  Let $A_1,...,A_N$ be complex self-adjoint 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 conjecture 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 natural number $N$ using the commutators $i[\rho,
A_h]$. The inequality has been proved in the cases $N=1,2$ by the joint efforts
of many authors. In this paper we prove the case N=3 for real matrices.
