Comments about "Bell's theorem" in Wikipedia

This document contains comments about the article "Bell's theorem" in Wikipedia
In the last paragraph I explain my own opinion.

Contents

Introduction

The article starts with the following sentence.
Bell's theorem is a "no-go theorem" that draws an important distinction between quantum mechanics (QM) and the world as described by classical mechanics.
The problem is: It can not be used as a yardstick between one or the other.
See also Reflection 4. Susskind lectures on Quantum Entanglements
In its simplest form, Bell's theorem states:
No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.
The problem is this sentence is to open, one sided.
Part of the problem is that experiments in the classical realm (involving probabilties) and experiments in the quantum realm (involving photons and electrons) are difficult to compare.
Bell summarized one of the least popular ways to address the theorem, superdeterminism, in a 1985 BBC Radio interview:
There is a way to escape the inference of superluminal speeds and spooky action at a distance. But it involves absolute determinism in the universe, the complete absence of free will.
In order to understand the processes that take place in the unverse or here on earth no superluminal speed (communication) is involved. Free will is a human quality and has nothing to do with quantum mechanics.

1. Historical background

In the vernacular of Einstein: locality meant no instantaneous ("spooky") action at a distance; realism meant the moon is there even when not being observed.
The thing is: we know that there is a moon. Generally speaking we do not know: where.
In his groundbreaking 1964 paper, "On the Einstein Podolsky Rosen paradox",physicist John Stewart Bell presented an analogy (based on spin measurements on pairs of entangled electrons) to EPR's hypothetical paradox.
Starting point are pairs of entangled electrons. This requires in advance knowledge i.e. experiments.
It is always tricky to discuss thought experiments.
Using their reasoning, he said, a choice of measurement setting here should not affect the outcome of a measurement there (and vice versa).
That seems comon sense, but has to be established by experiments.
After providing a mathematical formulation of locality and realism based on this, he showed specific cases where this would be inconsistent with the predictions of QM theory.
All of this is very soft and unclear. What is a mathematical formulation of locality?

2 Overview

Bell's theorem states that any physical theory that incorporates local realism cannot reproduce all the predictions of quantum mechanical theory.
Bell's theorem, in its simplicity, can not be used to explain all experiments (using elementary particles).
Because numerous experiments agree with the predictions of quantum mechanical theory, and show differences between correlations that could not be explained by local hidden variables, the experimental results have been taken by many as refuting the concept of local realism as an explanation of the physical phenomena under test.
In my view there exists no one for all set of rules which describe all experiments involving elementary particles. IMO it is more that each experiment requires its own mathematics to describe the results. Ofcourse there exists similarity between some.
In most cases the results of the experiments are in the source (reaction) where the particles (photons) were created.
For a hidden variable theory, if Bell's conditions are correct, the results that agree with quantum mechanical theory appear to indicate superluminal effects, in contradiction to the principle of locality.
As written before: The cause is in the reaction which creates both particles or photons. No superluminal effects are involved.
Finally, measurement at perpendicular directions has a 50% chance of matching, and the total set of measurements is uncorrelated. These basic cases are illustrated in the table below. Columns should be read as examples of pairs of values that could be recorded by Alice and Bob with time increasing going to the right.
The tabular table shows the experimental result with n=1(100% identical), n=-1(100% opposite) and n=0 (50% identical and 50% opposite)
Nothing wrong. So far so good.
With the measurements oriented at intermediate angles between these basic cases, the existence of local hidden variables could agree with a linear dependence of the correlation in the angle but, according to Bell's inequality (see below), could not agree with the dependence predicted by quantum mechanical theory, namely, that the correlation is the negative cosine of the angle.
There are two important issues:
  1. What for example are the raw values measured for an angle of 45 degrees?
  2. How is the correlation factor calculated using Quantum mechanics, based on the measured values.
Specific issue 1 is very important. Not knowing the actual results of this experiment with the angles of 45, 135, 225 and 315 degrees makes it impossible to decide if quantum mechanics or classical mechanics is right or wrong.

3. Importance of the theorem

The title of Bell's seminal article refers to the 1935 paper by Einstein, Podolsky and Rosen that challenged the completeness of quantum mechanics.
I expect that what Bell did is to challenge the 1935 paper i.e. that it was not complete (and possible even wrong)
In his paper, Bell started from the same two assumptions as did EPR, namely
(i) reality (that microscopic objects have real properties determining the outcomes of quantum mechanical measurements), and
(ii) locality (that reality in one location is not influenced by measurements performed simultaneously at a distant location).
Of these two the concept of reality is the most tricky, specific at individual particle level.
Bell was able to derive from those two assumptions an important result, namely Bell's inequality. The theoretical (and later experimental) violation of this inequality implies that at least one of the two assumptions must be false.
Again most probably the realistic concept.
The problem is that if you perform mathematics based on the concept that all the states have an equal chance and in reality the states are correlated than experimental results do not agree with the mathematical predictions.
n two respects Bell's 1964 paper was a step forward compared to the EPR paper: firstly, it considered more hidden variables than merely the element of physical reality in the EPR paper; and Bell's inequality was, in part, liable to be experimentally tested, thus raising the possibility of testing the local realism hypothesis.
When particles are correlated this is also realistic. You do not need hidden variables to explain that.
After the EPR paper, quantum mechanics was in an unsatisfactory position: either it was incomplete, in the sense that it failed to account for some elements of physical reality, or it violated the principle of a finite propagation speed of physical effects.
It is clearly the element that the physical reality is more complex as assumed.
In a modified version of the EPR thought experiment, two hypothetical observers, now commonly referred to as Alice and Bob, perform independent measurements of spin on a pair of electrons, prepared at a source in a special state called a spin singlet state.
First of all you can not really perform a thought experiment to define if any quantum theory is right or wrong.
When you prepare electrons, in of what you call a "singlet state", you also have to define experiments to demonstrate that this is the case.
It is the conclusion of EPR that once Alice measures spin in one direction (e.g. on the x axis), Bob's measurement in that direction is determined with certainty, as being the opposite outcome to that of Alice, whereas immediately before Alice's measurement Bob's outcome was only statistically determined (i.e., was only a probability, not a certainty); thus, either the spin in each direction is an element of physical reality, or the effects travel from Alice to Bob instantly.
This is completely the wrong reasoning. First of all you should start from the concept that the spin in the two particles are correlated. This is a direct consequence of what we call the "singlet state". This correlation between the two spins is part of what you could call the physical reality. As a direct result when one spin is measured the direction of the other spin is known. No communication is involved between the detectors and no instantaneous effects are involved.

4. Local realism

It should be remembered when reading the text in this paragraph that Local realism is not mathematics but physics. Too much attention is towards mathematics while it is an physical issue.
At the end of the paragraph we read:
The hidden parameter is often thought of as being associated with the source but it can just as well also contain components associated with the two measurement devices.
The cause of the correlation is in the source. It has nothing to do with the measurement devices. To call what causes the correlation "a hidden variable" is a misnomer. This name only causes confusion.

5. Bell inequalities

Ch(a,b) denote the correlation as predicted by any hidden variable theory.
It is important to mention that Ch(a,b) means correlation and not probability. The two concepts are "totally" different, but related. Probability comes first. Correlation second.

5.1 Original Bell's inequality

The inequality that Bell derived can then be written as:
Ch(a,c) - Ch(b,a) - Ch(b,c) =< 1,
where a, b and c refer to three arbitrary settings of the two analysers. This inequality is however restricted in its application to the rather special case in which the outcomes on both sides of the experiment are always exactly anticorrelated whenever the analysers are parallel.
The way this equation is written is difficult, because it is based on correlations and not on probabilities.
Suppose the two particles are perfectly anti-correlated—in the sense that whenever both measured in the same direction, one gets identically opposite outcomes, when both measured in opposite directions they always give the same outcome.
You cannot suppose this. When you suppose something this becomes a thought experiement. The physical reality can be different. What you should perform are experiments. Not one but many under different conditions. This becomes your raw data. The next step is to explain the results of the experiment.
What you can do perform is 1000 experiments where the angle between the two polarizers is zero. For example study: Reflection "two channel" Bell test. The result could be 502 times "+ +" and 498 times "- -". These results indicate a clear correlation between the test results. The explanation is that both photons are polarised in the same direction.
The only way to imagine how this works is that both particles leave their common source with, somehow, the outcomes they will deliver when measured in any possible direction. (How else could particle 1 know how to deliver the same answer as particle 2 when measured in the same direction? They don't know in advance how they are going to be measured...)
This is written from the human point of mind. Particles don't know something. The simplest explanation is that immediate when the particles are created, that they are (almost) identical. The only difference is that (in this particular reaction) they move in opposite directions. For the rest they are identical. This explanation implies that the particles don't have to know how they are measured nor when.

5.2 CHSH inequality

5.2.1 Derivation of CHSH inequality

6 Bell inequalities are violated by quantum mechanical predictions

7 Practical experiments testing Bell's theorem

Experimental tests can determine whether the Bell inequalities required by local realism hold up to the empirical evidence.
This is a very complex sentence. The first thing you should do with any experiment is to perform the experiment as much as possible under the different conditions that apply. Only after the results are know you can discuss if they are as expected or not. The not as expected case often is the most interesting.
When the polarization of both photons is measured in the same direction, both give the same outcome: perfect correlation.
Consider that the polarizer is in vertical direction. When a photon enters polarised in the vertical direction this photon will be detected.
Next consider that a photon enters the polariser under an angle of 10 degrees in the vertical direction, This photon will also be detected. That is a +.

The folowing table shows the results of all the angles between 0 and 180 when the polarizer is in vertical direction (has an angle of 0 degrees)
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170
+ + + + + - - - - - - - - - + + + +
Because the two photons are polarized in the same direction when in one direction the photon angle is 10 degrees also in the other direction the photon angle is 10 degrees. That means both are detected. This is een + + measurement.
In total going from 0 to 170 degrees there are 9 measurements + + and 9 measurements - - implying a total correlation of +1.

When measured at directions making an angle 45 degrees with one another, the outcomes are completely random (uncorrelated).
The folowing table shows the results of all the angles of the photon between 0 and 180 when the polarizer has an angle of 45 degrees
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170
+ + + + + + + + + - - - - - - - - -
When you compare all the angles between 0 and 170 degrees of case 0 degrees (first case) with case "45 degrees" you get:
5 times + +, 4 times - +, 5 times - - and 4 times - +.
That means the correlation is zero.
Measuring at directions at 90 degrees to one another, the two are perfectly anti-correlated.
The folowing table shows the results of all the angles of the photon between 0 and 180 when the polarizer has an angle of 90 degrees
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170
- - - - - + + + + + + + + + - - - -
When you compare all the angles between 0 and 170 degrees of case 0 degrees (first case) with case "90 degrees" you get:
18 times - + and 18 times - +. That means the correlation is -1
In general, when the polarizers are at an angle ? to one another, the correlation is cos(2?).
The folowing table shows the results of all the angles of the photon between 0 and 180 when the polarizer has an angle of 10 degrees
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170
+ + + + + + - - - - - - - - - + + +
When the angle is 10 degrees the results are:
8 times + +, 1 time - +, 8 times - - and 1 time - +. The correlation is 0.7778

The folowing table shows the results of all the angles of the photon between 0 and 180 when the polarizer has an angle of 20 degrees
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170
+ + + + + + + - - - - - - - - - + +
When the angle is 20 degrees the results are:
7 times + +, 2 times - +, 7 times - - and 2 times - +. The correlation is 0.5556
When you compare the correlation factors of 0, 10 and 20 degrees (i.e. 1, 0.7777 and 0.5555) the decrease is lineair and is not in accordance with the cos(2?) function
For more information and a discussion see Reflection 1: "two channel" Bell test
At the end of the paragraph we read:
Bell test experiments to date overwhelmingly violate Bell's inequality.
That is maybe true, however this paragraph does demonstrate that this claim is true.. What should be done is to show the possible results of the outcome of this test using the two theories and compare them with the actual results. That is not done.

7.1 Two classes of Bell inequalities

7.2 Practical challenges

10. See also

Following is a list with "Comments in Wikipedia" about related subjects

11. Notes

This article tries to explain the Bell Theorem by means of a thought experiment.
The experiment consist of a source and two identical boxes. Each box contains a switch with can be placed in three positions "1", "2" or "3". Each box has two lamps: a red lamp "R" and a green lamp "G".
At each run the source will emit two identical particles towards each box. Based on the state (variety) of the particle and the position of the switch either the "R" or the "G" lamp will lit up.
At page 6 we read:
Shortly after the experimenter pushed the button on the source in figure 1, the detectors flash one lamp each. The experimenter records the switch settings and the colors of the lamps and then repeats the experiment. Here, for example, the record reads 32RG – the switches are in positions 3 and 2 and the lamps flashed R and G, respectively 
 31RR 12GR 23GR 13RR 33RR 12RR 22RR 32RG 13GG 
 22GG 23GR 33RR 13GG 31RG 31RR 33RR 32RG 32RR
 31RG 33GG 11RR 12GR 33GG 21GR 21RR 22RR 31RG 
 33GG 11GG 23RR 32GR 12GR 12RG 11GG 31RG 21GR
 12RG 13GR 22GG 12RG 33RR 31GR 21RR 13GR 23GR
The above table shows the results of 5*9 = 45 runs.
Next we read:
I) If one examines only those runs in which the switches have the same setting (figure 4), then one finds that the lights always flash the same colors.
II) If one examines all runs , without any regard to how the switches are set (figure 5), then one finds that the pattern of flashing is completely random. In particular, half the time the lights flash the same colors, and half the time different colors
The next table shows a frequency of each combination.
11 12 13 21 22 23 31 32 33 sum
RR 11 1 2 2 1 2 1 4 15
GG 2 2 2 39
RG 3 4 2 0 9
GR 3 2 2 3 1 1 12
What this table shows that in 15+9 = 24 cases the colors are the same and in 9+12 = 21 cases the colors are different.
That means that is not exactly the same as the claim stated above that the chance for each is the same.
The general problem is that in order to evaluate this experiment more experiments of 45 runs should be evaluated.
At page 9 we read:
Consider a particular instruction set, for example, RRG. Should both particles be issued the instruction set RRG, then the detectors will flash the same colors when the switches are set to 11, 22, 33, 12, or 21; they will flash different colors for 13, 31, 23, or 32.
Because the switches at each detector are set randomly and independently, each of these nine cases is equally likely, so the instruction set RRG will result in the same colors flashing 5/9 of the time.
That is correct. Next we read:
Evidently the same conclusion holds for the sets RGR, GRR, GGR, GRG and RGG, because the argumentuses only the fact that one color appears twice and the other once. All six such instructions sets also result in the same colors flashing 5/9 of the time.
But the only instruction sets left are RRR and GGG, and these each result in the same colors flashing all ofthe time.
Therefore if instructions sets exist, the same colors will flash in at least 5/9 of all the runs, regardless of how the instruction sets are distributed from one run of the demonstration to the next.
This is Bell’s theorem (also known as Bell’s inequality) for the gedanken demonstration. But in the actual gedanken demonstration the same colors flash only 1/2 the time.
The data described above violate this Bell’s inequality, and therefore there can be no instruction sets.
To investigate this claim the following table shows all the combinations. Because there are 8 varieties and each box has 3 swithces, there are 8*3*3 = 72 combinations.
The next table shows all the 72 combinations and what the lamp settings are. Each combination has the same probability. 
     RRR  RRG  RGR  RGG    GRR  GRG  GGR  GGG
11    RR   RR   RR   RR     GG   GG   GG   GG
12    RR   RR   RG   RG     GR   GR   GG   GG
13    RR   RG   RR   RG     GR   GG   GR   GG    

21    RR   RR   GR   GR     RG   RG   GG   GG
22    RR   RR   GG   GG     RR   RR   GG   GG
23    RR   RG   GR   GG     RR   RG   GR   GG

31    RR   GR   RR   GR     RG   GG   RG   GG
32    RR   GR   RG   GG     RR   GR   RG   GG
33    RR   GG   RR   GG     RR   GG   RR   GG
The first combination RR shows the result of the variety RRR when each switch of both boxes is a "1".
The first line shows that when both switches are a "1" that in total there are 4 results RR and 4 results GG. The same result is also true when both switches are a "2" or a "3". All the other 6 lines shows the combinations when both switches are not the same.
The next table shows the frequency distribution of all 72 combinations.
11 12 13 21 22 23 31 32 33 sum
RR 422242 2 2424
GG 422242 2 2424
RG 2 22 2 2 2 12
GR 2 22 2 2 2 12
When you evaluate the total values there are 24+24=48 cases where there are equal colors and 12+12=24 when the colors are different.
The results of an actual experiment show that these numbers are 15+9=24 and 9+12 = 21. Part of the problem is that the number of runs 45 in one experiment is rather small, but in principle there is nothing wrong with it.

The following table shows the results of 100 simulations using the same logic.
1 0 2 1 2 8 9 8 12 10 13 11 6 3 4 6 1 3
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
The second line shows the total number of equal color pairs measured (so both red or both green). The first show how much each, The higest frequency is 13 with 31 equal pairs. This number of pairs is as expected because the average number should be with 2/3 * 45 = 30
The number of tests 1 with 24 equal pairs is small but not impossible.

The point is that when you observe the results of one experiment with 45 runs you cannot claim that the Bell's inequality is neither true or violated, nor that quatum physics is Right or Wrong.

12. References


This is an extremely interesting document and worthwhile reading.
At page 171 of the article you can read:
The rules of quantum mechanics can be employed to predict the results of the same experiment. I shall not give the details of how the predictions are derived from the mathematical formalism of the quantum theory; it can be stated however, that the procedure is completely explicit and is objective in the sense that anyone applying the rules correctly will get the same result.
In fact this is the weakest part of the whole article. This type of information is of uttermost importance if you want to understand to competing theories.
At page 174 of the article you can read (under the drawing):
The results not only violate the Bell inequality but also are in good agreement with the predictions of quantum mechanics, which fact adds to their credibility.
Unfortunate the predictions of quantum mechanics are missing.

12. External Links

Following is a list with additional information:

Reflection 1: "two channel" Bell test

The next table shows the combined results of 72 tests of the "two channel" Bell test.
The column angle shows the angle between the two polarizers.
The column correlation shows the correlation factor of the actual test results.
The column "++" shows the actual number of runs in which in both + detectors a photon is detected.
The column "+-" shows the actual number of runs in which in alice's + detectors and in bob's - detector a photon is detected. etc.
Angle + + - + - - + - Correlation
1 0 36 0 36 0 1
2 15 30 6 30 6 0.6666
3 22.5 27 9 27 9 0.5
4 30 24 12 24 12 0.3333
5 45 18 18 18 18 0
6 60 12 24 12 24 -0.3333
7 67.5 9 27 9 27 -0.5
8 75 6 30 6 30 -0.6666
9 90 0 36 0 36 -1
10 135 18 18 18 18 0
11 180 36 0 36 0 1
Correlation Table 72 runs
Graphical correlation 36 runs

What does the "Two channel Bell test" teaches us ?

It is important that you should consider this test from the view point that you have no previous knowledge. The only thing you know is that inside the source at regular intervals a reaction takes place which generates two photons. The two photons can be detected at 4 detectors. The only thing the experimenter can influence is the angle between the two polarizers.

One possible outcome of this experiment could have been that in all cases, independent of the angle between the polarizers, always you get the result of line 5. That means in all cases the outcome is completely random or unpredictable. In fact that is the most probable outcome in a general experiment which generates two photons.

The results of the table in fact show something else: The two photons are polarized in the same direction. This is also the only thing that the test learns us. Based on this model the observations are simulated. Using these simulated observations the correlation factor is calculated for each angle. The results show a lineair correlation relation.

The document also shows an other possible correlation: that of a cosinus function. That is inprinciple also possible.
In fact this is the biggest problem with the article "Bells theorem": The main attention is on mathematics and not on the results of actual experiments. They are missing.

When you ask 10 people to set up a different colour of hat and they all select the same colour you are suspicious. The next time the colour is different, but again all the same, and you become even more suspicious. When each time you repeat this experiment and they all have the same but a different colour, then you think by yourself: this is impossible there must be some form of communication between the 10 people.
In the "Two channel Bell test" the explanation is in the source. The two photons are always polarized in the same direction (in this case). The direction of both are random, but if you know one, you also know the other. There is no communication involved between the detectors.

There are no hidden variables involved.

Suppose that the results of the tests are different. That means that for the angle of 15 degrees different results of the "++", "+-" , "--" and "-+" values are observed.
Such actual experimental observed results are important. The cause between the results is of physical origin. The correlation factor is of minor importance, that is mathematics.
The explanation of the results is in the test setup, most probably is in the reaction which creates the particles.

The "Graphical Correlation" shows the relation between two correlation function. The blue one shows the correlation function based on simulated results. The pink line shows the correlation function based on the quantum theory. The results of the pink line are also based on actual experimental results. The biggest problem is that these results are not available. As a result it is very difficult to actual verify this.
The blue line is also supposed to demonstrate the "Bell inequality" theorem. The calculation to calculate the blue line is rather straight forward which more or less shows that there is no inequality involved. This raises doubts about the "Bell inequality" itself.

Reflection 2. Article in reference #5 by D'Espagnat (1979) in Scientific American.

It is interesting to compare the straight red line in the drawing at page 174 with the "Graphical correlation" displayed above. The problem is they "are" the same, but the meaning is completely different. They have nothing to do with each other. How come? Part of the problem is: what exactly is the Bell inequality.

Reflection 3. Article in reference #2 by N. David Mermin (1985) in Physics today

This article tries to explain Bell's inequality theorem by meaning of an experiment. The article starts by explaining how the aparatus used functions and discusses the results of the experiment. The problem with the results is when you perform a simulation of the logic used then the actual outome of the experiment and the simulation are in agreement. That means the actual outcome is as expected and there is no reason to claim that either clasical or quantum physics is either right or wrong.

When you study quatum physics type or experiment they can be clasified (among others) in two types:
one with involve one particle and one with involve two particles.

(*) The characteristic (variety) that the two particles are the same has to be established by experiment. That means both switches in position 1. In that case only 11RR and 11GG should be observed.

Reflection 4. Susskind lectures on Quantum Entanglements

In the document: http://www.lecture-notes.co.uk/susskind/quantum-entanglements/ Professor Leonard Susskind gives an excellent description of the mathematics involved in Quantum Mechanics.
In lecture 5 The "Violation of Bell's theorem" is discussed. See: http://www.lecture-notes.co.uk/susskind/quantum-entanglements/lecture-5/violation-of-bells-theorem/
In this lecture first the mathematics of Bell's theorem or Bell's inequality is discussed. Secondly the probabilities of two electrons each in the singlet state.
What the document shows is that these probabilties are in violation with the Bell's inequality.
The overall point is (not explicitly mentioned): that Quantum Mechanics is correct and Clasical physics is wrong.

The problem is that Bell's theorem can not be used as a yardstick to decide what is right or wrong. Bell's theorem describes certain experiments. Quantum mechanics describes other experiments in which entanglement is involved and generally speaking these experiments (reactions) cannot be compared with each other.

The same problem also is included in the following document: https://www.sciencenews.org/blog/context/quantum-spookiness-survives-its-toughest-tests
In this document we read:
To Einstein, that seemed possible only if the distant particle had acquired its property when the particles last encountered each other. So Bob’s coin would be locked into “heads only” when his and Alice’s coins parted ways. In Einstein’s view, both coins (or particles) possessed definite properties for their entire trip. But the math of quantum mechanics demands otherwise. In the quantum realm, particles do not possess precise properties until measured. A quantum particle’s spin axis, for instance, points neither up nor down until you measure it (like a spinning coin that is neither heads nor tails until you catch it).
The bigest problem with this paragraph is that they are mixing coins with (entangled) particles. That is wrong. The behaviour of coins is described by Bell's Theorem. The behaviour of entangled particles is not.
In retrospect Einstein's view is correct: both particles possessed definite properties for their entire trip i.e the correlation happens during the reaction in the source where they are created.
For an "endless" spinning coin this is different. The same situation also exist in the "lottary game" where they used numbered balls floating in the air. You have to stop the machine to make do a read out. The point is in these machines no entanglement is involved.

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