The Optimal Pure Gaussian State Associated with Joint Position-Momentum Measurements

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.