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 $[\rho,A_h]$.
Categories: math-ph math.MP math.ST stat.TH