Uncertainty Principle and Quantum Fisher Information - II

Abstract

Heisenberg and Schr{ö}dinger uncertainty principles give lower bounds for the product of variances $Var_ρ(A)\cdot Var_ρ(B)$, in a state $ρ$, if the observables $A,B$ are not compatible, namely if the commutator $[A,B]$ is not zero. In this paper we prove an uncertainty principle in Schr{ö}dinger form where the bound for the product of variances $Var_ρ(A)\cdot Var_ρ(B)$ depends on the area spanned by the commutators $[ρ,A]$ and $[ρ,B]$ with respect to an arbitrary quantum version of the Fisher information.