Balanced von Neumann entropy inequalities for bosonic and fermionic Gaussian states

When arbitrary finite local dimensions are allowed, bosonic Gaussian states and nondegenerate classical Gaussian block vectors satisfy exactly the same unconditional balanced linear entropy inequalities. We establish this equivalence for von Neumann entropy, extending the known Rényi-2 correspondence. Spectral entropy integrals and compatible realification transfer classical determinant inequalities to bosonic states and, in the forward direction, to even fermionic Gaussian states under complete-mode canonical anticommutation relation (CAR) restrictions. A high-temperature limit gives the bosonic converse. Both transfers include pure modes and imply the Zhang-Yeung non-Shannon inequality. We also derive quantitative independence bounds from conditional mutual information ε = I(A:C | B). For bosons, bounded covariance or total photon number gives a trace-distance bound C√ε for ρAC relative to ρA ⊗ ρC, with spectral and matrix Gram refinements of the prefactor. For fermions, an explicit square-root bound requires no energy or spectral gap assumption, and a complementary bound of order √[ε log(1/ε)] is uniform in the number of conditioning modes with A, C fixed. Known Gaussian continuity theorems convert the new covariance estimates to state distance. The fixed-mode square-root exponents are optimal. Combining fermionic rigidity with the established entropy-deficit identity yields an operational lower bound on non-Gaussianity from CMI and a probability difference for an even measurement event on AC. Version 1.0. Complete manuscript with Appendices A–G.