Quantum Mechanics formalism remains difficult to understand and sometimes is confusing, especially in the explanation of ERP paradox and of Bell’s inequalities with entanglement photons. So a chart of conversion, in which elements are named differently, is proposed. Next, experiment about Bell’s inequalities violation is described in another way, and we hope a clearer one. Main result is Bell’s inequalities would not be violated! The explanation would come from confusion between the definition of the correlation function S1, and a property S2. And consequently, Einstein, Podolski and Rosen would be right on the local “hidden” variable.
“God does not play dice” Einstein wrote. Where there appears to be random, there is only a cause we have not yet found or discriminated. EPR paradox [
Quantum Mechanics Formalism is difficult to understand [
・ A (or B) can mean Photon and/or Separator
・ x (or y) can mean Photon and/or Axis
・ Alice (or Bob) can mean a detector and/or its complementary detector
・ The same axis can be called either (Ox’) or (Oa)
Also some parameters are not precise enough, or are surprising. For example:
・ x,x> symbolizes the polarization status of a photon pair. The first x is for the first photon, the second x is for the second photon. But there are twodifferent photons. So the idea will be to distinguish the two photons with the letters X and Y.
・ When a photon is detected, the value of the measure is +1. When it is not detected, it value of the no measure is −1, which is quite surprising. Usually, when we cannot detect an object or a phenome, the value is 0. To avoid disturbing to much our reader, we will have here the convention to keep the convention −1 when a photon is not detected.
To correct the confusing term, we will use in this article the conversion Chart 1 hereafter.
Quantum mechanics introduces a wave function
with:
 I,J: axes; _{}
 X,Y: photons;_{ }
I> characterizes the polarity of the photon X according to the axis I.
Let us have a simple set up with a single separator, as seen in
1) First, separator A is adjusted with an angle
When we measure these photons on the detector DA, we will find {+} each time (and on the detector \DA, we will find nothing, i.e. {−})
So each time
It is said in QM the polarization (of this photon) parallel to I is exactly defined.
2) Then, separator A is adjusted with an angle
And let us call L the axis which has an angle of 45˚ with the axis I
These photons still have a polarity according to I (to simplify, we can say these are the same photons again)
When we measure these photons on the detector DA (with the new value of angle
It is the said in QM the polarization (of the photons) parallel to L is not defined.
Remark: it is “not” the polarization of the photons X which has changed; it is only the angle α to measure them which has changed. More generally, we can say that, except with the particular case of
Let us now use the proposed formalism to check the experimental set up used to prove Bell’s inequalities violation
Chart 1. Chart of conversion.
(
with
 a source which emits entangled pairs of photons (X,Y) in green;
 two channels separators A and B which direct photons according their polarization toward the detectors;
 detectors DA and DB which detect photons polarized according to α and β;
 detectors \DA and \DB which detect photons polarized according to \α and \β;
 a the measure done on DA;
 \a the measure done on \DA;
 b the measure done on DB;
 \b the measure done on \DB;
As a consequence of the wave function, there is a property associated
where
Another author [
A mathematical integral is nothing less than a sum (or an average sum).
This sum (or this average sum) S2 is called by Quantum mechanics theory: a correlation function property!
What is
Remark: it is a bit confusing to call “sum” an operation with a subtraction (minus “−”) in the formula. But we keep this QMdesignation of “sum”, adding the term “quantum”.
On the Chart 2 hereafter, we can check all the possible value for the quantum sum qs:
The quantum sum
In practice, c is the complementary measure of a. In other word,
We have seen on the previous paragraph on the quantum sum (§3.5) that
By mathematical construction, S2 cannot be greater than 2.
This property is called Bell’s inequalities
The correlation function S1 is defined as
with
E(α,β) the correlation coefficient.
We have to note that the correlation function is different of the correlation coefficient (and of the total corre lation). To define the correlation coefficient, we first need to explain what the probabilities in Quantum Mecha nics are.
Entanglement means the 2 photons of the pair emitted by the source have the same polarization.
Because produced by the same source, photons pairs are entangled photons.
1) Let us have a single photon pair (X, Y).
Chart 2. Possible values of the quantum sum qs.
It is supposed they have the polarity I,I>. Because they have the same polarity (I = I), they are entangled photons. They would also be entangled photons when they have the polarity J,J>.
When we measure these photon pairson DA (with the value of angle
2) Using a photons source, each pair of photons will have a different polarization (but the two photons of each pair have the same polarization). Because there is no preferential polarization at the source, 50% of these photons will be received on DA and 50% of these photons will be received on \DA. Please check Chart 4.
It is the said in QM there is total correlation between the measures on DA and on DB.
3) Then, when we measure these X photons on DA (with the new value of angle α = 45˚), we will find {+} in 50% of the cases and {−} in the other 50% cases (and on \DA we will find {−} in 50% of the cases and {+} in 50% of the cases). On the same time, when we measure these Y photons on DB (with the new value of angle β = 45˚), we will find {+} in 50% of the cases and {−} in the other 50% cases (and on \DB we will find {−} in 50% of the cases and {+} in 50% of the cases). So we will find on DA and DB: 25% of [+1, +1], 25% of [+1, −1], 25% of [−1, +1] and 25% of [−1, −1].
It is the said in QM there is total decorrelation between the measures on DA and on DB.
It has been transcribed in Chart 5 the results with the same 8 photon pairs.
Chart 3. Measure of a single pair.
Chart 4. Measures with α = 0˚ and β = 0˚.
Chart 5. Measures with α = 45˚ and β = 45˚.
Remark: we can do the same remark than this of the previous paragraph: it is not the polarization of the photons (X, Y) which has changed; it is only the angle α to measure them which has changed. More generally, we can say that, except with the particular case of α = 0 (or 180˚) and β = 0˚ (or 180˚), there would be between the measures on DA and DB decorrelation according to QM!
1) Let us first remind the definition of the probability of an event happening: it is the number of ways it can happen out of the total number of outcomes. In Quantum Mechanics,
2) If we take again the previous example of §4.3.b with 8 photon pairs, with the separators angles: α = 0˚ and β = 0˚
3) If we take again the previous example of §4.3.b with 8 photon pairs, but with the separators angles: α = 45˚ and β = 45˚
And then we cancheck that:
That means that with exactly the same 8 pairs of photons, probabilities will be different according we take a separator angle or another separator angle. The event the probability measure here is not the photons polarization, but it is the effect on photons detector by the separator angle.
In QM, it is defined a quantum correlation coefficient E(α;β) such as:
Remark: E(α,β) is different of the (mathematical) statistical correlation. E(α,β) can take any value in the interval [−1; +1]
When we apply this coefficient to:
 the first example (cf §4.4.b) with the 8 photons pairs:
it is said there is total “correlation” in QM.
 the second example (cf §4.4.c) with the “same” 8 photons pairs
it is said there is total “decorrelation” in QM.
With the examples (b) and (c), it is the same 8 photons pairs, but QM says on one hand there is total correlation and on the other hand there is total decorrelation about photons. In fact, it is depending of the angles of the separators. The quantum correlation coefficient E(α;β) does not measure the polarization of the photon, but the effect of the separators angles (α;β) on polarized photons.
1) Probability property
A property found in QM is
Results are shown in
2) Correlation coefficient property
Due to the random sequence:
So
And due to mathematical property on the angle:
And due to the randomsequence (remind −1 means no detection); by symmetry:
So
By definition of the Quantum correlation coefficient
Due to mathematical property on the angles
Results are shown in
Application: When α = β or when α = β + 180˚, E(α,β) = 1; it is said in QMthere is correlation.
Until here, all the results are determined; there is no uncertainty, no chance, no probabilities in the results.
3) Consequence on the correlation function S1:
Because,
Let us have the angles for the separator A which can take either the value 0˚, or the value 45˚. To clearly distinguish them, we will call the first possible value α = 0˚ and the second possible value γ = 45˚. Idem for the separator B with the angles: β = 22.5˚ or δ = 67.5˚. Please have a look on
If we use the property of Equation (25):
Chart 6. Values of the quantum correlation coefficient E.
So,
Definition of
It is only the QM theory which postulates that:
where (please check again the Conversion chart of §2.2)


This Equation (31) is not argued, not justified, and so it appears for us to be wrong!
Let us remind
 Equation (30) : S1 can reach the value 2.8
 Equations (2) and (4) called Bell’s inequality: S2 is equal or lower than 2
Then, due to the unjustified Equality (31) where
It is here proposed a conversion chart in order to better distinguish elements when necessary.
The main result of this paper is to distinguish the correlation function: the definition itself S1(α,γ,β,δ) on one side, and a property S2(a,c,b,d) on the other side. The equality between S1 and S2 is not obvious, not argued and maybe wrong.
The demonstration of the Bell’s inequalities violation could come from confusion between the definition S1 and a property S2 of the correlation function. And so it would mean that Bell’s inequalities violation is not strictly justified. Consequently, Albert Einstein, Boris Podolsky and Nathan Rosen with the ERP paradox could be right on the local hidden variable.
I would like to thank people who did the proofreading.
OlivierSerret, (2015) A Chart of Conversion Supporting EPR Paradox vs. Bell’s Inequalities Violation. Journal of Modern Physics,06,19501960. doi: 10.4236/jmp.2015.613201