Uncertainty principle with quantum Fisher information

TL;DR

This paper establishes a new lower bound for the determinant of the covariance matrix in quantum mechanics, linking it to the commutator and scalar product, advancing the understanding of quantum uncertainty principles.

Contribution

It proves a conjectured lower bound for the covariance matrix determinant using quantum Fisher information and operator monotone functions.

Findings

Derived a nontrivial lower bound for the covariance matrix determinant.

Connected the bound to the commutator and scalar product of quantum observables.

Validated the bound for arbitrary symmetric operator monotone functions.

Abstract

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, Isola and Imparato. 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.