Jensen-Shannon Divergence and Non-linear Quantum Dynamics

TL;DR

This paper derives a non-linear quantum dynamical equation using Jensen-Shannon divergence, showing it preserves Hamiltonian structure and extends to pure state space, offering new insights into quantum evolution.

Contribution

It introduces a novel non-linear Schrödinger equation derived via Jensen-Shannon divergence, maintaining symplectic and Hamiltonian properties in quantum dynamics.

Findings

The non-linear Schrödinger equation preserves symplectic structure.

Hamiltonian dynamics are maintained in the non-linear extension.

The dynamics are projected onto the space of pure states.

Abstract

Using the statistical inference method, a non-relativistic, spinless, non-linear quantum dynamical equation is derived with the Fisher information metric substituted by the Jensen-Shannon distance information. Among all possible implications, it is shown that the non-linear Schr\"odinger equation preserves the symplectic structure of the complex Hilbert space, hence a Hamiltonian dynamics. The canonically projected dynamics is obtained on the corresponding projective Hilbert space of pure state density operators.