ar X iv : 0 70 7 . 11 47 v 1 [ m at h - ph ] 8 J ul 2 00 7 Uncertainty principle with quantum Fisher information ∗

In this paper we prove a nontrivial lower bound for the determinant of the covariance matrix of quantum mechanical observables, which was conjectured by Gibilisco and Isola. The lower bound is given in terms of the commutator of the state and the observables and their scalar product, which is generated by an arbitrary symmetric operator monotone function. Introduction The basic object in the statistical description of a classical physical system is a probability space (Ω,B, μ), where the measure μ determines the state of the system and the physical quantities are measurable Ω → R functions. The covariance of the quantities X,Y ∈ L(Ω,R, μ) is defined as Covμ(X,Y ) = ∫