The simulations (of these digital EPR experiments) presented in this article allow obtaining a systematic violation of Bell’s inequalities, meaning that these simulations verify an idealization of a non-local reality. Unlike the theories with hidden variables, conventionally designed not to violate these inequalities (i.e. not to exceed a value of 2), in our model, we will only be able to have values greater than 2 (the violation predicted by the QM being of about 2.83). Figure 1 is the schematic diagram that corresponds to the ideal EPR experiment that we will model and simulate:

In the context of our simulation, the source will correspond to the random generation not of one direction but of a couple of arbitrary directions. Each direction will be characterized by an angle between 0˚ and 180˚. We will define all these angles with respect to an absolute frame of reference. The length of the segment carrying this direction is not important in our experience. We can assume it is equal to 1. We can thus represent our source objects in the form of crosses with branches that are more or less spaced (cf. Figure 2), the 2 branches corresponding to diameters are the couple of directions.

1^st important remark: The fact that the source provides 2 characteristics for the same source object is the 1^st important point of this idealization. As we will see later, it is certainly the keystone of this modeling class.

A detector will symbolically consist of 1 circle divided into 4 sectors of 90˚ each (see Figure 3) defined on a frame of reference fixed on this circle. The 4 sectors in this frame of reference are always between 0˚ and 90˚ for the 1^st sector, 90˚

and 180 for the 2^nd, 180˚ and 270˚ for the 3^rd and 270˚ and 360˚ for the 4^th. We assign a value +1 to the 1^st and 3^rd sectors and a value −1 to the 2^nd and 4^th sectors. This value will be used to define the output value of the detector. The orientation of the detector will be defined in the absolute frame of reference mentioned previously (i.e. the same as for the source objects) by the angle of its axis noted “0˚” in Figure 3. The detector will analyze the 2 directions of the source object by applying the following rule:

If the 2 directions of the source object are in one or two sectors of the same value, the detector outputs the value of the sector(s).

2^nd Important remark: This rule allows any orientation of the detector to potentially provide a value for any source object. On the other hand, this also means that for a fixed orientation of the detector, certain source objects may not provide a value (those whose directions are in 2 sectors of different values) and therefore seen as not detected. This is the 2^nd crucial point of our modeling (direct consequence of the previous “1^st Important remark”).

Because non-detections are certainly the most delicate point to accept at first glance, we will devote a dedicated section to them in which we will show that they:

・ Are impossible to avoid.

・ Cannot correspond to a 3^rd value (which would complete the possible outputs “+1” and “−1”).

・ Are of a different kind than the loopholes of real EPR experiments.

The 2 previous important remarks will be the fundamental characteristics of the modeling class represented by the model that we are going to study. But let’s continue to describe our modeling.

Our source objects are defined by a couple of direction. We will determine the 1^st direction of the couple by randomly choosing a 1^st angle between 0˚ and 180˚ with respect to the absolute frame of reference. We will then randomly choose a second angle between 0˚ and 180˚ to determine the deviation from the 2^nd direction to the 1^st direction (what is called spacing in the following). This second angle will therefore be added to the angle of the 1^st direction to define the 2^nd direction. We could very well have directly defined the 2^nd angle in the same way as the 2^nd direction (and that would not have changed anything to the results presented in this article). We define the 2^nd element of the couple rather as the spacing to the 1^st direction because our experiences will show that this spacing is a structuring parameter of our modeling. Indeed, the different values of violation of the Bell inequalities will depend on this parameter. This representation of the couple will therefore be more relevant.

Therefore, there are two ways to introduce this angular spacing as a parameter of our idealization (named SpacMax). First of all, the value of SpacMax is defined and fixed for all the trials of the experiment and is chosen between 0˚ and 180˚. It is this SpacMax parameter that will be structuring to obtain a variety of Bell inequality violation values. But it can be used either as a fixed spacing (at each trial, the spacing would be the same and would be SpacMax) or as a maximum angle of spacing (at each trial, the spacing would be randomly chosen in the interval [0˚; SpacMax] rather than [0˚; 180˚], it would then change at each trial). In our study, we mainly focus on the choice of the maximum angle of spacing. Once again, we could very well define a fixed spacing for the whole experiment, meaning that each source object would have 2 directions with always the same spacing and only the direction of this couple that would change at the source. We would then systematically obtain the violation of Bell’s inequalities again. We don’t make this choice for this modeling because, as we will see it hereafter, even if one obtains the expected QM results, this idealization (with fixed spacing) gives curves which appear slightly worse than those obtained with maximum spacing.

Let’s start by checking the expected statistics of basic QM experiments.

Concretely, Table 1 presents the implementation of the orientation of the detector (column D) and the definition of the source objects, 1^st direction (column E) and spacing (column F) in a spreadsheet. Line 1 gives the formulas used:

The role of the modulo (in the definition of the 2^nd direction, column F) is to provide a definition of the couple in the upper semicircle. When the angle defining the 2^nd direction (sum of the angle defining the position of the 1^st direction and the spacing) exceeds 180˚, the 2^nd direction (red line before modulo in Figure 4) is then

Table 1. Description of the implementation’s elements of the source objects.

defined by its semi-line (blue line after modulo) contained in the upper semi-circle (Figure 4).

Thus any couple has 2 ways of defining itself, for example in the case of Figure 4, (1^st dir = 145˚; Spacing = 50˚) which will be translated into (1^st dir = 145˚; 2^nd dir = 15˚) or (1^st dir = 15˚; Spacing = 130˚) which will be translated into (1^st dir = 15˚; 2^nd dir = 145˚). This “duplication” has no impact on statistics because it is shared by all source objects.

To determine the outputs of the detector (Table 2), we convert the 2 directions of the source object in the detector frame of reference (column G and column H) by removing the orientation of the detector (the 0˚ axis of Figure 3). Then, we look in which sectors these 2 directions are to obtain the result “+1” or “−1” or no result (column I) which corresponds to the rule mentioned previously. The implementation used is shown in line 1:

To analyze the results of the detector (Table 3), we separately count the “+1” (column J and O), the “−1” (column K and P) and the number of detected events, i.e. giving a result (column N). For information, we also count the number of trials (column L) and the number of undetected events (column M). Here is the implementation in line 1:

For a random choice of the orientation of the detector for each trial, we always obtain a probability of 0.5 to have “+1” and 0.5 to have “−1” and whatever the value of spacing (SpacMax) of the directions (Table 4) in agreement with the expected results of the QM:

We obtain the same results if the orientation of the detector is fixed for all the