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Let {\rho} the density matrix of a mixed Gaussian state. Assuming that one ofthe 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 Wignerdistribution is dominated by that of {\rho} and having the same variances andcovariance {\Delta}^{{\rho}}X_1,{\Delta}^{{\rho}}P_1, and{\Delta}^{{\rho}}(X_1,P_1) as {\rho}. This property can be viewed as ananalytic version of Gromov's non-squeezing theorem in the linear case, whichimplies that the intersection of a symplectic ball by a single plane ofconjugate coordinates determines the radius of this ball.

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