A volume inequality for quantum Fisher information and the uncertainty principle

Let $A_1,\mathrm{.}\mathrm{.}\mathrm{.},A_N$ be complex self-adjoint matrices and let $\rho$ be a density matrix. The Robertson uncertainty principle $$det(Co{v}_{\rho }(A_h,A_j))\ge det(-\frac{i}{2}Tr(\rho [A_h,A_j]))$$ 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 $$det(Co{v}_{\rho }(A_h,A_j))\ge det(\frac{f(0)}{2}<i[\rho ,A_h],i[\rho ,A_j]{>}_{\rho ,f})$$ that gives a non-trivial bound for any natural number $N$ using the commutators $i[\rho, A_h]$\mathrm{.}Theinequalityhasbeenprovedinthecases$N=1,2$ by the joint efforts of many authors. In this paper we prove the case N=3 for real matrices.