The Unique Pure Gaussian State Determined by the Partial Saturation of the Uncertainty Relations of a Mixed Gaussian State

Let {\rho} the density matrix of a mixed Gaussian state. Assuming that one of the Robertson--Schr\"odinger uncertainty inequalities is saturated by {\rho}, e.g. ({\Delta}^{{\rho}}X_1)^2({\Delta}^{{\rho}}P_1)^2={\Delta}^{{\rho}}(X_1,P_1)^2+(1/4)\hbar^2, we show that there exists a unique pure Gaussian state whose Wigner distribution is dominated by that of {\rho} and having the same variances and covariance {\Delta}^{{\rho}}X_1,{\Delta}^{{\rho}}P_1, and {\Delta}^{{\rho}}(X_1,P_1) as {\rho}. This property can be viewed as an analytic version of Gromov's non-squeezing theorem in the linear case, which implies that the intersection of a symplectic ball by a single plane of conjugate coordinates determines the radius of this ball.