The Quantum Theory and Reality

This document contains comments about the article The Quantum Theory and Reality by Bernard d'Espagnat in Scientific American of November 1979.
For the text of the article select:


(1) The article starts with the following sentence.
The doctrine that the world is made up of objects whose existence is independent of human consiousness turns out to be in conflict with quantum mechanics and with facts established by experiment.
This is a "beautifull" sentence. The main reason why I write that is because the sentence is very clear. Like the whole article.
The most important concept is independent of human consiousness , because that is the way we should performing science and studying physics.
At the same the time a theory which is in conflict with that idea should be very clearly specify what my mind has to do with the outcome of an experiment I perform.
See Remark 1 Next we read:
Any succesfull theory in physical science is expected to make accurate predictions.
That is 100% correct . Next we read:
Given some well-defined experiment, the theory should correctly specify the outcome or should at least assign the correct probabilities to all the correct outcomes.
Again this is 100% correct. However there are some problems...
Next page 128 center column:
This world view is based on three assumptions or premises
One is realism, the doctrine that regularities in observed phenomena are caused by some physical reality whose existence is independent of human observers.
IMO a slightly better sentence is:
One is realism, the doctrine that the physical reality contains regularities whose existence is independent of human observers.
This means that the observation process itself does not belong to the physical reality. We have to be carefull because observations can influence what is observed.
Next page 128 center column:
The second premise holds that inductive inference is a valid mode of reasoning and can be applied freely, so that legitimate conclusions can be drawn from consistent observations.
What we are discussing here is a toolbox which contains mathematics, logic, laws, and theories which are already accepted and demonstrated by experiments and many observations to be correct, implying that they can be used to discover new theories.
Next page 128 center column:
The third premise is called Einstein separability or Einstein locality and it states that no influence of any kind can propagate faster than the speed of light.
The "law" that no influence can propagate faster than the speed of light belongs to the second premise. Any claim that this law is not true should be clearly demonstrated.
Next page 133 right column:
For a vector associated with a macroscopic object in everyday life, one would assume as a matter of course, and with good reason, that all three components have definite values at all times; the value of a component might be unknown, but it cannot be undefined.
The value of any component might be unknown. period. There is nothing wrong with that as long as the particle is not disturbed i.e. measured. The sentence "it cannot be undefined" should be removed.
Immediate next page 133 right column:
When this assumption is applied to the spin vector of a particle, however, it becomes highly suspect and indeed in the conventional interpretation of quantum mechanics it is dismissed as an instance of a hidden parameter theory.
The spin vector defines a model of the inner structure of a proton. Quarks also fall in that category.
The major physical issue is to what extend the direction in space of this vector has any relation to the (direction of the) speed of the proton
Next page 133 right column:
A single instrument can measure only one spin component and in doing so it alters the values of the components. Hence in order to learn the values of three components three measurements would have to be made in succesion. By the time the particle emerged from the third instrument it would no longer have the same spin components it had when it entered the first instrument.
Any disturbance changes the spin.
Next page 134 left column:
He finds that if he measures component A for both protons in each pair some are A+ and others are A- but when one member is A+ the other is A-
Implying they are negative correlated.
If he measures component B for both protons in each pair some are B+ and others are B- but when one member is B+ the other is B-
Next page 134 left column:
Similary, a C+ proton is invariably accompanied by a C- one.
That is correct in theory, but how do you measure that in practice? When you observe the "Thought Experiment" on page 134 only the Y and Z axis are measured but not the X axis, which is the direction of movement of the particle.
Next page 134 right column:
The remaining pairs must then be made up of either one proton tested along axis A and one tested along axis B etc
For the sake of brevity I shall refer to the pairs in each of these polpulations as AB, etc
A pair that on testing yields the result A+ for one proton B+ for the other can be labeled an A+B+ pair.
When axis A and B are tested the following results are possible A+B+, A+B-, A-B+ and A-B-. However it is also possible that only one result is observed: A+, A-, B+ and B-
(2) At page 135 middle column:
The result is the inequality
n[A+B-] < 0r = n[A+C-] + n[B-C+]
Although this inequality is hereby formally derived, it cannot be tested directly by experiment because no instrument can independently measure two spin components of a single proton
This is a very important statement. See Remark 2
The experiments underconsideration, however, are carried out not on individual protons but on correlated pairs of them, and there is no need to make such impossible measurements.
The important issue is that in order to experimental verify this statement entangled particles are used. See Remark 3
To be explicit, the existence of a strict negative correlation implies that the first proton, which is already known by direct measurement to have the spin component A+ (assuming that second proton is B+) must also have the component B-
This implies a certain type of reasoning. What is true, is that this same reasoning applies for all combinations assuming that the same negative correlation is detected as described in Remark 3 . If the reasoning as such is correct is open for discussion.
At page 135 right column:
The resulting expression is
n[A+B+] < 0r = n[A+C+] + n[B+C+] (1)
This is the Bell inequality .
The following Bell inequality is also possible:
n[A+C+] < 0r = n[A+B+] + n[C+B+]
The B and C axis are reversed
This Bell inequality can also be written as:
n[A+C+] < 0r = n[A+B+] + n[B+C+] (2)
Comparing (1) and (2) shows that the first two parts are reversed and that the third term is the same. This shows that the "Bell inequality" is a rather tricky claim.
Of course the inequality is proved by this argument only if the three premises of local realistic theories are considered valid.
The Bell inequality is a mathematical statement. The only way to "prove" the statement, that means if it is a true description of the reality, is by performing the calibration experiments as outlined in Remark 3
The Bell inequality constitutes an explicit prediction of the outcome of an experiment.
That is partly true. It shows the tallied results of many tests of one experiment (setup).
In order to test the Bell inequality you have to test three populations AB, AC and BC 100 times.
Starting point is that the 3 axis are perpendicular to each other (90 degrees)
The following "Table" shows the results:
     A+B+  A+B- A-B+  A-B-   A+   A-   B+   B- 
AB    23    20   30    24     1    1        1       
     A+C+  A+C- A-C+  A-C-   A+   A-   C+   C- 
AC    20    22   28    27          1    1   1       
     B+C+  B+C- B-C+  B-C-   B+   B-   C+   C- 
BC    28    24   26    19     1    1        1       
What the table shows that the average number of pairs in each population is 100/4 = 25
What the first column shows is that 23 < (20+28) i.e. the Bell inequality is true in all the 4 cases.
IMO the table does not show the result of an experiment but the combined results of 300 experiments.
The rules of quantum mechanics can be employed to predict the same experiment
I shall not give the details of how the prediction is derived from the mathematical formalism of the quantum theory; it can be stated etc. Surprisingly the predictions of quantum mechanics differ from those of local realistic theories.
It is very unfortunate that the article does not show the predictions derived from the mathematical formalism of the quantum theory (at this point) for the the same experiment
It also does not give a detailed description of this experiment. This is in conflict which what is stated at the beginning of this article. See (1a)
Surprisingly the predictions of quantum mechanics differ from those of local realistic theories.
In particular quantum mechanics predicts that for some choices of the axis A, B and C the Bell inequality is violated so that there are more A+B+ pairs of protons than there are A+C+ and B+C+ pairs combined.
What this statement implies is that the Bell inequality function can be violated for some directions of the axis can be violated, but not for all.
These results are in contradiction with the results of Remark 3 which claims that the results of the preliminary tests are independent of the axis.
It is very unfortunate that the article does not show the details of this experiment which should demonstrate that the quantum theory is correct. For a more detailed critical evaluation see Reflection 1
The general conclusion is that you have to be very carefull to compare Bell inequality with Quantum Theory.
Thus local realistic theories and quantum mechanics are in conflict.
That is correct, assuming both are predicting different results for the same experiment.
The conflict raises two questions. First, what are the experimental facts of the situation? Is the Bell inequality satisfied or is it violated.
The most important experimental facts are the details of the experiment Does the experiment involve entanglement Yes or Not. This fact should be established before the Bell inequality test is performed.
See also: Reflection 10 - Remarks of 22 May 2018
At page 136, left column, we read:
The technical difficulty of the experiments should not pass unmentioned
In a thought experiment both protons of every pair always reach the instruments and the instruments themselves always yield an unambiguous measurement of the spin component along the chosen axis (*).
Real apparatus cannot reproduce these results.
To perform a thought experiment with elementary particles does not make sense. For example with entangled particles. The problem is the underlying physical reality. That is what you want to discover.
Next we read:
The detectors are never perfectly efficient: many protons are simply not registered at all.
There is in priniciple nothing wrong with that. Certain protons or photons should not be registered, because it is not clear in which direction they should go
At page 136 right column we read:
One experiment has measured the correlations of spin component of protons and therefore closely resembles the original thought experiment.
Each experiment has its own merit and should be studied independent of the thought experiment of page 134.
At page 138 middle column we read:
Most physicists concerned with these problems however have substantial confidence based on the five consistent results, that the issue has already been decided. For some choices of the axes A,B and C the "Bell inequality" is violated in nature and local realistic theories are therefore false.
The issue is what are the descriptions that describe the observations for each of those five experiments.
The picture at page 138 is a point in case.
The quantum mechanics curve is described by the function - cos (a - b).
The Bell inequality curve is described by - abs(a-b) / 90 with a and b in degrees.
The most important reason is because the model used to describe the "Bell inequality" is too simple.
At page 138 right column we read:
Another area that might be scrutinized for unacknowledged assumptions is the proof of the Bell inequality.
Page 135 shows the derivation of the Bell inequality. There is nothing wrong with that derivation. The issue is what type of physical process does the "Bell inequality" describe. That is the question.
At page 138 right column we read:
The entire series of experiments founded on the ideas of Einstein, Podalsky and Rosen is sometimes regarded as merely a test of hidden-parameter theories.
The name hidden-parameter theories is misleading. There is nothing wrong to envision a proton as a rotating object.
At page 139 middle column we read:
In the context of this experiment positivism asserts that it would be meaningless to attribute anything resembling a definite spin component to a particle before the component is measured; that the only quantity with any verifiable reality is the observation itself etc.
The problem is the other way around. When you do any measurement the physical state of what you want to measure, is modified meaning that you know more about the state before the measurement than after the measurement.
This can be helpfull if every measurement of a set of objects gives the same result implying that all the members of this set have the same measured characteristic independent if they are measured or not.
At page 139 right column we read:
It is induction that enabled the physicist to extrapolate from a series of obeserved negative correlations to the conclusion that any two protons in the singlet state have opposite values of any single spin component even if none of the components is measured.
See Reflection 5 - How important is the Bell inequality
Induction, our reasoning, is part of the problem.
Assuming that the particle moves in the X direction than based on experiments we know that photons and protons which are measured in the Y axis are correlated. The same if they are both measured in the Z axis. The problem is if you want to test the spin of a proton in the X axis, which is the direction of motion and which can influence the direction of the spin.
This is one of the reasons why tests with photons and protons which can be logical identical, physical are different.
At page 139 right column we read:
His (Bohr's) reasoning is that a particle and an instrument adjusted to make a specific measurement on it constitute in some respects a single system, which would be altered in an essentiel way if the setting of the instrument were changed.
What you want to know is the behaviour (law, theory) of a physical process in its most natural environment. This implies undisturbed by the process of observation. This is a problem for any process, being quantum mechanics or the Bell inequality.
If realism and the free use of induction are to be retained, the violation of the Bell inequality can be explained only by giving up the assumption of Einstein separability.
The problem of faster than light propagation has nothing specific to do with the Bell inequality as with quantum mechanics.

Reflection 1 - The simplest Experiment with photons

Page 134 shows a sketch of an thought experiment with protons. Picture 1 shows the simplest experiment based on that sketch using photons.

       Detector A <--- Source ---> Detector B ---------------------> Detector B

Source      E  E    E   E   E   E E   E   E     E     E   E   E  E  E     E E    E E    E
Detector A   X       X   X   X     X   X   X     X     X   X   X  X  X     X      X X    X
Detector B   X  X    X   X       X X   X   X     X     X   X   X  X  X     X X    X X    X        

				Picture 1
The above experiment consists solely of one source and two detectors. The source emits two particles simultaneous. Each detector is like a camera. That means each detector (camera) registers the arrival times of the incoming particles. For example: both photons or both protons.
The line marked with "Source" shows a sequence of the events E at the source over a period of time .
The line marked with "Detector A" shows a sequence of the photon detection events X at the Detector A over a period of time .
The line marked with "Detector B" shows a sequence of the photon detection events X at the Detector B over a period of time .
When you compare the two they are almost identical, except that at certain instances one is missing.

Reflection 2 - a slightly more complex experiment with photons

Page 134 shows a sketch of an thought experiment with protons. Picture 4 shows a slightly simpler experiment with photons.

                                               Detector B+     
                 Detector A <--- Source ---> Analyzer 2
                                               Detector B-

Source      E  E    E   E   E   E E   E   E     E     E   E   E  E  E     E E    E E    E
Detector A  X       X   X   X     X   X   X     X     X   X   X  X  X     X      X X    X
Detector B+ X       X                 X   X           X       X  X          X      X             
Detector B-    X        X       X X             X         X         X     X      X      X        

				Picture 2
Compared with Picture 1, Picure 2 has one analyzer instead of Detector B. The analyzer services as a beam splitter that means each photon goes to either Detector B+ or Detector B-
The result is that the arrival times pattern of Analyzer 2 is now divided in two: One for Detector B+ and one for Detector B-
When you compare the two (that is Detector A with the combined pattern of the detectors B+ and B-) they are identical, except that at certain instances one is missing.

How ever what is important: The analyzer 2 and the Detectors B+ and B- can be turned over a certain angle.
First the operator should turn the analyzer in increments of 90 degrees and observe what happens.
Secondly the operator should turn the analyzer in increments of 10 degrees and observe what happens.
Generally speaking in all these test the results should be the same.
That means:

Again also here you should place analyzer 2 at different positions.
There is one overall message: There is no faster than light communication involved

Reflection 3 - The experiment of page 134 with protons

Picture 3 shows a sketch of an thought experiment at page 134 of the Scientific American article of 1979 with protons.
                 Detector A+                       Detector B+     
                     _                               _
                      |\                               /|
                        \                             /
                         \                           /
                      Analyzer 1 <--- Source ---> Analyzer 2
                         /                           \
                        /                             \
                      |/_                             _\|
                 Detector A-                        Detector B-

Source      E  E    E   E   E   E E   E   E     E     E   E   E  E  E     E E    E E    E
Detector A+             X         X             X         X         X     X      X      X
Detector A- X       X       X         X   X           X       X  X                 X     
Detector B+ X       X                 X   X           X       X  X          X      X             
Detector B-    X        X       X X             X         X         X     X      X      X        

                                     Picture 3
In the experiment we have one source and two analyzers and 4 detectors all the same plane. That means the angle betwee the analyzers is zero
In this particular experiment when in one test (one pair) one proton has property A+ the other proton has B-
Also when one proton has property A- the other proton has property B+.
What this experiment shows that the particles are (negative) correlated.
However there are also possible outcomes: The importance is that all these outcomes do not directly reflect problems in how the particles are measured. They can also be caused, because the reality is not so "simple" as "we" assume.

The next step in the experiment is that Analyzer 2 in increments of 10 degrees rotates. That means first 10 degrees then 20 degrees untill 90 degrees.
What happens is that the correlation slowly increases from -1 to 0. The correlation at 90 degrees is zero. This means there is no correlation. In fact what you measure is over a complete different axis.
When the angle is zero both analyzers measure the spin in the X direction. When the angle is 90 degrees one measure in the X direction and the other in the Y direction.
Also in this case it is possible that both analyzers measure nothing. That means the spin is in the Z direction.

What is more important is that to explain this behaviour you have to use the quantum theory or better a model which depicts this 3 dimensional behaviour.
IMO the "Bell inequality" can not be used to predict the outcome of this experiment and is of not much use to describe this experiment.

Reflection 4 - Physical considerations

The most important part in Picture 134 is the source. The source, dependent about the details, can generate simultaneous two photons or two protons.
In the case of photons, the photons it self after leaving the source move in the +X and -X direction.
In the analyzers the polarization of photons is measured in the +Y and - Y direction (at the left side) and the +Z and -Z direction (at the right side) which is perpendicular to the direction of motion.
In fact that are the only two axis measured. When you rotate the source horizontal the same is measured and when you position the source vertical also. What this means that with the experiment in Picture 134 you cannot test the Bell inequality with photons.

However what is more before you do the real test you should first perform some calibration tests with polarized photons. The idea is that when you use standard photons when you turn the left analyzer the number of photons detected with each angle should be identical. With polarized photons you should observe that the number of photons detected should vary. That means when one is at maximum the other should be zero.

Turning the polarizers acurately around the axis of rotation is technically very difficult because each polarizer services like a knife which "splits" the photons.

The case of protons is phyical completely different as photons. Protons have a spin, which can not be said of photons. The spin of a proton has an x,y and z direction. For a photon which moves in the x direction we can only speak of a Y and Z direction. The consequent is that the laws which describe the behaviour of protons is different. That means the "Bell inequality" theorem does not make sense for photons.

Reflection 5 - How important is the "Bell inequality"

The problem with the "Bell inequality" is that it is a mathematical equation. The question is which physical process does it describe correctly.
IMO the "Bell inequality" theorem is not a description of the experiment of picture 134 specific in relation to photons.
In order to understand the experiment of picture 134 you have to perform thousands of experiments. Only then you start to understand the concept of entanglement, which means that the two photons which leave the source have something in common i.e. there exists a correlation.
It is important that the source, the analyzer and the detector are all part of what is called a process and should be performed in an automatical way without any human intervention. That means the detector should be a camara or geiger counter which registers all events

Reflection 6 - Faster than light signals

           Detector A+                                        Detector B+     
                  _                                               _
                 |\                                                /|
                   \                                              /
                    \                                1           /
                 Analyzer 1 <--------- Source -----SW-----> Analyzer 2
                    /                               |2           \
                   /                                |             \
                 |/_                                |             _\|
            Detetector A-                       Analyzer 3     Detector B-
                                                  /   \
                                                 /     \
                                     Detector   C-     C+     

                                                Picture 4
The most important part of the experiment is the optical switch. When the switch is in position 1 the Analyzers 1 and 2 are used. When the switch is in position 2 the Analyzers 1 and 3 are used.
In order to perform the above experiment it is important that the angle between Analyzer 1 and Analyzer 2 is zero. The same between Analyzer 1 and Analyzer 3.
Before starting the real experiment it is important first to establish that with the optical switch in position 1 that the correlation between Analyzer 1 and Analyzer 2 is -1. That means that the detection of A+ coincides with B- and A- coincides with B+
The same with the optical switch in position 2: that the correlation between Analyzer 1 and Analyzer 3 is -1. That means that the detection of A+ coincides with C- and A- coincides with C+

What is the cause of this correlation or the lack of it ?
What is the cause of this correlation or the lack of it ?
  • For some the cause is in the source i.e. the chemical reaction which generates (decays) the two particles i.e. protons or photons.
  • For some the cause is in the analyzers or the detectors. That means measuring (observing) one influences the other.
That means communication is involved between the two sides. Even faster than light signals could be involved.
That means in this case:
  • That there is communication involved between Analyzer 1 and Analyzer 2 (when the switch is in position 1)
  • But this also means that there is communication between Analyzer 1 and Analyzer 3 (when the switch is in position 2)
The theory behind this experiment to change the switch rapidly between position 1 and position 2. This switching should be almost instantaneous.
You start with the switch in position 1 and immediate after the photon is generated you switch to position 2 and you will observe either a C- or a C+
Immediate after the next photon is generated you switch back to position 1 and you will observe either a B+ or a B-
The problem is you do not know when the events that generate the two photons happen
The only solution is to switch at a constant frequency (the same as the number of events counted, divided by the total time of each experiment) and observe what happens. The longer the distance between switch and the source the better, because the higher the chance that there is the possiblity that the switch is activated inbetween the source and the switch. Ofcourse there is a higher chance on errors.

But what does this test demonstrate ?
Most probably the result will be that even with such a fast switching mechanisme their will be the same -1 correlation. That means that when you measure 100 A+ you will also measure (approximately) 50 B- and 50 C-
But suppose completely the opposite happens. That means that when you measure 100 A+ you will also measure (approximately) 25B+ 25B- 25C+ and 25 C-
The central concept of the experiment is that it is symetric, meaning no preference for either analyzer and that the behaviour of each analyzer is identical.

IMO it is impossible to demonstrate that there is any communication between the two analyzers involved

Reflection 7 - Two Bell inequality with dice

This reflection is important in order to understand the importance of the Bell inequality.
The Bell inequality with dice consists of six flavours :
n(1) <= n(2) + n(3) + n(4) + n(5) + n(6)
n(2) <= n(1) + n(3) + n(4) + n(5) + n(6)
n(3) <= n(1) + n(2) + n(4) + n(5) + n(6)
n(4) <= n(1) + n(2) + n(4) + n(5) + n(6)
n(5) <= n(1) + n(2) + n(3) + n(4) + n(6)
n(6) <= n(1) + n(2) + n(3) + n(4) + n(5)
With n(1) meaning the total numbers of one's thrown in a certain test run consisting of n throws. n(2) means the total number of two's thrown etc.
Lets study those 6 equations with an example.

Reflection 8 - Two photon emission

  1. 10 October 2013
    Two-photon absorption and emission by Rydberg atoms in coupled cavities
    In the INTRODUCTION chapter of this document we read:
    In this context, the two-photon process, namely, the atoms transit from one energy level to another through an intermediate energy level that simultaneously involves two photons of the same frequency (the degenerate two-photon transition) or of different frequencies (the nondegenerate two-photon transition), has attracted great interest because it provides great opportunity for producing light with nonclassical properties
    Important is the text "great opportunity" which does not mean that they have succeeded.
  2. 30 November 2011
    Spontaneous Two-Photon Emission from a Single Quantum Dot
  3. See "Nature photonics" 2 March 2008:
    Observation of two-photon emission from semiconductors
    In the abstract of this document we read:
    Two-photon absorption in semiconductors has been extensively investigated; however, spontaneous semiconductor two-photon emission has not been observed, nor has it been fully analysed theoretically so far. We report the first experimental observations of two-photon emission from semiconductors and develop a corresponding theory.
    IMO the most important part are the experimental observations. This is an article of 2008 !!!
  4. The following document discusses two photon emission in general in paragraph 5.1.3: Chapter 5 Emission process
    It is not clear from this document to what extend the two photons are identical.
  5. 17 May 2007
    High-Rate Entanglement Source via Two-Photon Emission from Semiconductor Quantum Wells by Alex Hayat, Pavel Ginzburg and Meir Orenstein.
    This is the best article I have found which explains "two-photon emission"
    At page 4 of this document we read:
    The cavity with higher reflectivity top mirror will launch the majority of photon pairs to be emitted collinearly downwards in the -z direction. Photon pairs emitted collinearly downwards will have therefore opposite polarizations and can be separated by a polarization beam splitter.
    The reason while they have opposite polarizations is not obvious.

Reflection 9 - Remarks of 30 December 2015

  1. Remark 1. See (1) All physical laws, all physical process are completely independent of humans. Humans have no influence (in general) regarding the evolution of the world and how everything evolves. Specific human consious have nothing to do with this.
    A different issue is when humans perform experiments in order to discover the laws of nature. Humans can influence the outcome of these experiments. The general phylosophy ofcourse should be to perform the experiments independent of any human influence, because otherwise the laws of nature become biased or in error.
  2. Remark 2. See (2) What means n[A+B-]? n[A+B-] are the total number of protons of an experiment which have spin components A+ and B-. In such an experiment the spin components along the A axis and B axis are measured. When you perform such an exepriment with 100 protons n[A+B-] the theoretical value is 25. The same for n[A+C-] and n[B-C+]
  3. Remark 3. See (3) When entangled particles are involved you first have to demonstrate that the particles are entangled i.e. that negative correlation is involved.
    This requires a sequence of calibration experiments which each consists of at least 100 individual tests in order to see how accurate the correlation is.
There are three important conclusions of these calibration experiments:
  1. That the direction of the spin of the protons at one side are completely random and unpredictable.
  2. The the correlation between the spin of two protons at each instant is negative or -1.
  3. That these calibration experiments have nothing to do with the Bell inequality. They are merely a reflection of processes which can happen in the physical world, in which we live.

Reflection 10 - Remarks of 22 May 2018

When you study the Bell equality it is based around three parameters: the X,Y,Z values of spin components of protons or other elementary particles. Each parameter can have two values i.e. +X and -X and the chance of each is the same.
The result are 8 possible combinations i.e (+X,+Y,+Z) , (+X,+Y,-Z) untill (-X,-Y,-Z). The chance of each is 1/8.
In reality none of these combinations can directly be tested. Only the 12 combinations of two parameters. : (+X,+Y), (+X,+Z), (+Y,+Z), (+X,-Y), (+X,-Z), (+Y,-Z), etc until (-Y,-Z). Each of these combinaties has a chance of 1/4.
Combining the first three combinations you get the Bell inequality: (+X,+Y) < (+X,+Z) + (+Y,+Z)
Which makes sense, because on average 1/4 < 1/4 + 1/4 assuming that all individual chances are equal.

However, and that is important, when for a certain experiments the chances are not equal for all axis, this inequality could be invalidated, which is the case when the particles are entangled.
However, and that is important such an invalidation does not explain why or how these particles are entangled. For more information see for example:

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Created: 27 July 2014
Modified: 8 January 2016
Modified:22 May 2018

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