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 NA:B between the two modes, while external entanglement is quantified by the negativity NAB:CD across a Gaussian purification. Numerical sampling suggests the boundary structure of the attainable region in the (NAB:CD,NA:B) plane, which we then determine analytically. Parametrizing the global symplectic spectrum yields a closed admissible interval for the remaining spectral parameter at fixed NAB:CD. We prove constructively that every point in this interval corresponds to a physical covariance matrix, implying that all intermediate values of NA: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.