Robertson-Type Uncertainty Principles and Generalized Symmetric and Antisymmetric Covariances

Abstract

Uncertainty principles are one of the basic relations of quantum mechanics. Robertson has discovered first that the Schrödinger uncertainty principle can be interpreted as a determinant inequality. Generalized quantum covariance has been previously presented by Gibilisco, Hiai and Petz. Gibilisco and Isola have proved that among these covariances the usual quantum covariance introduced by Schrödinger gives the sharpest inequalities for the determinants of covariance matrices. We have introduced the concept of symmetric and antisymmetric quantum f-covariances which give better uncertainty inequalities. In this paper we generalize the concept of symmetric and antisymmetric covariances considering a continuous path between them, called $\alpha ,f$-covariances. We derive uncertainty relations for $\alpha ,f$-covariances. Moreover, using a simple matrix analytical framework, we present here a short and tractable proof for the celebrated Robertson uncertainty principle. In our setting Robertson inequality is a special case of a determinant inequality, namely that the determinant of the (element-wise) real part of a positive self-adjoint matrix is greater or equal to the determinant of its imaginary part.