Refinement of Robertson-type uncertainty principles with geometric interpretation

A generalisation of the classical covariance for quantum mechanical observables has previously been presented by Gibilisco, Hiai and Petz. Gibilisco and Isola has proved that the usual quantum covariance gives the sharpest inequalities for the determinants of covariance matrices. We introduce a new generalisation of the classical covariance which gives better inequalities than the classical one furthermore it has a direct geometric interpretation.