Fig. 1: EPR entanglement from elastic collision.

– Before the collision, at time t₀ (left half), particle A with mass m_A rests at 〈x_A(t₀)〉 = 0 with a large Gaussian position uncertainty and negligible, strongly squeezed momentum uncertainty. The entanglement is produced by a single collision with particle B having the mass m_B = 3m_A, a high momentum \(\langle {\hat{p}}_{{\rm{B}}}({t}_{0})\rangle \,\gg\, \Delta {\hat{p}}_{{\rm{B}}}({t}_{0})\,\gg\, 0\), and a position \({\hat{x}}_{{\rm{B}}}({t}_{0})=-{x}_{0}\) with negligible, strongly squeezed position uncertainty. After the collision, at time t₁ (right half), measurements are performed. The two masses have the same momentum \(\langle {\hat{p}}_{{\rm{B}}}({t}_{1})\rangle \,=\,\langle {\hat{p}}_{{\rm{A}}}({t}_{1})\rangle\) due to the mass ratio 1:3 and the conservations of momentum and energy. The momenta are even identical for any individual ensemble measurement ‘i’ since the momentum uncertainties (almost) exclusively originate from particle B. The measured values \(\langle {\hat{p}}_{{\rm{A,B}}}({t}_{1})\rangle +\delta {p}_{{\rm{A,B,i}}}({t}_{1})\) (vertical dashed lines in the Gaussian distributions top right) are always identical, i.e. the differential values show no quantum uncertainty. The momentum uncertainties of the two particles are quantum correlated. The initial position uncertainty \(\Delta {\hat{x}}_{{\rm{A}}}({t}_{0})\,\gg\, 0\) gets also distributed onto both particles. The right half of the Gaussian uncertainty corresponds to a statistically later collision, which results in a later establishment of new velocities. The collision halves the velocity of mass B, and mass A is accelerated to 3/2 of the initial velocity of B. The position uncertainty of A is therefore mirrored at its centre line and compressed by a factor of 1/2 due to A's uncertain initial position, see Eq. (11) while B takes over the other half of A's position uncertainty without a change of sign (see supplement). In conclusion, the position uncertainties of particles A and B after the collision are quantum anti-correlated. By measuring either A or B we are in a position to predict with certainty, and without in any way disturbing the second system either the value of the quantity [x] or the value of the quantity [p]. My complemented version of the EPR thought experiment makes obvious that the description by the wave function is complete. Hidden variables are not motivated by the EPR thought experiment.