(resp. particle 2) is in the state (resp. ) according to the direction n⃗ ; in the second state, it is the reverse.

A fundamental feature of the state |Ψ⟩, called the singlet spin state, is that the expression (9) remains valid regardless of the direction n⃗ chosen for the spin pro-jection, whether it is the directions n⃗ = x⃗, y⃗ or z⃗, or any linear combination of these. Another fundamental feature of the state (9) is that the particles are in what is called an entangled state. Mathematically, this means that this state cannot be factored as follows:

|Ψ⟩ ≠|Ψ1⟩ ⊗ |Ψ2⟩(10)

where |Ψ1⟩ and |Ψ2⟩ would be states associated with particles 1 and 2, respectively. The consequence of this entanglement is that measurements of the spin of par-ticles 1 and 2, performed in a given direction, show strict (anti-)correlations: if the measurement of the spin of particle 1 in a direction n⃗ equals (resp. ), then the measurement of the spin of particle 2 in the same direction will, with probability 1, equal (resp. ).

Now imagine that particles 1 and 2 move apart in opposite directions. Two observ-ers, O1 and O2, then decide to measure the spin projection in a direction n⃗, while the particles are in sufficiently distant regions and at such moments that no signal could have propagated from one to the other during the measurement process. This ensures that a measurement made on one of the particles, say particle 1, cannot influence the physical situation of particle 2. This hypothesis, known as Einstein’s separability or locality, is based on the principle that no interaction can propagate faster than the speed of light.

Now, let’s consider a set of such systems and the corresponding sequence of meas-urements. Observer O1 measures the spin of particle 1 in an arbitrary direction n⃗ and obtains the following sequence38:

(11)

For this observer, the sequence of results shows no regularity; it appears, in accordance with the principles of quantum mechanics, as completely random. The form of state (9) only indicates that, for observer 1, unaware of the fate of particle 2, particle 1 is in a superposition of states | ↑1,n⃗⟩ and | ↓1,n⟩, with eigenvalues and , and that the probabilities of measuring these two values are identical, each being equal to .

On the other hand, observer O2, measuring the spin of particle 2 in the same direc-tion n⃗, obtains the following sequence:

(12)

This sequence is obviously equally random. There is no issue as long as we limit ourselves to performing measurements and recording the results.

However, if now the two observers, O1 and O2, decide to compare their results, they will observe the existence of strict anti-correlations between the spin meas-urements of the two particles along the direction n⃗: every time the measurement of the spin of particle 1 gives (or ), the measurement of particle 2 gives (or ).

What will they conclude from this series of experiments, where the spin meas-urement results of particles 1 and 2, which are believed to be random, are system-atically anti-correlated? They will conclude that the state |Ψ⟩ certainly does not