A positive de...nite symmetric matrix quali...es as a quantum mechanical covariance matrix if and only if + 1 2 i~ 0 where is the standard symplectic matrix. This well-known condition is a strong version of the uncertainty principle, which can be reinterpreted in terms of the topological notion of symplectic capacity, closely related to Gromov's non-squeezing theorem. We show that a recent re...nement of the latter leads to a new class of geometric invariants. These are the volumes of the

more »

... onal projections of the covariance ellipsoid on symplectic subspaces of the phase space. We compare these geometric invariants to the algebraic "universal quantum invariants"of Dodonov and Sera...ni.