Exact finite-dimensional reduction of Wigner dynamics for open quantum systems using EGQPDs
Simulating non-Gaussian quantum dynamics in open continuous-variable systems is notoriously challenging: exact methods scale exponentially, while approximations sacrifice non-classical features such as Wigner negativity. Here we introduce an exact finite-dimensional reduction that overcomes this curse of dimensionality. We prove that the algebra of extended Gaussian quasi-probability densities (EGQPDs) is closed under Lindblad evolution with linear jump operators, reducing the infinite-dimensional Wigner PDE to a closed system of ODEs that scales polynomially with the number of modes. A necessary and sufficient condition for closure is established: the algebra remains closed iff every jump operator is at most linear; otherwise the extended algebra with polynomial prefactors is required. We further show that the minimal number of Gaussian components defines a non-Gaussian rank—a discrete complexity measure monotonically non-increasing under the dynamics. The framework yields explicit ODEs for the component parameters, admits a rigorous density theorem and error propagation bound, and provides closed-form algebraic criteria for Gaussianity, purity, and entanglement. Seven numerical experiments validate the method across Gaussian and non-Gaussian states, entanglement decay, PT-symmetric exceptional-point dynamics, and complexity scaling, demonstrating machine-precision accuracy and exponential speedups over grid and Fock methods. The EGQPD framework provides a systematic, exact, and efficient toolbox for non-Gaussian open quantum dynamics.