Complete characterization of internal-external entanglement trade-offs in two-mode Gaussian states

We characterize the exact trade-off between internal and external entanglement in asymmetric two-mode Gaussian states at fixed local symplectic eigenvalues. Internal entanglement is quantified by the negativity $\mathcal{N}_{A:B}$ between the two modes, while external entanglement is quantified by the negativity $\mathcal{N}_{AB:CD}$ across a Gaussian purification. Numerical sampling suggests the boundary structure of the attainable region in the $(\mathcal{N}_{AB:CD},\mathcal{N}_{A:B})$ plane, which we then determine analytically. Parametrizing the global symplectic spectrum yields a closed admissible interval for the remaining spectral parameter at fixed $\mathcal{N}_{AB:CD}$. We prove constructively that every point in this interval corresponds to a physical covariance matrix, implying that all intermediate values of $\mathcal{N}_{A:B}$ between the resulting sharp bounds are achievable. The upper bound is attained for opposite-sign correlations of equal magnitude, while the positive branch of the lower bound occurs when one global symplectic eigenvalue equals unity. We also identify two external-entanglement thresholds, derive a saturated monogamy-type inequality, and recover the symmetric-state limit.