A Robertson-type uncertainty principle and quantum Fisher information

Let A1,…,ANA1,…,AN be complex self-adjoint matrices and let ρρ be a density matrix. The Robertson uncertainty principledet{Covρ(Ah,Aj)}⩾det-i2Tr(ρ[Ah,Aj])gives a bound for the quantum generalized variance in terms of the commutators [Ah,Aj][Ah,Aj]. The right side matrix is antisymmetric and therefore the bound is trivial (equal to zero) in the odd case N=2m+1N=2m+1.Let f   be an arbitrary normalized symmetric operator monotone function and let 〈·,·〉ρ,f〈·,·〉ρ,f be the associated quantum Fisher information. We have conjectured the inequalitydet{Covρ(Ah,Aj)}⩾detf(0)2〈i[ρ,Ah],i[ρ,Aj]〉ρ,fthat gives a non-trivial bound for any N∈NN∈N using the commutators [ρ,Ah][ρ,Ah]. In the present paper the conjecture is proved by mean of the Kubo–Ando mean inequality.