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A positive definite symmetric matrix {\sigma} qualifies as a quantummechanical covariance matrix if and only if {\sigma}+(1/2)i\hbar{\Omega}\geq0where {\Omega} is the standard symplectic matrix. This well-known condition isa strong version of the uncertainty principle, which can be reinterpreted interms of the topological notion of symplectic capacity, closely related toGromov's non-squeezing theorem. We show that a recent refinement of the latterleads to a new class of geometric invariants. These are the volumes of theorthogonal projections of the covariance ellipsoid on symplectic subspaces ofthe phase space. We compare these geometric invariants to the algebraic"universal quantum invariants" of Dodonov and Serafini.

The Symplectic Camel and Quantum Universal Invariants: the Angel of Geometry vs. the Demon of Algebra