Effective Mode Approximation for Probabilistic Verification of Collective Hamiltonians in Large Continuous-Variable Quantum Systems

The Effective Mode Approximation (EMA) is a verification-oriented framework for characterizing collective Hamiltonian dynamics in large continuous-variable (CV) quantum systems from experimentally accessible collective measurements. Rather than reconstructing a full mode-resolved Hamiltonian, EMA maps the observed dynamics onto a canonically normalized collective mode and tests whether summed quadrature trajectories are consistent with an effective harmonic description. We validate EMA using time-resolved homodyne sampling in Gaussian simulations of ring-coupled multi-qu-mode optical systems with N=8,16,32, and 64 modes. One-tone and two-tone sinusoidal models, selected using the Akaike Information Criterion (AIC), recover a stable dominant collective frequency across system size and produce residuals that remain centred near zero. The results show that EMA can verify dominant collective behaviour with a fixed number of effective parameters even when full microscopic reconstruction is impractical. EMA is therefore best understood not as a full-state ansatz, but as a low-overhead tool for validating collective dynamics under realistic measurement constraints in scalable CV hardware.