This paper unifies deterministic and stochastic Electromagnetic Information Theory (EIT) through symplectic geometry. For spatially incoherent sources, both formulations yield identical eigenvalues and spatial Degrees of Freedom (NDF). This equivalence is shown to be a structural necessity: the radiometric étendue, the Hamiltonian phase-space volume, and the NDF are the same symplectic invariant of the source--observer configuration. Liouville's theorem guarantees conservation of the NDF under lossless propagation; Gromov's Non-Squeezing Theorem establishes an irreducible minimum phase-space cell, setting a fundamental geometric bound on resolving power. The physical manifestation of this symplectic structure is the formation of \textit{Spatial Information Flows} (SIFs) -- level-set curves of high mutual information which, for convex sources with rotational symmetry, coincide with the optimal sampling curves of Bucci et al. Spatial information in electromagnetic fields is therefore governed by the geometry of the source--observer configuration, providing the foundation for a geometric theory of electromagnetic information.

翻译：本文通过辛几何统一了确定性理论与随机理论下的电磁信息理论。对于空间非相干源，两种理论可导出相同的特征值和空间自由度。这种等价性源于结构必然性：辐射光学扩展量、哈密顿相空间体积与空间自由度均属于源-观测构型的同一辛不变量。刘维尔定理保证了无损传播过程中空间自由度的守恒性；格罗莫夫非压缩定理则确定了不可约的最小相空间胞元，从而为分辨能力建立了基本几何界限。该辛结构的物理表现为“空间信息流”的生成——即高互信息的水平集曲线，对于具有旋转对称性的凸源，该曲线与布奇等人的最优采样曲线完全重合。因此，电磁场中的空间信息由源-观测构型的几何特性所支配，这为电磁信息的几何理论奠定了坚实基础。