Uncertainty Principle and Quantum Fisher Information - II

TL;DR

This paper establishes a Schr{"o}dinger-type uncertainty principle where the lower bound for the product of variances depends on the area spanned by commutators involving the state and observables, linked to quantum Fisher information.

Contribution

It introduces a new uncertainty relation in Schr{"o}dinger form based on quantum Fisher information and the geometric area of commutators.

Findings

Derived a Schr{"o}dinger uncertainty principle with a Fisher information-dependent bound.

Connected the uncertainty bound to the geometric area spanned by specific commutators.

Extended the understanding of uncertainty principles in quantum mechanics.

Abstract

Heisenberg and Schr{\"o}dinger uncertainty principles give lower bounds for the product of variances $Var_{\rho}(A)\cdot Var_{\rho}(B)$, in a state $\rho$, 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{\"o}dinger form where the bound for the product of variances $Var_{\rho}(A)\cdot Var_{\rho}(B)$ depends on the area spanned by the commutators $[\rho,A]$ and $[\rho,B]$ with respect to an arbitrary quantum version of the Fisher information.