:R, C:G) (Alice:A, Bob:B) Result:(GR)Diff Wave:203.67598891160878°
(A:R, B:R, C:G) (Alice:C, Bob:A) Result:(GR)Diff Wave:236.18228462020627°
(A:R, B:R, C:G) (Alice:A, Bob:C) Result:(RG)Diff Wave:254.44418076225924°
(A:R, B:G, C:R) (Alice:C, Bob:A) Result:(RR)Same Wave:317.2556548792309° <-
(A:R, B:G, C:R) (Alice:B, Bob:C) Result:(GR)Diff Wave:135.19957046552892°
(A:G, B:R, C:G) (Alice:A, Bob:C) Result:(GG)Same Wave:38.16741779834976° <-
(A:G, B:G, C:R) (Alice:B, Bob:C) Result:(GR)Diff Wave:147.42837822243075°
(A:G, B:R, C:R) (Alice:A, Bob:C) Result:(GR)Diff Wave:197.3206120463011°
(A:G, B:G, C:R) (Alice:A, Bob:C) Result:(GR)Diff Wave:143.86133740366955°
(A:R, B:R, C:G) (Alice:C, Bob:B) Result:(GR)Diff Wave:72.24531042336667°
(A:G, B:R, C:R) (Alice:A, Bob:B) Result:(GR)Diff Wave:174.84695084948635°
(A:R, B:R, C:G) (Alice:B, Bob:A) Result:(RR)Same Wave:79.03564053287022° <-
(A:G, B:G, C:R) (Alice:A, Bob:C) Result:(GR)Diff Wave:332.278541241043°
(A:R, B:R, C:G) (Alice:C, Bob:A) Result:(GR)Diff Wave:249.933870179784°
(A:G, B:R, C:R) (Alice:B, Bob:A) Result:(RG)Diff Wave:170.61355527100176°
(A:G, B:G, C:R) (Alice:A, Bob:C) Result:(GR)Diff Wave:324.4486209365025°
(A:G, B:R, C:R) (Alice:B, Bob:C) Result:(RR)Same Wave:176.41203368780063° <-
(A:G, B:R, C:G) (Alice:C, Bob:B) Result:(GR)Diff Wave:37.198388580297745°
(A:G, B:G, C:R) (Alice:C, Bob:A) Result:(RG)Diff Wave:339.3091989363497°
(A:G, B:R, C:G) (Alice:B, Bob:B) Result:(RR)Same Wave:22.312254673238424°
(A:R, B:G, C:R) (Alice:A, Bob:B) Result:(RG)Diff Wave:316.95728140409244°
(A:R, B:R, C:G) (Alice:C, Bob:B) Result:(GR)Diff Wave:49.25931829314021°
(A:R, B:G, C:R) (Alice:A, Bob:C) Result:(RR)Same Wave:306.7749934488405° <-

Of these 46, there were 16 trials where they got the same result--that's about 0.35 of all the trials in this subset, which is greater than 1/3. Of course you can only be confident about whether or not a given data set is violating a Bell inequality when you look at a much larger number of trials, but from the data you've given so far the indication is that it isn't. So what I would suggest is doing something like the above for a larger run, this time having two counters that ignore trials such as (Alice:A, Bob:A) where they both chose the same setting, and on trials where they didn't choose the same setting the counter for "diff detector settings" is increased by one (so you're counting the total number of trials where Alice and Bob chose different settings), and the counter for "diff detector settings, same result" is increased by one only if the result was "Same" rather than "Diff" (counting the total number of trials where Alice and Bob chose different settings AND got the same result). After a large number of trials you can compare the two counters--if your program is working correctly, you should find that the value of the second counter is 1/3 or more the value of the first counter.

Meanwhile, I would really appreciate it if you would look over my proof of the Bell inequality, and tell me if you can see any reason why it shouldn't apply to your program if it's working correctly.

 

[Hydr0matic]

First of all, after your last post I realized I'd made quite a big mistake. Instead of counting the trials with results equal or opposite in a subset of angles same or different, I counted trials where both conditions matched in the total set.

To your questions...
Yes, RR means red red, no detection at alice or bob.
The total non-detections are removed since it wasn't relevant.
I replaced the 11, 12.. with AA, AB.. respectively.

But provided we assume that on every trial the source sends them objects with identical states (whether homogeneous or inhomogeneous), it is not possible for the probability of getting the same result when they choose different detector settings to be less than 1/3; that's all the Bell inequality here is saying.

I have looked through your proof carefully and there is indeed no reason why it shouldn't apply. I've also tested it, and I got the following results:

With photon polarization Same and cut-off cos(20), when Alice and Bob chose same angle:

-------- RESULT: 3364/3364 (1)
-------- Non-detections (RR):2618/3364(77.82%)
-------- Angle distribution: BB:1185, AB:1044, AC:1140, BC:1105, AA:1096, CB:1104, CC:1083, BA:1150, CA:1093,
-------- Angles: A:0° B:120° C:240°
-------- Pairs in subset: 3364/10000

ResultCounts
Alice: (same as Bob)
RRR:3290
RGR:2255
GRR:2220
RRG:2235

When they chose the same angle I get a 100% correlation regardless of the cut-off value. The cut-off only affects the number of non-detections.

When they chose different angles, I get:

-------- RESULT: 3744/6636 (0.5641952983725136)
-------- Non-detections (RR):3744/6636 (56.42%)
-------- Angle distribution: BC:1110, AA:1077, CA:1092, BB:1149, AB:1036, CB:1115, CC:1138, BA:1179, AC:1104,
-------- Angles: A:0° B:120° C:240°
-------- Pairs in subset: 6636/10000

ResultCounts
Alice: (same as Bob)
GRR:2213
RRG:2187
RGR:2200
RRR:3400

Now, clearly this is NOT a violation of Bell's inequality. HOWEVER, as you can see, 100% of the result are non-detections. In fact, unless the cut-off value is less than cos(30), ALL correlations are non-detections. Now, I've read a couple of times that one of the main objections to the Bell tests is the low detector efficiency. So I would assume that in real tests, detectors cut-off value would range within cos(0) and cos(30). In which case, whenever they checked results at different angles, they were all non-detections.
I would imagine that this is the reason why they are using sources that emit photons with opposite polarization instead of the same.

However, it really is a lot simpler conceptually to assume both waves have the same angle, and Alice and Bob always get the same result on every trial where they pick the same detector angle. Unless you really prefer the idea of a 90-degree offset between the two waves.

Judging from the results above, it would probably be more appropriate to use sources with opposite polarization.

So, if we change the polarization in my application to Opposite, and do the above tests again. First counting opposite results when Alice and Bob chose the same angle:

cut-off°
10 --- RESULT: 728/3295 (0.220) --- Non-detections (RR):2567/3295 (728+2567=3295)
20 --- RESULT: 1487/3367 (0.441) --- Non-detections (RR):1880/3367 (1487+1880=3367)
30 --- RESULT: 2214/3310 (0.668) --- Non-detections (RR):1096/3310 (2214+1096=3310)
40 --- RESULT: 2926/3288 (0.889) --- Non-detections (RR):362/3288 (2926+362=3288)
45 --- RESULT: 3269/3269 (1.000) --- Non-detections (RR):0/3269
50 --- RESULT: 2991/3389 (0.882) --- Non-detections (RR):0/3389 (2991=3389-398) All RRs turn into GGs (>45)
60 --- RESULT: 2200/3338 (0.659) --- Non-detections (RR):0/3338 (2200=3338-1138)

Analysing these results we see that condition 1 is only truly fulfilled with cut-off value cos(45). However, when the cut-off angle is less than 45° all non-opposite results are non-detections. So, whenever Alice or Bob detects something, the other one always get opposite result.

Counting opposite results when Alice and Bob chose different angles:

cut-off°
5 ----- RESULT: 759/6712 (0.113) --- Non-detections (RR):5953/6712 (759+5953=6712)
10 --- RESULT: 1450/6710 (0.216) --- Non-detections (RR):5260/6710 (1450+5260=6710)
15 --- RESULT: 2264/6676 (0.339) --- Non-detections (RR):4412/6676 (2264+4412=6676)
20 --- RESULT: 2287/6664 (0.343)---- Non-detections (RR):3984/6664 (2287+3984=6664-393) (393 GGs)
30 --- RESULT: 2199/6732 (0.326) --- Non-detections (RR):3412/6732 (2199+3412=6732-1121)
40 --- RESULT: 2184/6669 (0.327) --- Non-detections (RR):2619/6669 (2184+2619=6669-1866)
45 --- RESULT: 2273/6720 (0.338) --- Non-detections (RR):2277/6720 (2273+2277=6720-2170)
50 --- RESULT: 2250/6637 (0.339) --- Non-detections (RR):1803/6637 (2250+1803=6637-2584)
60 --- RESULT: 2241/6646 (0.337) --- Non-detections (RR):1077/6646 (2241+1077=6646-3328)
70 --- RESULT: 2284/6675 (0.342) --- Non-detections (RR):385/6675 (2284+385=6675-4006)
75 --- RESULT: 2217/6519 (0.340) --- Non-detections (RR):0/6519 (2217+0=6675-4458)
80 --- RESULT: 1487/6745 (0.220) --- Non-detections (RR):0/6745 (1487+0=6745-5258)
85 --- RESULT: 743/6661 (0.111) --- Non-detections (RR):0/6661 (743+0=6661-5918)

First of all, notice that cut-offs 5,10,80 & 85 give results that violate Bell's inequality(*). More tests reveal that results from cut-off values higher than cos(15) or lower than cos(75) violate Bell's(*).
Second, notice that for all other cut-off values, the result is approximately 1/3. So the classical lightwave model predicts that, when Alice and Bob measure different angles of pairs with opposite polarization, they will get opposite results with probability 1/3 OR LESS, regardless of cut-off value.

* Since it's only cut-off cos(45) that truly satisfies condition 1, one cannot say that cut-offs at 5,10,80 & 85 violate Bell's inequality.

Examining cut-off values with other ABC angles, for example 0°,45° and 90°, one finds that the boundry for violation of Bell's* is changed to 22.5°. i.e the boundry is half of the difference between the measured angles. (Or equal to, if you consider the whole range -22.5° <> 22.5°)

Here is a complete result with 100 trials from the above test at cut-off cos(10):
Lines marked with <- at the end are part of the subset.

-------- RESULT: 1552/6613 (0.23468924845002267)
-------- Non-detections (RR):5061/6613(76.53%)
-------- Angle distribution: AA:1138, CA:1102, AB:1118, BA:1102, AC:1058, CC:1119, CB:1092, BC:1141, BB:1130,
-------- Angles: A:0° B:120° C:240°
-------- Pairs in subset: 6613/10000

ResultCounts
Alice:
RRG:1079
RRR:6759
RGR:1094
GRR:1068

Bob:
RRR:6598
RGR:1137
GRR:1140
RRG:1125


Alice: Bob:
(A:R, B:R, C:G) (A:R, B:R, C:R) (Alice:A, Bob:A) Result:(RR)Same Wave1:234.1378800941786° Wave2: 324.1378800941786°
(A:R, B:R, C:R) (A:R, B:G, C:R) (Alice:C, Bob:A) Result:(RR)Same Wave1:35.72264460666506° Wave2: 125.72264460666506° <-
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:A, Bob:B) Result:(RR)Same Wave1:42.20000826206056° Wave2: 132.20000826206055° <-
(A:R, B:G, C:R) (A:R, B:R, C:R) (Alice:A, Bob:B) Result:(RR)Same Wave1:123.5108071256557° Wave2: 213.5108071256557° <-
(A:R, B:G, C:R) (A:R, B:R, C:R) (Alice:B, Bob:A) Result:(GR)Diff Wave1:123.5276037607258° Wave2: 213.5276037607258° <-
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:A, Bob:A) Result:(RR)Same Wave1:160.73075126416177° Wave2: 250.73075126416177°
(A:R, B:G, C:R) (A:R, B:R, C:R) (Alice:A, Bob:C) Result:(RR)Same Wave1:118.53166896417368° Wave2: 208.53166896417366° <-
(A:R, B:R, C:R) (A:G, B:R, C:R) (Alice:C, Bob:C) Result:(RR)Same Wave1:267.1078710578241° Wave2: 357.1078710578241°
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:B, Bob:A) Result:(RR)Same Wave1:225.4604050311087° Wave2: 315.46040503110873° <-
(A:R, B:R, C:R) (A:R, B:G, C:R) (Alice:B, Bob:A) Result:(RR)Same Wave1:37.572428656288245° Wave2: 127.57242865628825° <-
(A:R, B:G, C:R) (A:R, B:R, C:R) (Alice:C, Bob:A) Result:(RR)Same Wave1:291.2279827790366° Wave2: 21.22798277903661° <-
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:C, Bob:C) Result:(RR)Same Wave1:71.79740547206602° Wave2: 161.79740547206603°
(A:R, B:G, C:R) (A:R, B:R, C:R) (Alice:C, Bob:B) Result:(RR)Same Wave1:293.993078538492° Wave2: 23.993078538492° <-
(A:G, B:R, C:R) (A:R, B:R, C:R) (Alice:B, Bob:C) Result:(RR)Same Wave1:3.710932337109001° Wave2: 93.710932337109° <-
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:A, Bob:B) Result:(RR)Same Wave1:192.85550819474088° Wave2: 282.8555081947409° <-
(A:R, B:R, C:R) (A:R, B:R, C:G) (Alice:C, Bob:C) Result:(RG)Diff Wave1:155.10432541233536° Wave2: 245.10432541233536°
(A:R, B:R, C:G) (A:R, B:R, C:R) (Alice:B, Bob:C) Result:(RR)Same Wave1:53.92845630720172° Wave2: 143.9284563072017° <-
(A:R, B:R, C:R) (A:R, B:R, C:G) (Alice:A, Bob:B) Result:(RR)Same Wave1:321.7942596025368° Wave2: 51.79425960253678° <-
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:B, Bob:A) Result:(RR)Same Wave1:340.74542582860863° Wave2: 70.74542582860863° <-
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:B, Bob:B) Result:(RR)Same Wave1:346.45056727244116° Wave2: 76.45056727244116°
(A:R, B:G, C:R) (A:R, B:R, C:R) (Alice:B, Bob:B) Result:(GR)Diff Wave1:301.45060740427596° Wave2: 31.450607404275956°
(A:R, B:G, C:R) (A:R, B:R, C:R) (Alice:B, Bob:A) Result:(GR)Diff Wave1:292.3347606939048° Wave2: 22.334760693904798° <-
(A:G, B:R, C:R) (A:R, B:R, C:R) (Alice:A, Bob:A) Result:(GR)Diff Wave1:3.3515104072192603° Wave2: 93.35151040721927°
(A:R, B:R, C:R) (A:G, B:R, C:R) (Alice:C, Bob:A) Result:(RG)Diff Wave1:91.84621678526328° Wave2: 181.84621678526327° <-
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:C, Bob:C) Result:(RR)Same Wave1:41.90960393497796° Wave2: 131.90960393497795°
(A:R, B:R, C:R) (A:G, B:R, C:R) (Alice:B, Bob:A) Result:(RG)Diff Wave1:269.8585535605929° Wave2: 359.8585535605929° <-
(A:R, B:R, C:R) (A:G, B:R, C:R) (Alice:A, Bob:A) Result:(RG)Diff Wave1:277.70946204703966° Wave2: 7.709462047039665°
(A:G, B:R, C:R) (A:R, B:R, C:R) (Alice:B, Bob:A) Result:(RR)Same Wave1:353.13632396738507° Wave2: 83.13632396738507° <-
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:A, Bob:B) Result:(RR)Same Wave1:319.85682382439944° Wave2: 49.85682382439944° <-
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:C, Bob:C) Result:(RR)Same Wave1:194.31557633767088° Wave2: 284.3155763376709°
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:C, Bob:A) Result:(RR)Same Wave1:192.96231128480113° Wave2: 282.9623112848011° <-
(A:R, B:G, C:R) (A:R, B:R, C:R) (Alice:A, Bob:A) Result:(RR)Same Wave1:294.5512712344834° Wave2: 24.55127123448341°
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:A, Bob:A) Result:(RR)Same Wave1:109.59643598677565° Wave2: 199.59643598677565°
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:B, Bob:A) Result:(RR)Same Wave1:17.39269320572545° Wave2: 107.39269320572545° <-
(A:G, B:R, C:R) (A:R, B:R, C:R) (Alice:A, Bob:B) Result:(GR)Diff Wave1:359.95549344857426° Wave2: 89.95549344857426° <-
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:C, Bob:B) Result:(RR)Same Wave1:314.0349337884509° Wave2: 44.03493378845093° <-
(A:G, B:R, C:R) (A:R, B:R, C:R) (Alice:B, Bob:A) Result:(RR)Same Wave1:181.4324188574362° Wave2: 271.43241885743623° <-
(A:R, B:R, C:R) (A:R, B:G, C:R) (Alice:B, Bob:C) Result:(RR)Same Wave1:214.97368166443803° Wave2: 304.973681664438° <-
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:C, Bob:B) Result:(RR)Same Wave1:165.69429826353536° Wave2: 255.69429826353536° <-
(A:R, B:R, C:R) (A:R, B:R, C:G) (Alice:A, Bob:B) Result:(RR)Same Wave1:145.301710799591° Wave2: 235.301710799591° <-
(A:R, B:R, C:G) (A:R, B:R, C:R) (Alice:A, Bob:C) Result:(RR)Same Wave1:239.45457214043003° Wave2: 329.45457214043006° <-
(A:G, B:R, C:R) (A:R, B:R, C:R) (Alice:A, Bob:C) Result:(GR)Diff Wave1:352.3414943341509° Wave2: 82.34149433415092° <-
(A:R, B:R, C:R) (A:R, B:G, C:R) (Alice:A, Bob:C) Result:(RR)Same Wave1:203.99552547397013° Wave2: 293.99552547397013° <-
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:A, Bob:B) Result:(RR)Same Wave1:348.79981944898356° Wave2: 78.79981944898356° <-
(A:R, B:R, C:R) (A:R, B:G, C:R) (Alice:A, Bob:A) Result:(RR)Same Wave1:201.12117214272135° Wave2: 291.1211721427213°
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:B, Bob:A) Result:(RR)Same Wave1:47.82000276013689° Wave2: 137.8200027601369° <-
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:C, Bob:A) Result:(RR)Same Wave1:19.95420944676516° Wave2: 109.95420944676516° <-
(A:R, B:G, C:R) (A:R, B:R, C:R) (Alice:C, Bob:B) Result:(RR)Same Wave1:125.7349113218338° Wave2: 215.7349113218338° <-
(A:R, B:R, C:R) (A:R, B:R, C:G) (Alice:A, Bob:B) Result:(RR)Same Wave1:337.0598713583604° Wave2: 67.0598713583604° <-
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:A, Bob:C) Result:(RR)Same Wave1:223.4035984747488° Wave2: 313.40359847474883° <-
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:B, Bob:B) Result:(RR)Same Wave1:165.88525275241784° Wave2: 255.88525275241784°
(A:R, B:R, C:R) (A:G, B:R, C:R) (Alice:B, Bob:C) Result:(RR)Same Wave1:269.0346627988359° Wave2: 359.0346627988359° <-
(A:R, B:G, C:R) (A:R, B:R, C:R) (Alice:B, Bob:C) Result:(GR)Diff Wave1:111.42251244920784° Wave2: 201.42251244920783° <-
(A:R, B:R, C:R) (A:R, B:R, C:G) (Alice:C, Bob:B) Result:(RR)Same Wave1:157.5249478861851° Wave2: 247.5249478861851° <-
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:B, Bob:C) Result:(RR)Same Wave1:340.1889934539382° Wave2: 70.18899345393822° <-
(A:R, B:R, C:G) (A:R, B:R, C:R) (Alice:B, Bob:C) Result:(RR)Same Wave1:248.61150797931379° Wave2: 338.6115079793138° <-
(A:G, B:R, C:R) (A:R, B:R, C:R) (Alice:A, Bob:C) Result:(GR)Diff Wave1:358.79360815426685° Wave2: 88.79360815426685° <-
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:B, Bob:B) Result:(RR)Same Wave1:165.4625205759179° Wave2: 255.4625205759179°
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:B, Bob:A) Result:(RR)Same Wave1:48.12877458474593° Wave2: 138.12877458474594° <-
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:B, Bob:B) Result:(RR)Same Wave1:315.5072208466679° Wave2: 45.507220846667906°
(A:R, B:R, C:R) (A:G, B:R, C:R) (Alice:C, Bob:A) Result:(RG)Diff Wave1:263.06639095605215° Wave2: 353.06639095605215° <-
(A:R, B:R, C:R) (A:R, B:R, C:G) (Alice:C, Bob:A) Result:(RR)Same Wave1:326.99124381048114° Wave2: 56.99124381048114° <-
(A:R, B:R, C:G) (A:R, B:R, C:R) (Alice:C, Bob:C) Result:(GR)Diff Wave1:64.79753534354637° Wave2: 154.79753534354637°
(A:R, B:G, C:R) (A:R, B:R, C:R) (Alice:C, Bob:C) Result:(RR)Same Wave1:300.47969331972365° Wave2: 30.47969331972365°
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:C, Bob:C) Result:(RR)Same Wave1:226.86417866602403° Wave2: 316.86417866602403°
(A:R, B:R, C:G) (A:R, B:R, C:R) (Alice:C, Bob:C) Result:(GR)Diff Wave1:238.73102312436706° Wave2: 328.7310231243671°
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:B, Bob:A) Result:(RR)Same Wave1:75.58966665762914° Wave2: 165.58966665762915° <-
(A:R, B:R, C:R) (A:G, B:R, C:R) (Alice:A, Bob:B) Result:(RR)Same Wave1:277.31875612414126° Wave2: 7.3187561241412595° <-
(A:R, B:R, C:R) (A:R, B:R, C:G) (Alice:A, Bob:B) Result:(RR)Same Wave1:336.9991552178912° Wave2: 66.9991552178912° <-
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:C, Bob:A) Result:(RR)Same Wave1:10.24960171915506° Wave2: 100.24960171915507° <-
(A:R, B:G, C:R) (A:R, B:R, C:R) (Alice:C, Bob:C) Result:(RR)Same Wave1:298.2832574224113° Wave2: 28.283257422411282°
(A:R, B:R, C:R) (A:G, B:R, C:R) (Alice:A, Bob:A) Result:(RG)Diff Wave1:99.6544700072415° Wave2: 189.6544700072415°
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:A, Bob:A) Result:(RR)Same Wave1:222.12437116324668° Wave2: 312.1243711632467°
(A:R, B:R, C:G) (A:R, B:R, C:R) (Alice:C, Bob:B) Result:(GR)Diff Wave1:248.3276771483694° Wave2: 338.3276771483694° <-
(A:G, B:R, C:R) (A:R, B:R, C:R) (Alice:C, Bob:B) Result:(RR)Same Wave1:173.37716366170937° Wave2: 263.37716366170935° <-
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:B, Bob:C) Result:(RR)Same Wave1:282.2478201393047° Wave2: 12.247820139304679° <-
(A:R, B:R, C:R) (A:G, B:R, C:R) (Alice:A, Bob:B) Result:(RR)Same Wave1:263.1492525837448° Wave2: 353.1492525837448° <-
(A:R, B:R, C:R) (A:R, B:G, C:R) (Alice:A, Bob:A) Result:(RR)Same Wave1:22.8958136578825° Wave2: 112.8958136578825°
(A:R, B:R, C:G) (A:R, B:R, C:R) (Alice:A, Bob:A) Result:(RR)Same Wave1:65.61605792399588° Wave2: 155.61605792399587°
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:A, Bob:A) Result:(RR)Same Wave1:223.73056283271416° Wave2: 313.73056283271416°
(A:R, B:G, C:R) (A:R, B:R, C:R) (Alice:B, Bob:C) Result:(GR)Diff Wave1:115.15263077987316° Wave2: 205.15263077987316° <-
(A:R, B:R, C:R) (A:R, B:G, C:R) (Alice:C, Bob:B) Result:(RG)Diff Wave1:21.004246593556807° Wave2: 111.00424659355681° <-
(A:R, B:R, C:R) (A:R, B:R, C:G) (Alice:C, Bob:A) Result:(RR)Same Wave1:149.6761176325398° Wave2: 239.6761176325398° <-
(A:R, B:R, C:R) (A:R, B:R, C:G) (Alice:B, Bob:A) Result:(RR)Same Wave1:157.48701199037342° Wave2: 247.48701199037342° <-
(A:R, B:R, C:R) (A:R, B:G, C:R) (Alice:A, Bob:C) Result:(RR)Same Wave1:211.44143224704325° Wave2: 301.4414322470433° <-
(A:R, B:R, C:R) (A:R, B:R, C:G) (Alice:B, Bob:C) Result:(RG)Diff Wave1:151.7881553779931° Wave2: 241.7881553779931° <-
(A:R, B:R, C:R) (A:R, B:R, C:G) (Alice:B, Bob:A) Result:(RR)Same Wave1:336.77253467714985° Wave2: 66.77253467714985° <-
(A:R, B:R, C:R) (A:R, B:R, C:G) (Alice:C, Bob:B) Result:(RR)Same Wave1:150.84780853014178° Wave2: 240.84780853014178° <-
(A:R, B:R, C:G) (A:R, B:R, C:R) (Alice:A, Bob:B) Result:(RR)Same Wave1:66.06742015546814° Wave2: 156.06742015546814° <-
(A:R, B:G, C:R) (A:R, B:R, C:R) (Alice:B, Bob:A) Result:(GR)Diff Wave1:125.74272701948371° Wave2: 215.7427270194837° <-
(A:R, B:R, C:G) (A:R, B:R, C:R) (Alice:C, Bob:C) Result:(GR)Diff Wave1:236.48965566643° Wave2: 326.48965566643°
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:A, Bob:B) Result:(RR)Same Wave1:109.9916145019468° Wave2: 199.9916145019468° <-
(A:R, B:R, C:R) (A:R, B:G, C:R) (Alice:B, Bob:C) Result:(RR)Same Wave1:210.52822909816123° Wave2: 300.5282290981612° <-
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:B, Bob:B) Result:(RR)Same Wave1:315.72372067606165° Wave2: 45.723720676061646°
(A:R, B:R, C:R) (A:R, B:R, C:G) (Alice:B, Bob:B) Result:(RR)Same Wave1:339.4231793713175° Wave2: 69.42317937131747°
(A:R, B:R, C:R) (A:R, B:G, C:R) (Alice:B, Bob:B) Result:(RG)Diff Wave1:208.92772526638552° Wave2: 298.92772526638555°
(A:R, B:G, C:R) (A:R, B:R, C:R) (Alice:C, Bob:C) Result:(RR)Same Wave1:293.54305481361735° Wave2: 23.543054813617346°
(A:R, B:R, C:R) (A:G, B:R, C:R) (Alice:A, Bob:C) Result:(RR)Same Wave1:274.94079044002507° Wave2: 4.940790440025069° <-
(A:R, B:R, C:R) (A:G, B:R, C:R) (Alice:B, Bob:B) Result:(RR)Same Wave1:91.26311868162189° Wave2: 181.2631186816219°
(A:R, B:R, C:R) (A:R, B:R, C:R) (Alice:B, Bob:A) Result:(RR)Same Wave1:165.65882919660118° Wave2: 255.65882919660118° <-

 

[JesseM]

-------- RESULT: 3744/6636 (0.5641952983725136)
-------- Non-detections (RR):3744/6636 (56.42%)
-------- Angle distribution: BC:1110, AA:1077, CA:1092, BB:1149, AB:1036, CB:1115, CC:1138, BA:1179, AC:1104,
-------- Angles: A:0° B:120° C:240°
-------- Pairs in subset: 6636/10000

ResultCounts
Alice: (same as Bob)
GRR:2213
RRG:2187
RGR:2200
RRR:3400

Now, clearly this is NOT a violation of Bell's inequality. HOWEVER, as you can see, 100% of the result are non-detections.

I don't understand, what part of the listed results am I supposed to see that in? Does the 6636/10000 you got for "pairs in subset" refer to the subset of trials where Alice and Bob chose different angles? If so, then if they got RR on 3744/6636, doesn't this mean that 56.42% of the results in this subset are non-detections? Or does "non-detections" mean something different from RR? Maybe you just mean that at least one of them got R, so that on 100% of the trials in the subset where they chose different angles, their results were either RR or RG or GR, but never GG? Even if that's it, I don't understand how I could see this from what you wrote.

In fact, unless the cut-off value is less than cos(30), ALL correlations are non-detections.

You mean, in the subset of trials where they choose different angles, all correlations are non-detections? This makes sense, since if they choose different angles their two detectors are either 120 or 240 apart...if the wave's angle is between 0 and 30 away from Alice's angle, then it's got be in one of four possible ranges away from Bob's: 120-150, 120-90, 240-270, or 240-210. In the ranges 120-90 and 240-270 there's no way the cos^2 of Bob's angle could be as large as cos^2(30), and in the ranges 120-150 or 240-210, the cos^2 of Bob's angle would only reach cos^2(30) if his angle was exactly 150 or 210 away from the wave's angle...and if you're choosing the wave's angle randomly, the chances of this are infinitesimal.

Now, I've read a couple of times that one of the main objections to the Bell tests is the low detector efficiency. So I would assume that in real tests, detectors cut-off value would range within cos(0) and cos(30). In which case, whenever they checked results at different angles, they were all non-detections.

But in real tests, the detectors don't work the way they do in your classical model...if the polarization is random, each detector should ideally get R 50% of the time and G 50% of the time, but in QM the probability of the two detectors getting the same result on a given trial is simply equal to the cos^2 of the angle between the two detectors. So if Alice gets G when hers is set to 120, and Bob's is set to 240 on that trial, there should be a cos^2(240-120) = cos^2(120) = 0.25 probability that Bob also gets G.

And I think "detector efficiency" in QM doesn't work like in your model where there's a cutoff, I would guess it just uniformly drops the probability that a given photon gets through...for example if the detector efficiency were 0.2, that would mean even if their detectors were aligned, if one got G there would only have an 0.2 probability of also getting G, while if their detectors were 120 degrees apart, if one got G the other would only have an 0.25*0.2 = 0.05 probability of also getting G.

So, if we change the polarization in my application to Opposite, and do the above tests again. First counting opposite results when Alice and Bob chose the same angle:

cut-off°
10 --- RESULT: 728/3295 (0.220) --- Non-detections (RR):2567/3295 (728+2567=3295)
20 --- RESULT: 1487/3367 (0.441) --- Non-detections (RR):1880/3367 (1487+1880=3367)
30 --- RESULT: 2214/3310 (0.668) --- Non-detections (RR):1096/3310 (2214+1096=3310)
40 --- RESULT: 2926/3288 (0.889) --- Non-detections (RR):362/3288 (2926+362=3288)
45 --- RESULT: 3269/3269 (1.000) --- Non-detections (RR):0/3269
50 --- RESULT: 2991/3389 (0.882) --- Non-detections (RR):0/3389 (2991=3389-398) All RRs turn into GGs (>45)
60 --- RESULT: 2200/3338 (0.659) --- Non-detections (RR):0/3338 (2200=3338-1138)

Analysing these results we see that condition 1 is only truly fulfilled with cut-off value cos(45). However, when the cut-off angle is less than 45° all non-opposite results are non-detections. So, whenever Alice or Bob detects something, the other one always get opposite result.

Counting opposite results when Alice and Bob chose different angles:

cut-off°
5 ----- RESULT: 759/6712 (0.113) --- Non-detections (RR):5953/6712 (759+5953=6712)
10 --- RESULT: 1450/6710 (0.216) --- Non-detections (RR):5260/6710 (1450+5260=6710)
15 --- RESULT: 2264/6676 (0.339) --- Non-detections (RR):4412/6676 (2264+4412=6676)
20 --- RESULT: 2287/6664 (0.343)---- Non-detections (RR):3984/6664 (2287+3984=6664-393) (393 GGs)
30 --- RESULT: 2199/6732 (0.326) --- Non-detections (RR):3412/6732 (2199+3412=6732-1121)
40 --- RESULT: 2184/6669 (0.327) --- Non-detections (RR):2619/6669 (2184+2619=6669-1866)
45 --- RESULT: 2273/6720 (0.338) --- Non-detections (RR):2277/6720 (2273+2277=6720-2170)
50 --- RESULT: 2250/6637 (0.339) --- Non-detections (RR):1803/6637 (2250+1803=6637-2584)
60 --- RESULT: 2241/6646 (0.337) --- Non-detections (RR):1077/6646 (2241+1077=6646-3328)
70 --- RESULT: 2284/6675 (0.342) --- Non-detections (RR):385/6675 (2284+385=6675-4006)
75 --- RESULT: 2217/6519 (0.340) --- Non-detections (RR):0/6519 (2217+0=6675-4458)
80 --- RESULT: 1487/6745 (0.220) --- Non-detections (RR):0/6745 (1487+0=6745-5258)
85 --- RESULT: 743/6661 (0.111) --- Non-detections (RR):0/6661 (743+0=6661-5918)

First of all, notice that cut-offs 5,10,80 & 85 give results that violate Bell's inequality(*). More tests reveal that results from cut-off values higher than cos(15) or lower than cos(75) violate Bell's(*).
Second, notice that for all other cut-off values, the result is approximately 1/3. So the classical lightwave model predicts that, when Alice and Bob measure different angles of pairs with opposite polarization, they will get opposite results with probability 1/3 OR LESS, regardless of cut-off value.

* Since it's only cut-off cos(45) that truly satisfies condition 1, one cannot say that cut-offs at 5,10,80 & 85 violate Bell's inequality.

Yes, and notice that when you make the cutoff cos^2(45) (not cos(45) as you say...I know what you meant, but the difference is important!), the fraction of trials where they get the same result is closer to 1/3 than for any other angle. You had said earlier that in the case where the two waves were 90 degrees apart, then the predetermined states of the waves were never (A:R, B:R, C:R), or (A:G, B:G, C:G)...the predetermined states were always "inhomogeneous". And as I showed in the proof, for inhomogeneous predetermined states, if Alice and Bob choose their angles randomly, then on the subset of trials where they choose different angles, the probability of getting opposite results should be exactly 1/3 (assuming that when they pick the same angle they always get opposite results). So I think if you did an even larger number of trials, the fraction in the cutoff = cos^2(45) case would tend to converge towards 0.33333...

 

[DrChinese]

I think if you did an even larger number of trials, the fraction in the cutoff = cos^2(45) case would tend to converge towards 0.33333...

That's true of course, if the underlying algorithm is correct - which I question anyway. The real issue is that there are no samples of A, B and C - regardless of how you decide to seed it - that meet the criteria that:

a) Alice and Bob always get the same result when AA, BB, or CC are selected; and
b) The sample converges on any number less than .333 when different combos are selected; and
c) Non-detections are random.

The last item is the "fair-sampling hypothesis" (JesseM is already well aware of this). This is an experiemental issue that deserves a separate discussion. Suffice it to say that no evidence of a violation of the "fair sampling hypothesis" has ever been demonstrated.

However, there are a few people who believe it that "fair sampling" does NOT occur, and that explains the overall violation of Bell's Inequality. The problem with that concept is that it points to a different (and very strange) relationship than the observed cos^2 rule. The other big problem is that as samples get relatively larger due to improvements in technology, the results converge ever more closely on the Quantum predictions.

 

[Hydr0matic]

I don't understand, what part of the listed results am I supposed to see that in? Does the 6636/10000 you got for "pairs in subset" refer to the subset of trials where Alice and Bob chose different angles? If so, then if they got RR on 3744/6636, doesn't this mean that 56.42% of the results in this subset are non-detections? Or does "non-detections" mean something different from RR? Maybe you just mean that at least one of them got R, so that on 100% of the trials in the subset where they chose different angles, their results were either RR or RG or GR, but never GG? Even if that's it, I don't understand how I could see this from what you wrote.

6636 is the subset where Alice and Bob chose different angles, yes. According to Bell, given condition 1, when we look at the results where they chose different angles, they would get the Same result at least 1/3 of the trials. So the number 3744 represents the trials (in the subset) where they got the Same result. Which were 56.42% of the trials (more than 1/3), which is not a violation.
Non-detection means RR, yes. And since we're only counting RRs and GGs (Same result), if I get 3744 RR's then ALL of the counts must be RRs.

You mean, in the subset of trials where they choose different angles, all correlations are non-detections? This makes sense, since if they choose different angles their two detectors are either 120 or 240 apart...if the wave's angle is between 0 and 30 away from Alice's angle, then it's got be in one of four possible ranges away from Bob's: 120-150, 120-90, 240-270, or 240-210. In the ranges 120-90 and 240-270 there's no way the cos^2 of Bob's angle could be as large as cos^2(30), and in the ranges 120-150 or 240-210, the cos^2 of Bob's angle would only reach cos^2(30) if his angle was exactly 150 or 210 away from the wave's angle...and if you're choosing the wave's angle randomly, the chances of this are infinitesimal.

Exactly.

But in real tests, the detectors don't work the way they do in your classical model...if the polarization is random, each detector should ideally get R 50% of the time and G 50% of the time, but in QM the probability of the two detectors getting the same result on a given trial is simply equal to the cos^2 of the angle between the two detectors. So if Alice gets G when hers is set to 120, and Bob's is set to 240 on that trial, there should be a cos^2(240-120) = cos^2(120) = 0.25 probability that Bob also gets G.

Ok, this is what bothers me. How can they assume how the polarizers work? Is there any experiment made that proves polarizers are governed by chance? Maybe I've missed it. How can physicists be more open to tossing out locality, than to reconsider assumptions about how polarizers work?

Yes, and notice that when you make the cutoff cos^2(45) (not cos(45) as you say...I know what you meant, but the difference is important!)

The choice of cos and not cos² seems more reasonable to me. Intensity is a concept relating to lots of light. I mean, what's the intensity of one photon?
If you use the absolute value of cos there's no difference anyway (cos²(a) > cos²(b) => |cos(a)| > |cos(b)|, same for =, <). But maybe I'm wrong.

You had said earlier that in the case where the two waves were 90 degrees apart, then the predetermined states of the waves were never (A:R, B:R, C:R), or (A:G, B:G, C:G)...the predetermined states were always "inhomogeneous".

For a specific case, yes. This is completely dependant on the cut-off value. A high cut-off cos(20) gives only (RRR, GRR, RGR, RRG), a cut-off cos(45) gives all states except (RRR, GGG), and cut-off cos(70) gives cos(20) inversed (GGG, RGG, GRG, GGR).

And as I showed in the proof, for inhomogeneous predetermined states, if Alice and Bob choose their angles randomly, then on the subset of trials where they choose different angles, the probability of getting opposite results should be exactly 1/3 (assuming that when they pick the same angle they always get opposite results). So I think if you did an even larger number of trials, the fraction in the cutoff = cos^2(45) case would tend to converge towards 0.33333...

You're absolutely right. From 100,000 generated pairs I got a subset of 66557, out of which 22309 were opposite results (0.33518...). It indeed converges towards 1/3.

Simple question: Does my model emulate actual results or not? Not with fair sampling, subtraction of accidentals and other stuff. All the data. Because unlike the other models, this one actually predicts that there will be lots of non-detections.

So does it emulate the results or not? If not, why?

If you think there's something wrong in my application, what tests can I run to further test it?

 

[DrChinese]

Originally Posted by JesseM
But in real tests, the detectors don't work the way they do in your classical model...if the polarization is random, each detector should ideally get R 50% of the time and G 50% of the time, but in QM the probability of the two detectors getting the same result on a given trial is simply equal to the cos^2 of the angle between the two detectors. So if Alice gets G when hers is set to 120, and Bob's is set to 240 on that trial, there should be a cos^2(240-120) = cos^2(120) = 0.25 probability that Bob also gets G.

Hydromatic:
Ok, this is what bothers me. How can they assume how the polarizers work? Is there any experiment made that proves polarizers are governed by chance? Maybe I've missed it. How can physicists be more open to tossing out locality, than to reconsider assumptions about how polarizers work?

So does it emulate the results or not? If not, why?

If you think there's something wrong in my application, what tests can I run to further test it?

You are hung up on your algorithm, and it does NOT - as JesseM points out - match known experiments. The EPR Paradox STARTS with the fact when measuring the same attribute for Alice and Bob (i.e. same angle setting), that knowing the results for Alice implies what the results will be for Bob. Under local realism, the implication is that the results were predetermined. Since your formula does not do this, you are wrong before you get any further. You must change your formula so that Alice and Bob's results are perfectly correlated when measuring the same angle. Further, to match experiment, they must be R 50% of the time and G 50% of the time which I think your formula does properly.

Next comes the difficult part. You must have an algorithm that works independently on Alice and Bob. And still gives detections on both sides that ALWAYS have a detection probability not only that is independent of the other side's setting - it must also be independent of the polarization of the photon itself! This is in direct contradiction to your formula! It has been known for over 200 years (Malus, 1807) that the correct formula for detection is cos^2 theta when there is a polarizer. When it is a beamsplitter instead of a polarizer, it goes to either one or the other detectors. SO THERE IS NO CUTOFF! (sorry for shouting, but this should have been obvious to all of us earlier).

In real experiments, these elements are checked carefully when the alignments are performed to insure that the results make sense. PDC crystals actually generate pairs with known polarization when the polarization of the input pump is known. So your purported cut-off effect is actually checked during the setup, and it does not exist. A beam-splitter is used, and it is rotated to determine optimal alignments as photons pass through the entire apparati. There is no cut-off, but Malus' Law is seen in how the splitting occurs.

You also wonder if assumptions have been made about how a polarizer works? Hmmm. As mentioned, beam splitters are used which either transmit or deflect the incident beam. We are simply trying to predict results, and those predictions work according to Malus' law. And that is simply a classical requirement which happens to exactly match quantum predictions. You will want to follow that too. I personally would not characterize matching theory and experiment as "assumption". More of a requirement.

So my point is: NO, your results do not match reality. I hope you can see that your formula needs rework. Because of Bell's Theorem, you cannot have a local realistic theory that gives results identical to quantum theory. So your mission is quite impossible, since this was your original objective.

By the way, there are theorists who cling to the idea that such a formula IS possible. Emilio Santos comes to mind. He has pursued stochastic solutions (that is the kind of formula you are developing) and is convinced one is possible. Although he does get published, I can't think of anyone (outside of perhaps his co-authors) who holds out hope for such a solution in face of Bell's Theorem and subsequent tests.

 

[Hydr0matic]

DrChinese, I appreciate you joining the discussion, but I would prefer if you ask me questions - like JesseM - if there's something I'm not clear about. Shouting about "no cutoff" when you've misunderstood the meaning is obviously not very constructive.

Clarifications:

1. I am NOT trying to emulate Local Realism. My idea is nothing like Local realism. I think the idea of hidden variables is very silly. I believe that particles and waves have defined properties (polarization, position..), but the notion of predetermened outcomes of a random process, lightspeed communication between photons and so forth, is indeed very silly.

2. The cut-off value we are discussing refers to the detector, not the polarizer. The polarizers indeed follow Malus' law. The cut-off value comes into play when you replace the idea of a photon with the idea of an electromagnetic wave. Because, if you consider light as a wave, there is no limit to how small the amplitude can be. It can be infinitesimal. However, there IS a limit to what we can detect. We don't have detectors that can register infinitely weak lightwaves. There is a lightwave amplitude cut-off value, below which we cannot detect any lightwaves.

3. I don't know how your programming skills are, but it seems like you're having issues interpreting my code (which I admit is quite messy. Too lazy to fix it). My application computes random angles A,B or C for Alice and Bob independantly. It generates a photon with random polarization, then assigns either same or opposite polarization to the other photon. It then calculates - independantly for Alice and Bob - whether the relative angle between photon and polarizer is small enough to output a detectable lightwave.

You are more than welcome to continue the discussion and/or ask me questions if something needs further/better explaining...

 

[JesseM]

Sorry, forgot to answer this earlier...

But in real tests, the detectors don't work the way they do in your classical model...if the polarization is random, each detector should ideally get R 50% of the time and G 50% of the time, but in QM the probability of the two detectors getting the same result on a given trial is simply equal to the cos^2 of the angle between the two detectors. So if Alice gets G when hers is set to 120, and Bob's is set to 240 on that trial, there should be a cos^2(240-120) = cos^2(120) = 0.25 probability that Bob also gets G.

Ok, this is what bothers me. How can they assume how the polarizers work? Is there any experiment made that proves polarizers are governed by chance? Maybe I've missed it. How can physicists be more open to tossing out locality, than to reconsider assumptions about how polarizers work?

I don't understand what you mean by "how polarizers work" or "governed by chance". My statement above refers only to statistics--if we do a large number of trials, we will find that at each detector we get R on about 50% of trials and G on about 50%, but if we look at the subset of trials where the difference between detector angles was 120 degrees we'll find that only on about 25% of the trials do they get the same results (or opposite results, if we're doing an experiment where they always get opposite results when they choose the same detector angle). This is just a statement about observed results as predicted by QM, nothing about what is "really" going on with the particles or whether their behavior involves any true randomness (either at the detectors or at the moment the pair is created). The use of probability to describe the statistics just reflects our own practical lack of ability to predict the behavior in any deterministic way, it doesn't mean there couldn't be some underlying deterministic rule that explains what happens on any trial (similarly, in classical statistical mechanics you can still talk about probabilities for things like an increase or decrease in entropy, based on your ignorance of the precise microstate of a large system--the precise position and momentum of every particle making it up--even though it's assumed that given the exact microstate its behavior would be perfectly predictable). The point here is just that no local realist rule, including yours, can reproduce these statistics.

Yes, and notice that when you make the cutoff cos^2(45) (not cos(45) as you say...I know what you meant, but the difference is important!)

The choice of cos and not cos² seems more reasonable to me. Intensity is a concept relating to lots of light. I mean, what's the intensity of one photon?
If you use the absolute value of cos there's no difference anyway (cos²(a) > cos²(b) => |cos(a)| > |cos(b)|, same for =, <). But maybe I'm wrong.

But I thought you weren't talking about photons in your experiment, you were talking about a classical experiment where we sent classical waves and the detectors only went off if the intensity was above some threshold. Anyway, intensity is a concept related to the energy of light, so if we're using light of a uniform frequency like a laser, than all the photons should have the same energy, therefore intensity in a given region is directly proportional to the number of photons that hit that region over many trials (which is proportional to the probability that a single photon will hit that region in a single trial).

Simple question: Does my model emulate actual results or not? Not with fair sampling, subtraction of accidentals and other stuff. All the data. Because unlike the other models, this one actually predicts that there will be lots of non-detections.

Keep in mind that one can do experiments where "red" does not represent a non-detection like a photon failing to make it through a polarizer, but instead red and green represent two different outcomes for a particle that is measured, like an electron either being deflected up (spin-up) or down (spin-down) by a magnetic field. In this case at least, I assume one could just throw out all the cases where one detector registered an electron but the other didn't. In the case of photons, I'm not actually sure how they make sure that they are actually measuring members of an entangled pair rather than just random background photons, and I don't know if the "red" result here would just correspond to one detector failing to go off or if it would mean something else, basically I don't know that much about the experimental side of these Bell tests. Maybe someone else can answer this question.

 

[DrChinese]

DrChinese, I appreciate you joining the discussion, but I would prefer if you ask me questions - like JesseM - if there's something I'm not clear about. Shouting about "no cutoff" when you've misunderstood the meaning is obviously not very constructive.

Clarifications:

1. I am NOT trying to emulate Local Realism. My idea is nothing like Local realism. I think the idea of hidden variables is very silly. I believe that particles and waves have defined properties (polarization, position..), but the notion of predetermened outcomes of a random process, lightspeed communication between photons and so forth, is indeed very silly.

2. The cut-off value we are discussing refers to the detector, not the polarizer. The polarizers indeed follow Malus' law. The cut-off value comes into play when you replace the idea of a photon with the idea of an electromagnetic wave. Because, if you consider light as a wave, there is no limit to how small the amplitude can be. It can be infinitesimal. However, there IS a limit to what we can detect. We don't have detectors that can register infinitely weak lightwaves. There is a lightwave amplitude cut-off value, below which we cannot detect any lightwaves.

3. I don't know how your programming skills are, but it seems like you're having issues interpreting my code (which I admit is quite messy. Too lazy to fix it). My application computes random angles A,B or C for Alice and Bob independantly. It generates a photon with random polarization, then assigns either same or opposite polarization to the other photon. It then calculates - independantly for Alice and Bob - whether the relative angle between photon and polarizer is small enough to output a detectable lightwave.

You are more than welcome to continue the discussion and/or ask me questions if something needs further/better explaining...

As said, sorry for the shouting. Also, I am a software developer (actually manager), so I think I can follow OK.

The issue is what you are saying about amplitudes and detection. This really can't apply here. Specifically, you say "whether the relative angle between photon and polarizer is small enough to output a detectable lightwave". This is demonstrably false. We know that there is no angular difference at which detection probability is affected.

You say that you believe that particles have defined properties. That is the basis for realism. You also say that communication between photons is silly, which is equivalent to saying that you believe in locality. That sounds like local realism, which is what Bell says is inconsistent with quantum mechanics.

As I said earlier, you cannot produce a formula via computer program that reproduces all of the predictions of quantum mechanics with random selection of settings for Alice and Bob. Your results will always need to depend on the relative settings for Alice and Bob, and all the other factors will essentially cancel out.

 

[Hydr0matic]

Apologies for the ridiculously late reply.

I don't understand what you mean by "how polarizers work" or "governed by chance".

This is just a statement about observed results as predicted by QM, nothing about what is "really" going on with the particles or whether their behavior involves any true randomness (either at the detectors or at the moment the pair is created). The use of probability to describe the statistics just reflects our own practical lack of ability to predict the behavior in any deterministic way, it doesn't mean there couldn't be some underlying deterministic rule that explains what happens on any trial

I'm actually not debating underlying determinism or whether there is true randomness or not. I'm saying there isn't any randomness at all. This is the key difference. In both QM and Local realism the polarizers are goverened by chance. By this I mean that, when a photon hits the polarizer there is a cos²(θ) probability that it goes through. This is not the case in classical physics. In classical physics ALL photons (lightwaves) go through, but they do so with an amplitude A = A₀ cos(θ).

Let's take an example.
Say we have a light-source that emits all photons with the same polarization. A polarizer is set with a 45° angle difference to the photon polarization. Now, in QM, all photons will hit the polarizer at a 45° difference, and therefore each photon has a cos²(45°) = 50% chance of passing through. In classical physics on the other hand, all photons (lightwaves) will pass through with an amplitude A = A₀ cos(45°) of the emitted one. IF this new amplitude A is below the detectors cut-off value, NO photons are detected.

As you can see, this is not just a matter of interpretation or underlying mechanisms. Although the theories give equivalent results for lots of light, in single-photon setups they do not.

But I thought you weren't talking about photons in your experiment, you were talking about a classical experiment where we sent classical waves and the detectors only went off if the intensity was above some threshold. Anyway, intensity is a concept related to the energy of light, so ../ /.. (which is proportional to the probability that a single photon will hit that region in a single trial).

True, my setup is completely classical. So it doesn't matter if I compare amplitude or intensity, they give the same results. To me at least, amplitude makes more sense.

basically I don't know that much about the experimental side of these Bell tests. Maybe someone else can answer this question.

Neither do I. Would appreciate that very much.

The issue is what you are saying about amplitudes and detection. This really can't apply here. Specifically, you say "whether the relative angle between photon and polarizer is small enough to output a detectable lightwave". This is demonstrably false. We know that there is no angular difference at which detection probability is affected.

Ofcourse there is. If a photon has polarization ϕ and the polarizer is set to angle θ, the probability of detection is cos²(ϕ-θ). (in QM & LR)

You say that you believe that particles have defined properties. That is the basis for realism. You also say that communication between photons is silly, which is equivalent to saying that you believe in locality. That sounds like local realism, which is what Bell says is inconsistent with quantum mechanics.

Hehe, true, I believe in realism and locality. The only difference from Local Realism, and also the key difference, is how the polarizers behave [see above]. If you remove the randomness from the polarizers and view the polarization from a classical wave perspective, I believe the experimental results will make perfect sense in a local realists world.

 

[DrChinese]

1. Ofcourse there is. If a photon has polarization ϕ and the polarizer is set to angle θ, the probability of detection is cos²(ϕ-θ). (in QM & LR)

2. Hehe, true, I believe in realism and locality. The only difference from Local Realism, and also the key difference, is how the polarizers behave [see above]. If you remove the randomness from the polarizers and view the polarization from a classical wave perspective, I believe the experimental results will make perfect sense in a local realists world.

1. The probability of passing through the polarizer is per Malus' Law. But the likelihood of being detected by the detector itself is not affected. Usually, Bell tests use polarizing beamsplitters that cause the vertical and hortizontal components to be split. There are 2 detectors, and the photon is seen at one or the other. In your formulas, it looks as if you are using something you call "cut-off" to bias your results. This is part of what I am objecting to.

But the other part is the idea that the photon had a definite polarization prior to being observed. This is one of the assumptions that Bell challenges.

2. The experimental results do NOT make sense from a classical perspective, this has already been explained many times to you and others. A classical analysis yields a different formula altogether - namely Match=(cos^2(theta)/2)+.25, if you assume Malus (i.e. the classical) as a fundamental starting point - which is what you assert. That formula has a range of .25 to .75, while the quantum formula has a range of 0 to 1. Of course, we observe the 0 to 1 range.

-DrC