Since 1927, although Einstein was considered to have played a fundamental pioneer role in the birth of quantum physics and was himself recognizing the quantum theory’s significant achievements, he started to have some reservations with some of its fondamental aspects and constantly argued against the pretention of its founders and proponents to have settled a definitive theory^{[1]}.

How could the great scientist accept the **essential statistical nature** of the quantum theory, which ceases to be deterministic as soon as we try to measure an observable^{[2]}? How could he give a full credit to a theory which gives such an important place to **the observation ** or more precisely to** the act of measuring**?

Einstein started to wonder whether it was possible, at least in principle, to ascribe certain properties to a quantum system i**n the absence of measurement**: both indeterminism and irrealism so far inherent to the quantum interpretation would then begin to crack.

The so called 1935 **EPR**^{[3]}** paper** Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? (Phys. Rev. 47, 777 (1935) – Published 15 May 1935) challenges more particularly the prediction of quantum mechanics that it is impossible to know both the position and the momentum of a quantum particle.

The articles begins with the logical disjunction of these two following assertions as a first premise; one or another of these must hold:

- (1) the description of reality given by the wave function in quantum mechanics is **not complete**

- (2) two physical quantities described by non-commuting operators cannot have simultaneous **reality**

According to the Copenhagen interpretation of quantum mechanics, (1) is false^{[4]} and (2) is true.

The aim of the EPR article is for the authors to show, on the contrary, that** (1) is true and (2) is false**.

To demonstrate that the assertion (2) is false, the two important terms of the precedent assertions have to be preliminarily defined.

The authors offer only a necessary condition for **completeness**; namely, that for a complete theory

"Every element of the physical reality must have a counterpart in the physical theory"

And for the purpose to determine when a quantity has **reality**, or a definite value, they offer a minimal sufficient condition:

"If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of reality corresponding to that quantity."

Then, in paragraph 2, they proceed by sketching an iconic thought experiment whose variations continue to be important and widely discussed. The experiment concerns two quantum systems I and II that are spatially distant from one another, perhaps quite far apart, but such that **the total wave function for the pair links both the positions of the systems as well as their linear momenta**.

In the EPR example the total linear momentum is zero along the x-axis. Thus if the linear momentum of one of the systems along the x-axis were found to be p, the x-momentum of the other system would be found to be −p. At the same time their positions along x are also strictly correlated so that determining the position of one system on the x-axis allows us to infer the position of the other system along x. The paper constructs an explicit wave function for the combined (I + II) system that embodies these links even when the systems are widely separated in space.

So what the authors have just shown is that it is possible to find an example of two non-commuting operators (namely position and momentum)^{[5]} that can be considered to have a definite value for the same particule. Hence the proposition (2) is false and necessarily (1) holds true. They are

"thus forced to conclude that the quantum-mechanical description of physical reality given by wave functions is not complete"

However, in the penultimate paragraph of EPR they address the final problem of getting real values for incompatible quantities **simultaneously**.

Indeed one would not arrive at our conclusion if one insisted that two or more physical quantities can be regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted. … This makes the reality [on the second system] depend upon the process of measurement carried out on the first system, which does not in any way disturb the second system. No reasonable definition of reality could be expected to permit this.

After completeness and reality, the paper needs to define a third fundamental notion, **locality**, from which simultaneity can be saved. Actually, if systems are spatially separate,

the process of measurement carried out on the first system does not disturb the second system in any way.

In summary, separated systems as described by EPR have **definite position and momentum values simultaneously**. Since this cannot be inferred from any state vector, the quantum mechanical description of systems by means state vectors is incomplete.

[1] "*Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the 'old one'. I, at any rate, am convinced that He is not playing at dice*." Einstein to Max Born, letter 52, 4th december 1926.

[2] We should recall here that between observations, the state of a system evolves in a perfectly definite way, according to the time-dependent Schrödinger equation. But even if **we know the state-vector exactly**, we **don't know the result of any given measurement**.

[3] After the names of its co-authors Albert Einstein, Boris Podolsky and Nathan Rosen.

[4] In quantum mechanics, it is usually assumed that the wave function **does contain** a complete description of the physical reality of the system in the state to which it corresponds.

[5] According to the Heisenberg Uncertainty Principle, we know that **[X,P] = XP - PX** = ih/2π (so [X,P] != 0 and those operators do not commute)