Entanglement as complex astigmatism: geometry of the multimode Gaussian ray

Abstract We present a geometric formulation of multimode Gaussian dynamics based on the complex matrix trajectory Z ( t ), governed by the generalized Newtonian equation Z ̈ + Ω 2 ( t ) Z = 0 subject to the symplectic area constraint. While physically equivalent to the covariance matrix formalism, this approach reduces the computational complexity for time-dependent potentials. We identify Gaussian entanglement with complex astigmatism —the off-diagonal imaginary components of Z —and prove that these components constitute a geometric invariant under local symplectic transformations. Applying this framework, we derive: (1) the Hong–Ou–Mandel dip as a consequence of symplectic volume exclusion in the antisymmetric mode; (2) EPR steering correlations as a necessary condition of the global purity constraint; and (3) an analytic relation for the Schmidt number K = 1 + 4 χ 2 / ℏ 2 , linking high-dimensional entanglement directly to phase-space eccentricity. This provides a deterministic geometric method for analyzing coherence and correlations in continuous-variable systems.