  Polar duality is a well-known concept from convex geometry and analysis. In
the present paper, we study two symplectically covariant versions of polar
duality keeping in mind their applications to quantum mechanics. The first
variant makes use of the symplectic form on phase space and allows a precise
study of the covariance matrix of a density operator. The latter is a
fundamental object in quantum information theory., The second variant is a
symplectically covariant version of the usual polar duality highlighting the
role played by Lagrangian planes. It allows us to define the notion of
"geometric quantum states" with are in bijection with generalized Gaussians.
