A Robertson-type Uncertainty Principle and Quantum Fisher Information

TL;DR

This paper establishes a new determinant inequality linking quantum covariance and Fisher information, providing non-trivial bounds for any number of observables, extending the Robertson uncertainty principle in quantum information theory.

Contribution

It introduces a generalized determinant inequality involving quantum Fisher information and commutators, applicable for any number of observables, extending prior uncertainty principles.

Findings

Proves a determinant inequality relating covariance and Fisher information.

Provides non-trivial bounds for any number of observables.

Extends Robertson's uncertainty principle to a broader quantum setting.

Abstract

Let $A_1,...,A_N$ be complex selfadjoint matrices and let $\rho$ be a density matrix. The Robertson uncertainty principle $$ det (Cov_\rho(A_h,A_j)) \geq det (- \frac{i}{2} Tr (\rho [A_h,A_j])) $$ gives a bound for the quantum generalized covariance in terms of the commutators $ [A_h,A_j]$. The right side matrix is antisymmetric and therefore the bound is trivial (equal to zero) in the odd case $N=2m+1$. Let $f$ be an arbitrary normalized symmetric operator monotone function and let $<\cdot, \cdot >_{\rho,f}$ be the associated quantum Fisher information. In this paper we prove the inequality $$ det (Cov_\rho (A_h,A_j)) \geq det (\frac{f(0)}{2} < i[\rho, A_h],i[\rho,A_j] >_{\rho,f}) $$ that gives a non-trivial bound for any $N \in {\mathbb N}$ using the commutators $[\rho,A_h]$.