  We show that given experimental data obtained from joint position and
momentum measurements one can construct an optimal pure state approximating the
observed quantum state. For that purpose we use a tool from multivariate
statistical analysis (the MVE method), which relies on the existence of a
minimum volume ellipsoid (the John--L\"owner ellipsoid) containing a given
convex set. This method allows us to determine the shape of the covariance
ellipsoid, which is thereafter calibrated using the expression of the
uncertainty principle in terms of the notion of symplectic capacity. We finally
use the Wigner formalism to produce the best approximating Gaussian state.
