constitute a complete description of the spin states of the two particles but that it is an incomplete description of the spin state of a set of particle pairs39. Within this set, each pair of particles should be describable by a state of the form | ↑1,n⟩ ⊗ | ↓2,n⃗⟩ or | ↓1,n⃗⟩ ⊗ | ↑2,n⃗⟩ in which the spin of particles 1 and 2 along a given direction n⃗ is not random but predetermined by hidden variables.

Einstein and his colleagues also want to demonstrate that the type of state (9) allows an internal contradiction in quantum mechanics to be revealed. For this, they define the concept of an element of reality: “If, without disturbing a system, one can predict with certainty (that is, with a probability equal to one) the value of a physical quantity, then there exists an element of reality corresponding to that physical quantity”.

In the experiment described above, due to the strict anti-correlations between the spin measurements of the two particles along the direction n⃗, it is possible, by measuring the spin of particle 1 along this direction, to predict with certainty the result of the measurement of the spin of particle 2 along the same direction, without disturbing it. According to the definition given by EPR, this demonstrates that there exists an element of reality associated with this physical quantity.

But, as mentioned above, the use of a spin singlet state makes the direction n⃗ arbitrary, so it seems possible to extend the reasoning to any component of the spin of particle 2, and to associate it with an element of reality. However, this reasoning clashes with a fundamental rule of quantum mechanics, formalized by relations (8), according to which two (and, a fortiori, three) components of the spin cannot be simultaneously well-defined. In other words, it is impossible for them to correspond, at the same time, to elements of reality.

For supporters of realism, this thought experiment highlights a fundamental flaw in quantum mechanics, opening the way to the idea that it could be completed by a hidden variable theory. However, in the orthodox interpretation, or Copenhagen interpretation, and in particular according to Niels Bohr, this experiment demon-strates nothing, as its premises are considered incorrect. The state |Ψ⟩ in no way represents an objective reality of the system formed by particle 1 and particle 2 but is merely a tool for predicting the possible results of measurements made in a specific experimental context. Now, the considerations of Einstein and his colleagues concern various hypothetical40, mutually exclusive measurements relative to a single state41; they are, therefore, without foundation. Bohr rejects thus the notion of an element of reality in the sense of EPR and, even more so, their conclusion of the incompleteness of quantum mechanics.

An absolutely remarkable fact is that physicist John Bell, who, like Einstein, believed that quantum mechanics was an incomplete theory, was able to, based on the EPR thought experiment, conceive of a test in the form of inequalities, known as Bell’s inequalities, which must be respected by a deterministic and local hidden variables theory. Another remarkable fact is that this test does not depend on the nature or structure of the hidden variables, other than the fact that they respect the aforementioned properties. The derivation of these inequalities is somewhat technical: for that, the reader is referred to Bell’s collection of enlight-ening articles42. The experiments, equally remarkable, testing these inequalities were primarily conducted by Alain Aspect and his collaborators between 1980 and 198243—based on a proposal made by Clauser, Horne, Shimony, and Holt in 196944—and then, in a much more sophisticated manner, by Hensen and collab-orators in 201545, to exclude a number of possible loopholes, referred to as "dead zones." These experiments all show a significant violation of Bell’s inequalities and thus exclude the possibility of local hidden variables. This means that a deter-ministic hidden variables theory must inevitably exhibit properties incompatible with relativistic invariance.