A volume inequality for quantum Fisher information and the uncertainty principle
Let $A_1,...,A_N$ be complex self-adjoint matrices and let $ρ$ be a density matrix. The Robertson uncertainty principle $$ det(Cov_ρ(A_h,A_j)) \geq det(- \frac{i}{2} Tr(ρ[A_h,A_j])) $$ 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 $<\cdot, \cdot >_{ρ,f}$ be the associated quantum Fisher information. In this paper we conjecture the inequality $$ det (Cov_ρ(A_h,A_j)) \geq det (\frac{f(0)}{2} < i[ρ, A_h],i[ρ,A_j] >_{ρ,f}) $$ that gives a non-trivial bound for any natural number $N$ using the commutators $i[ρ, A_h]$. 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.