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In a recent thread, I tried to show the error in Bell/CHSH type inequalities and their alleged violation by QM and experiments. Some have been unable to understand the argument. I will attempt again to illustrate the issue. We will use an analogy of coin-tossing which maintain all the key features of the Bell-CHSH discussion. To my knowledge, nobody has presented the argument in this manner before. I will present the argument in two parts.

Feel free to ask clarifying questions as we go along. If there is interest, I may even discuss exactly how all the loopholes fit into this analogy, and why they are irrelevant, including super-determinism, and show exactly the flaw in Gill's most recent paper.

Part 1: The Bell argument

Assumptions:
1. Coins are local realistic
2. Coins have only two possible outcomes H = +1, T=-1

Derivation of the equality (cf inequality):
- Let A be the outcome we get when we toss a single coin.
- We assume that the other outcome we could have gotten also exists, even though we did not get it, call it B.
- Tossing a single coin produces either H or T and not both. Possible outcomes for AB are (HT, TH)
- Therefore A + B = 0, and E(A) + E(B) = 0 for all local realistic coins.

QM predictions:
- QM predicts E(A) = E(B) = 0.25.

Bell's theorem:
- Since from QM E(A) + E(B) = 0.5 =/= 0, it means QM is not local realistic.

Aspect-type experiments:
- We need to test experimentally whether QM or local realism is correct. Unfortunately we can only read one outcome at a time from a single coin. However, if we toss two coins, we can still obtain accurate estimates of E(A) and E(B) in the form of <A> and <B>, where A is the outcome we get from the first coin, and <B> is the outcome we get from the second coin. We should get similar results because the two coins are drawn from the same population.
- After the experiment we observe <A> = <B> = 0.25, <A> + <B> = 0.5, exactly what QM predicted.
- Therefore QM is correct and local realism is wrong.

The example illustrate nothing about Bell's inequality. Its like a study of a single point as an illustration of the triangle inequality.

Bell and CHSH are an illusion and talking about them is a waste of time. There must be no talk any more of inequalities or violations.

Michel is mired down in semantic issues.

Let's talk about physics.

Suppose we want to do an experiment to distinguish between two theories.

Q1. Is that possible, yes or no?

Michel thinks it is impossible in the situation that either theory could lead to any data. eg four independently measured correlations can always be anything between -1 and +1 so it is, according to him, a waste of time even to try to measure them. (Why he keeps writing programs to simulate them is a mystery to me. Those programs prove nothing since the only thing anyone can say is that each correlation is between -1 and +1. I knew that in advance.)

If however we disagree with Michel that experiment is senseless, we should ask ourselves:

Q2. In what sense can experiment distinguish between theories?

We usually don't get certain proof from an experiment that some theory is true or false. Usually we only get statistical evidence pointing more or less strongly to one or the other.

Q3. If we can only hope for statistical proof, then what is the best experiment to do?

Take a look at

http://rpubs.com/gill1109/Wireframe

How can we best choose between two different theoretical correlation surfaces E(a, b), when we only have limited experimental resources?

I am still waiting for Michel to contribute to the experiment of the century by helping prepare the computer software which we'll use to evaluate the result. It's very important that this is done by different people with different opinions. People of strong independent mind, and people of intelligence. People with expertise in computing.

Part 2: The unraveling
--------------------------
The argument I have made many times is that the reason for the violation is due to an erroneous substitution of actual results from two separate systems, into an expression derived using actual & counter-factual results from a single system. It sounds so simple but why has this been difficult to understand so far? This discussion between Heinera and Richard in a related thread illustrates why:

Heinera wrote:So if the original correlations (all computed on the whole set) didn't violate the CHSH inequality (CHSH<2), and the correlations computed on four disjoint random subset would not change much, we can now conclude that the four latter correlations would still not significantly violate the CHSH inequality, since term by term, they are approximately equal to the original correlations?

gill1109 wrote:Yes Heinera, you are home. They might violate it a little, but in all probability they won't violate it by much

Let us translate these innocent looking argument to our coin toss system:

If we measure <A> on a fair sample, of the population, we should get almost the same result as the population.
If we measure <B> on a fair sample, of the population, we should get almost the same result as the population.
Just because we measured <A> on one fair sample, and <B> on a different fair sample, does not change the fact that E(A) + E(B) = 0 for the population. You might violate it a little due to experimental error, but it shouldn't matter whether we use a single population or two disjoint fair samples of the population. Then the expectation values from the two coins should still not significantly violate the expression we obtained from the single coin

So what is wrong with this argument:
-------------------------------------------
First, let us present a counter-example which shoes that the QM prediction is fully consistent with local-realism:

- We have a local realistic coin with a (H,T) probability distribution of [0.625,0.375]. E(A) = E(B) = 0.625(1)+0.375(-1) = 0.25, E(A) + E(B) = 0.5, fully agreeing with the experiments and QM.
- What about our local-realistic relationship E(A) + E(B) = 0 which we derived. Is it still valid for our local-realistic coins? Yes of course, provide we throw only single coins. Let us throw the coins on a glass table and read A from above, and B from below, and create a spreadsheet. Each row of the spreadsheet will necessarily sum to 0, and therefore <A> + <B> = 0. Agreeing with E(A) + E(B) = 0.
- How can the same local-realistic coins produce <A> + <B> = 0 in one experiment and <A> + <B> = 0.5, in another experiment? Because values calculated on two coins are not the same as values calculated on a single coin.

- Further illustration of the problem:
-------------------------------------------
Most people do not understand that counter-factual results do not have the same correlations as actual results. Take my coins for example, if the probability of H is 0.625, then the counter-factual probability for H can not also be 0.625. To illustrate even further, the actual result is what we get, the counter-factual result is what we did not get but could have gotten. If the likelihood of getting H is 0.625, the of course the likelihood of not getting H is 1-0.625 = 0.375. So the counter-factual probability is (1 - actual probability).

Failure to understand this elementary point is at the root of Bells theorem.

In the CHSH expression, S = E(a, b) − E(a, b′) + E(a′, b) + E(a′ b′), only one of the expectation values is actual, and the others are counterfactual. Bell assumed erroneously that each of those correlations should get the same functional form as the actual ones, not realising that because they are counterfactual, we must flip the probability distributions.

If the actual E(a,b) = -cos(a-b)**2, then the counterfactual E(a,b) must be flipped to 1 - cos(a-b)**2.

Do you guys get it now?

Minkwe wrote:Do you guys get it now?

No, because I nowhere make any erroneous substitution. So everything you say is pretty irrelevant. Anyway, I am interested in an experiment which has nothing to do with CHSH but which will either give Joy and the experimenter the Nobel prize, or all three of us the igNobel prize. And I'm looking for people who will help that experiment to happen.

gill1109 wrote:

Minkwe wrote:Do you guys get it now?

No, because I nowhere make any erroneous substitution. So everything you say is pretty irrelevant. Anyway, I am interested in an experiment which has nothing to do with CHSH but which will either give Joy and the experimenter the Nobel prize, or all three of us the igNobel prize. And I'm looking for people who will help that experiment to happen.

Richard,
What part of it do you not understand. Point it out and I will explain. If you are not interested in the content of the thread, do not participate in it. Stay on topic.

minkwe wrote:Richard,
What part of it do you not understand. Point it out and I will explain.

I understand it all, perfectly well, thank you.

minkwe wrote:The argument I have made many times is that the reason for the violation is due to an erroneous substitution of actual results from two separate systems

There is no erroneous substitition of actual results from two separate systems. Nowhere. This has been explained to you a hundred times by many different people. Seems no-one can get it across.

I offered to explain to you the mistakes in Adenaur, de Raedt and Rosinger's papers, but only if you might deign to waste half an hour trying a little experiment of mine.

If you want to learn about their mistakes, which are similar to yours, you had better try that experiment. Because I am hoping that when you do it, and see the results, the scales may fall from your eyes. If you are not open to new experiences you probably will not learn much, ever. If you want to stay blind, well just keep on posing the same question till the sheep come home, or till Kingdom come, or hell freezes over.

In the meantime I am learning Python so we will be able to run Joy's experiment without your help after all, thank you. We already have perl and Excel and R versions. Others will come soon. Mathematica is close to being ready.

gill1109 wrote:

minkwe wrote:The argument I have made many times is that the reason for the violation is due to an erroneous substitution of actual results from two separate systems, into an expression derived using actual & counter-factual results from a single system.

There is no erroneous substitution of actual results from two separate systems.

Seriously Richard?

Astounding!
Do you want me to show you where it is done in your papers and in the literature by ALL experimentalists who have done Bell-type experiments?

If you had a legitimate argument you would have stated it. You've had plenty of opportunity. Yet for some reason, your claims are unable to be stated in English, they can only be experienced by running a silly piece of R-code. Unconvincing. Hopefully, somebody else who has wasted their time to run the code will help you to express your point in English, and explain why my argument in this thread is wrong.

minkwe wrote:

gill1109 wrote:

minkwe wrote:The argument I have made many times is that the reason for the violation is due to an erroneous substitution of actual results from two separate systems, into an expression derived using actual & counter-factual results from a single system.

There is no erroneous substitution of actual results from two separate systems.

Seriously Richard?

Astounding!
Do you want me to show you where it is done in your papers and in the literature by ALL experimentalists who have done Bell-type experiments?

Seriously. Regarding my papers, you made some feeble attempts and failed miserably. You never even got as far as finding out what the main mathematical results actually are! It was quite incredible.

As to the literature of all experimentalists, I am afraid that they are often pretty fuzzy about the logic of what they are doing. They just see the chance of getting famous by doing a difficult experiment (it is difficult, you know!) which some theoretician has said is important. They don't understand the theoretician's reasoning.

I'm afraid the literature is a big big mess. A lot of people are very confused, including, I'm afraid to say, many famous experimentalists. I can show you some truly incredible nonsense, if you are interested. But only after you have done my experiment. It might teach you something useful, something you don't seem to know about yet.

gill1109 wrote:As to the literature of all experimentalists, I am afraid that they are often pretty fuzzy about the logic of what they are doing. They just see the chance of getting famous by doing a difficult experiment (it is difficult, you know!) which some theoretician has said is important. They don't understand the theoretician's reasoning.

I'm afraid the literature is a big big mess. A lot of people are very confused, including, I'm afraid to say, many famous experimentalists. I can show you some truly incredible nonsense, if you are interested. But only after you have done my experiment. It might teach you something useful, something you don't seem to know about yet.

Richard, I'm talking about the same experimentalists you have cited in your papers, whose claims you have repeated in your papers, and with whom you have jointly written papers. Are you now throwing them under the bus? You did not say if you wanted me to show you were you did infact make the substitution in your papers. Do you want me to show it, or are you ready to retract the claim that nobody is doing that.

You still haven't mentioned where my argument above is wrong. Can it not be expressed in English?

minkwe wrote:

gill1109 wrote:As to the literature of all experimentalists, I am afraid that they are often pretty fuzzy about the logic of what they are doing. They just see the chance of getting famous by doing a difficult experiment (it is difficult, you know!) which some theoretician has said is important. They don't understand the theoretician's reasoning.

I'm afraid the literature is a big big mess. A lot of people are very confused, including, I'm afraid to say, many famous experimentalists. I can show you some truly incredible nonsense, if you are interested. But only after you have done my experiment. It might teach you something useful, something you don't seem to know about yet.

Richard, I'm talking about the same experimentalists you have cited in your papers, whose claims you have repeated in your papers, and with whom you have jointly written papers. Are you now throwing them under the bus? You did not say if you wanted me to show you were you did infact make the substitution in your papers. Do you want me to show it, or are you ready to retract the claim that nobody is doing that.

You still haven't mentioned where my argument above is wrong. Can it not be expressed in English?

I am not answering any more questions from you till you have (a) done my experiment, (b) translated the code for determing the bet outcome into Python.

There is a lot of work to do. Because of your lack of cooperation I have to learn Python. Oh well...

I have some busy days ahead. You can email me when you have done your homework.

Good night (or good day), and if I don't hear from you again, fare well! Peace!

I am not answering any more questions from you till you have (a) done my experiment, (b) translated the code for determine the bet outcome into Python.

That'll be your loss then. I guess that means you are retracting the claim that nobody is substituting actual results for counter-factual results. Since the claim is obviously false.

There is a lot of work to do. Because of your lack of cooperation I have to learn Python. Oh well...

You do not need Python for your bet. If you want to learn python, that is fine for you but don't deceive yourself that you need it for the bet. You don't.

Michel, I have switched to this new thread for my reply.

I have worked through Case 1 and 2 and Experiments I and II with the biased coin. In Case 2, for five throws [using SIGN(RAND()-0.375) for each throw in Excel] I obtained <A> = 1 and <B> = 0.6. So <A> + <B> = 1.6 which still does not exceed 2. As to be expected, because neither <A> nor <B> can exceed 1.

Next, looking at some of your more recent posts, you note that the counterfactual of 0.625 is 0.375 for the biased coin that I used. And you say that the counterfactual of 0.625 is taken as 0.625 in CHSH calculations. I had assumed that the four pairs of angles produced outcomes based on four different sets of balls, so that no counterfactuals were needed, and no ball was measured twice. (Although, as these are macroscopic entities it would not matter if the same ball was measured many times at many different angles? A measurement of a macroscopic object cannot be wrong like a counterfactual calculation?)

I can see some sense in what you are doing as two particles in a pair always give opposite results when A and B are equally aligned. So there is a feeling that the two particles are always exact opposites. Almost as two sides of the same particle. In fact the electron is the exact antiparticle of the positron in the pair.

The outcomes at A and B depend not only on the particles but on the detector angles. And I therefore agree that the outcome for a particle at A is equivalent to a biased coin. For a general angle at A the outcome is not likely to be 50/50 odds for this biased coin (but see note).

I do not see a 'counterfactual' problem if all correlations are based on measurements, so the four sets are all treated equally. And the target to be reached would be an average abs correlation of 0.6.


Note: for every coin with a bias, there is a complementary coin with the opposite bias at the same detector and detector angle. I.e. electrons have spin 1 or spin -1 and only 'one at a time' of these two types of electron can be in one individual pair. So in a long run of outcomes at A, with a constant angle, <A> = 0 because there are likely to be as many electrons biased towards - as to +.

Ben6993 wrote:Michel, I have switched to this new thread for my reply.

I have worked through Case 1 and 2 and Experiments I and II with the biased coin. In Case 2, for five throws [using SIGN(RAND()-0.375) for each throw in Excel] I obtained <A> = 1 and <B> = 0.6. So <A> + <B> = 1.6 which still does not exceed 2. As to be expected, because neither <A> nor <B> can exceed 1.

So, for the two coins, experiment you got <A> + <B> = 1.6, What did you get for the single coin experiment using the same biased coin? Did it violate <A> + <B> = 0, even by experimental error. You can even use the same data, you don't have to repeat the tosses. Just filp the result of the first column and call it B, then calculate <A> + <B> and verify that it is exactly 0.

The point is that the same biased coin can violate the <A> + <B> = 0 based on whether we are using counter-factual outcomes or actual outcomes from different coins. But the violation is only apparent because we shouldn't even be using <A> + <B> = 0 for two separate coins as you identify.

I had assumed that the four pairs of angles produced outcomes based on four different sets of balls, so that no counterfactuals were needed, and no ball was measured twice.

Yes and No. Yes because that is what is done in EPR experiments, four different sets of particles. No as concerns Joy's experiment because Richard insists on measuring all the correlations on the same set of balls.

The derivation of the inequality as explained viewtopic.php?f=6&t=39 uses counterfactual correlations. Counter-factual correlations can not violate it. Not even by spooky action (Same as E(A) + E(B) = 0 can never be violated by a single coin). The reason they do not use counterfactuals in experiments is because it is impossible to recover the particle pair and re-measure them and they figure the probability distribution should not change so they just use another set. They are wrong, as we just found out, the inequality for actual outcomes is different from the counterfactual one. I'm arguing that actual correlations can violate a counterfactual inequality. So if Richard insists on measuring on a single set of balls, he would be cheating.

(Although, as these are macroscopic entities it would not matter if the same ball was measured many times at many different angles? A measurement of a macroscopic object cannot be wrong like a counterfactual calculation?)

The coins are also macroscopic aren't they. Yet measuring on a single coin obeys the equality, while measuring on separate coins violates it.

I do not see a 'counterfactual' problem if all correlations are based on measurements, so the four sets are all treated equally.

That is because you are thinking only about the experiments and forgetting the reason we are even discussing this to start with. Somebody derived some inequalities, using certain assumptions. One of those assumptions was that the counterfactual results of a single set of particles, exists along side the actual results. The inequality they got was S <= 2 (similar to our E(A) + E(B) = 0). Unfortunately, it was not possible to measure the counterfactual results on the same set, so they assumed that the results obtained from separate sets of particles could be used instead. So they performed real experiments which produced actual results and they observed that S > 2. (similar to your experiment II with <A> + <B> = 1.6). They concluded that the experiment had violated the original inequality (similar to violation of E(A) + E(B) = 0). They forgot to remember that the actual correlations they got from the experiment, could not be directly compared with the inequality which was derived with counterfactual terms. They had to use a different inequality derived for actual terms only (similar to our E(A) + E(B) <= 2). Which means their experiment did not really violate any inequality. Just because their actual results appeared to violate S <= 2 does not mean the counterfactual results would not have obeyed it. (similar to the biased coin obeying <A> + <B> = 0 for experiment 1, and yet producing <A> + <B> = 1.6 for experiment II.

Ben,
Here is the python code for the coin toss example:

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import numpy

N = 100000  # number of tosses
t = 0.375   # Tail bias

print "Case 1: Single coin toss (Experiment I)"
A = numpy.random.choice([-1, 1], p=(t, 1-t), size=N)
B = -A
A_B = A + B
print "%0.2f <= E(A) + E(B) <= %0.2f" % (A_B.min(), A_B.max())
print "<A> + <B> = %0.2f + %0.2f = %0.2f" % (A.mean(), B.mean(), A_B.mean())

print
print "Case 2: Two coin tosses (Experiment II)"
A = numpy.random.choice([-1, 1], p=(t, 1-t), size=N)
B = numpy.random.choice([-1, 1], p=(t, 1-t), size=N)
A_B = A + B
print "%0.2f <= E(A) + E(B) <= %0.2f" % (A_B.min(), A_B.max())
print "<A> + <B> = %0.2f + %0.2f = %0.2f" % (A.mean(), B.mean(), A_B.mean())

print
print "Case 3: Re-use just one data stream from Experiment II + its counterfactual"
C = - A
A_B = A + C
print "%0.2f <= E(A) + E(B) <= %0.2f" % (A_B.min(), A_B.max())
print "<A> + <B> = %0.2f + %0.2f = %0.2f" % (A.mean(), C.mean(), A_B.mean())

print
print "Case 3: Re-use the other data stream from Experiment II + its counterfactual"
D = - B
A_B = D + B
print "%0.2f <= E(A) + E(B) <= %0.2f" % (A_B.min(), A_B.max())
print "<A> + <B> = %0.2f + %0.2f = %0.2f" % (B.mean(), D.mean(), A_B.mean())



Results

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Case 1: Single coin toss (Experiment I)
0.00 <= E(A) + E(B) <= 0.00
<A> + <B> = 0.25 + -0.25 = 0.00

Case 2: Two coin tosses (Experiment II)
-2.00 <= E(A) + E(B) <= 2.00
<A> + <B> = 0.25 + 0.25 = 0.50

Case 3: Re-use just one data stream from Experiment II + its counterfactual
0.00 <= E(A) + E(B) <= 0.00
<A> + <B> = 0.25 + -0.25 = 0.00

Case 3: Re-use the other data stream from Experiment II + its counterfactual
0.00 <= E(A) + E(B) <= 0.00
<A> + <B> = 0.25 + -0.25 = 0.00


Michel I see you have posted twice recently here and several times in other threads, but I have to tell you that I am not reading your posts till you've emailed me with your observations concerning the results of my little R experiment, and a Python version of the draft program for evaluating the outcome of Joy's experiment.

I furthermore suggest we erase the words "proof", "inequality", "bound", and "violation" from the vocabulary of experimental physics. They are superfluous and apparently cause a lot of confusion.

More precisely: the words can be avoided; but if they are used, they need very careful qualification. EG "statistical proof" might be better than "proof", but still one needs to explain what is meant by "statistical proof" ... apparently a lot of people have never even thought about the concept.

I've tried to explain what is wrong with your idea of a Bell illusion here: http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=31&start=190#p1467

Please take a good look at the two graphics.

gill1109 wrote:I furthermore suggest we erase the words "proof", "inequality", "bound", and "violation" from the vocabulary of experimental physics. They are superfluous and apparently cause a lot of confusion.

I suggest you write to Nature and Physical Review letters and tell them. Don't forget all those experimentalists with whom you've written articles. So long as they keep spreading falsehood about locality and realism. I'll keep pointing out their silly mistakes about bounds, inequalities and fake violations. Even you too. Do you still believe that realism is untenable? Do you still believe that counterfactual definiteness is untenable? If you do, then you still share the illusion.

More precisely: the words can be avoided; but if they are used, they need very careful qualification. EG "statistical proof" might be better than "proof", but still one needs to explain what is meant by "statistical proof" ... apparently a lot of people have never even thought about the concept.

I've told you already. Statistics and error are irrelevant. If you disagree, produce a 4xN spreadsheet of any size, from any source, introducing as much error as you like, which violates the CHSH by just 0.000000000001. You can't do it, so why are you deceiving people that statistical error is relevant to violations of the CHSH.

Michel, I have done more calculations following your reply.

CHSH statistic = E(a, b) − E(a, b′) + E(a′, b) + E(a′ b′). The settings a, a′, b and b′ are 0°, 90°, 45° and 135°, respectively

For the trivial case of a CHSH run of one pair of particles, CHSH = AB - AB' + A'B + A'B'

If A and B are measurement outcomes, and A' (90°) and B'(135°) are counterfactual estimates based on A (0°) and B(45°) such that A' = -A and B' = -B.

Then
CHSH = -2 when A = 1 and B = -1
CHSH = -2 when A = -1 and B = 1
CHSH = 2 when A = 1 and B = 1
CHSH = 2 when A = -1 and B = -1.
These are the only four possible outcomes
.

But if A, A', B and B' are all different and independent particles, the possible outcomes are greater and can be any one of 16 combinations ranging from A = B = A' = B' = 1 through to A = B = A' = B' = -1. In each case the CHSH statistic is again either 2 or -2.

I think that also limits the CHSH statistic to be bounded by 2 for expectations based on n runs rather than on one run only. (But have been too idle to check that with data.) That applies to counterfactual data and to independent data alike. So results based on the CHSH statistic will fail to beat 2.

Using counterfactual measurements reduces the scope of the outcomes from 16 combinations to four combinations. And I agree that would happen even with macroscopic measurements where the same ball was measured at two different angles.

I think that a new statistic =
[abs{E(a, b)} + abs{E(a, b′)} + abs{E(a′, b)} + abs{E(a′ b′)}]/4
might avoid a CHSH problem. Where only one measurement per ball is to be used as an outcome, and each of the four calculations is based on a completely different set of balls.

I do not expect this statistic (average abs expectation = 0.5) to be exceeded either, but that is a different matter.

Ben6993 wrote:Michel, I have done more calculations following your reply.

CHSH statistic = E(a, b) − E(a, b′) + E(a′, b) + E(a′ b′). The settings a, a′, b and b′ are 0°, 90°, 45° and 135°, respectively

For the trivial case of a CHSH run of one pair of particles, CHSH = AB - AB' + A'B + A'B'

If A and B are measurement outcomes, and A' (90°) and B'(135°) are counterfactual estimates based on A (0°) and B(45°) such that A' = -A and B' = -B.

Then
CHSH = -2 when A = 1 and B = -1
CHSH = -2 when A = -1 and B = 1
CHSH = 2 when A = 1 and B = 1
CHSH = 2 when A = -1 and B = -1.
These are the only four possible outcomes
.

But if A, A', B and B' are all different and independent particles, the possible outcomes are greater and can be any one of 16 combinations ranging from A = B = A' = B' = 1 through to A = B = A' = B' = -1. In each case the CHSH statistic is again either 2 or -2.

Ben, you are on the right track, but you missed something. In your second try with independent particles, the *pairs* should be independent, not the singles. So instead of independent A, A', B and B', you should generate independent A1, B1, A2, B2', A3', B3, A4', B4' (a total of 8 in 4 pairs). Then calculate all the possibilities for CHSH = A1B1 - A2B2' + A3'B3 + A4'B4'. Note the numbers. Each paired correlation should not share a term with another in the indpendent case. By generating only 4 singles, you did not really enforce the independence.

There should be 256 possibilities, not 16. 1/8 of them should violate the CHSH upper bound, 1/8 should violate the lower bound. If you use the version of the CHSH which takes absolute values, 1/4 of them should violate it.

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import itertools
[''.join(v) for v in itertools.product('+-', repeat=8)]


Code: Select all

['++++++++', '+++++++-', '++++++-+', '++++++--', '+++++-++', '+++++-+-', '+++++--+', '+++++---', '++++-+++', '++++-++-', '++++-+-+', '++++-+--', '++++--++', '++++--+-', '++++---+', '++++----', '+++-++++', '+++-+++-', '+++-++-+', '+++-++--', '+++-+-++', '+++-+-+-', '+++-+--+', '+++-+---', '+++--+++', '+++--++-', '+++--+-+', '+++--+--', '+++---++', '+++---+-', '+++----+', '+++-----', '++-+++++', '++-++++-', '++-+++-+', '++-+++--', '++-++-++', '++-++-+-', '++-++--+', '++-++---', '++-+-+++', '++-+-++-', '++-+-+-+', '++-+-+--', '++-+--++', '++-+--+-', '++-+---+', '++-+----', '++--++++', '++--+++-', '++--++-+', '++--++--', '++--+-++', '++--+-+-', '++--+--+', '++--+---', '++---+++', '++---++-', '++---+-+', '++---+--', '++----++', '++----+-', '++-----+', '++------', '+-++++++', '+-+++++-', '+-++++-+', '+-++++--', '+-+++-++', '+-+++-+-', '+-+++--+', '+-+++---', '+-++-+++', '+-++-++-', '+-++-+-+', '+-++-+--', '+-++--++', '+-++--+-', '+-++---+', '+-++----', '+-+-++++', '+-+-+++-', '+-+-++-+', '+-+-++--', '+-+-+-++', '+-+-+-+-', '+-+-+--+', '+-+-+---', '+-+--+++', '+-+--++-', '+-+--+-+', '+-+--+--', '+-+---++', '+-+---+-', '+-+----+', '+-+-----', '+--+++++', '+--++++-', '+--+++-+', '+--+++--', '+--++-++', '+--++-+-', '+--++--+', '+--++---', '+--+-+++', '+--+-++-', '+--+-+-+', '+--+-+--', '+--+--++', '+--+--+-', '+--+---+', '+--+----', '+---++++', '+---+++-', '+---++-+', '+---++--', '+---+-++', '+---+-+-', '+---+--+', '+---+---', '+----+++', '+----++-', '+----+-+', '+----+--', '+-----++', '+-----+-', '+------+', '+-------', '-+++++++', '-++++++-', '-+++++-+', '-+++++--', '-++++-++', '-++++-+-', '-++++--+', '-++++---', '-+++-+++', '-+++-++-', '-+++-+-+', '-+++-+--', '-+++--++', '-+++--+-', '-+++---+', '-+++----', '-++-++++', '-++-+++-', '-++-++-+', '-++-++--', '-++-+-++', '-++-+-+-', '-++-+--+', '-++-+---', '-++--+++', '-++--++-', '-++--+-+', '-++--+--', '-++---++', '-++---+-', '-++----+', '-++-----', '-+-+++++', '-+-++++-', '-+-+++-+', '-+-+++--', '-+-++-++', '-+-++-+-', '-+-++--+', '-+-++---', '-+-+-+++', '-+-+-++-', '-+-+-+-+', '-+-+-+--', '-+-+--++', '-+-+--+-', '-+-+---+', '-+-+----', '-+--++++', '-+--+++-', '-+--++-+', '-+--++--', '-+--+-++', '-+--+-+-', '-+--+--+', '-+--+---', '-+---+++', '-+---++-', '-+---+-+', '-+---+--', '-+----++', '-+----+-', '-+-----+', '-+------', '--++++++', '--+++++-', '--++++-+', '--++++--', '--+++-++', '--+++-+-', '--+++--+', '--+++---', '--++-+++', '--++-++-', '--++-+-+', '--++-+--', '--++--++', '--++--+-', '--++---+', '--++----', '--+-++++', '--+-+++-', '--+-++-+', '--+-++--', '--+-+-++', '--+-+-+-', '--+-+--+', '--+-+---', '--+--+++', '--+--++-', '--+--+-+', '--+--+--', '--+---++', '--+---+-', '--+----+', '--+-----', '---+++++', '---++++-', '---+++-+', '---+++--', '---++-++', '---++-+-', '---++--+', '---++---', '---+-+++', '---+-++-', '---+-+-+', '---+-+--', '---+--++', '---+--+-', '---+---+', '---+----', '----++++', '----+++-', '----++-+', '----++--', '----+-++', '----+-+-', '----+--+', '----+---', '-----+++', '-----++-', '-----+-+', '-----+--', '------++', '------+-', '-------+', '--------']

Ben, you'll also note that for counterfactual values on a single set, the expression

S = |AB - AB'| + |A'B + A'B'| must be equal to 2, an equality (conceptually similar to our A + B = 0 for a single coin). It is impossible to have any value different from 2 for a single set.