Entanglement as Complex Astigmatism: Geometry of the Multimode Gaussian Ray

We present a geometric formulation of multimode Gaussian dynamics based on the complexmatrix trajectory Z(t), governed by the generalized Newtonian equation ¨Z + Ω2(t)Z = 0 subjectto the symplectic area constraint. While physically equivalent to the covariance matrix formalism,this approach reduces the computational complexity for time-dependent potentials. We identifyGaussian entanglement with complex astigmatism—the off-diagonal imaginary componentsof Z—and prove that these components constitute a geometric invariant under local symplectictransformations. Applying this framework, we derive: (1) the Hong–Ou–Mandel dip as a consequenceof symplectic volume exclusion in the antisymmetric mode; (2) EPR steering correlations as anecessary condition of the global purity constraint; and (3) an analytic relation for the Schmidtnumber K = p1 + 4χ2/ℏ2, linking high-dimensional entanglement directly to phase-space eccentricity.This provides a deterministic geometric method for analyzing coherence and correlations in continuous-variable systems.