Information Dynamics in Quantum Harmonic Systems: Insights from Toy Models

This study explores quantum information dynamics using a toy model of coupled harmonic oscillators, focusing on the interplay between mutual information, synchronization, and circuit complexity. We examine how variations in coupling strength, detuning and external factors, such as a magnetic field, influence information flow and computational metrics. Through exact Gaussian methods we determine the circuit depth for generating target states, examine time-dependent effects, and show that increased fidelity corresponds to more regular behavior. In the context of ion transport, we compare sudden and adiabatic protocols, quantifying their fidelity-complexity through a nonadiabaticity metric. This analysis demonstrates the superior performance of smooth control sequences in minimizing operational errors. We also establish synchronization and mutual information as complementary but distinct measures of quantum correlations, with particularly divergent behavior in nonlinear regimes.