A Robertson-type Uncertainty Principle and Quantum Fisher Information

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 $[\rho,A_h]$.