On Covariance and Quantum Fisher Information

For a quantum state and a set of observables, we can construct an associated covariance matrix and a natural quantum Fisher information matrix. These two matrices characterize the uncertainty and information content of the observables in the relevant state. An inequality between these two matrices is established. This inequality may be interpreted as a general quantification of the Heisenberg uncertainty principle from a statistical estimation perspective. In particular, it implies a new uncertainty relation which refines the celebrated Schrödinger uncertainty relation.