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

Abstract

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. Because the external negativity depends only on the spectrum of the reduced two-mode state, the bounds hold for any purification, including non-Gaussian purifications.

The paper determines the attainable trade-off between entanglement inside a two-mode Gaussian state and entanglement with its purifying environment, fixing the local symplectic eigenvalues $a,b\geq1$. Internal entanglement is measured by $\mathcal N_{A:B}$ and external entanglement by $n=\mathcal N_{AB:E}$ for a pure extension. It extends the previously studied symmetric case to arbitrary local asymmetry. The optimization uses global symplectic eigenvalues $\nu_\pm=\cosh r_\pm$, with $C=\cosh[\ln(2n+1)]$ and $z=\cosh(r_+-r_-)$. Writing $s=a+b$ and $d=|a-b|$, the exact admissible interval is $1+d^2/(C-1)\leq z\leq\min\{C,s^2/(C+1)-1\}$, with $1+d\leq C\leq ab+\sqrt{(a^2-1)(b^2-1)}$; the symmetric pure endpoint is treated separately. Internal negativity decreases with $z$. Explicit covariance-matrix reconstruction proves that every point in this interval is physical, establishing sharp bounds and attainability of every intermediate negativity. The upper bound is attained by $c_1=-c_2$; the positive lower bound occurs at $\nu_-=1$. Internal entanglement is unavoidable below $\alpha=[\sqrt{a^2+b^2-1}+\sqrt{a^2+b^2-2}-1]/2$. Above a second, explicitly derived threshold $\beta$, all compatible states are separable. The upper envelope also yields a saturated monogamy-type inequality. The external negativity depends only on the reduced-state spectrum, so the bounds apply to arbitrary purifications, including non-Gaussian ones. They do not cover non-Gaussian reduced states or multimode systems. Applications to curved-spacetime field modes require separate covariance-matrix calculations.