There is a growing interest, both from the theoretical as well as experimental side, to test the
validity of the quantum superposition principle, and of theories which explicitly violate it by
adding nonlinear terms to the Schrödinger equation. We review the original argument elaborated by
Gisin (1989 Helv. Phys. Acta 62 363), which shows that the non-superluminal-signaling condition
implies that the dynamics of the density matrix must be linear. This places very strong constraints
on the permissible modifications of the Schrödinger equation, since they have to give rise, at the
statistical level, to a linear evolution for the density matrix. The derivation is done in a
heuristic way here and is appropriate for the students familiar with the textbook quantum mechanics
and the language of density matrices.