No-faster-than-light-signaling implies linear evolutions. A re-derivation

TL;DR

This paper reviews how the principle of no faster-than-light signaling constrains quantum dynamics, showing that any modifications to the Schrödinger equation must produce linear density matrix evolution to prevent superluminal communication.

Contribution

It re-derives Gisin's argument demonstrating that non-linear quantum evolutions violate no-signaling, emphasizing the necessity of linearity in density matrix dynamics.

Findings

Nonlinear modifications lead to superluminal signaling.

Linearity of density matrix evolution is necessary for no faster-than-light communication.

The derivation clarifies constraints on alternative quantum theories.

Abstract

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 Schr\"odinger 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 Schr\"odinger 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.