A Robertson-type uncertainty principle and quantum Fisher information

Let A1,...,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))} >= det{ -1/2Tr(rho[A(h), A(j)])} gives a bound for the quantum generalized variance 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 (.,.)(rho,f) be the associated quantum Fisher information. We have conjectured the inequality det{Cov rho(A(h), A(j))} >= det {f(0)/2 < i[rho, A(h)], i[rho, Aj]>rho,f} that gives a non-trivial bound for any N is an element of N using the commutators [rho, A(h)]. In the present paper the conjecture is proved by mean of the Kubo-Ando mean inequality