A Robertson-type uncertainty principle and quantum Fisher information Original Research Article

Let A1,…,AN be complex self-adjoint matrices and let ρ be a density matrix. The Robertson uncertainty principleimagegives a bound for the quantum generalized variance in terms of the commutators [Ah,Aj]. 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 left angle bracket·,·right-pointing angle bracketρ,f be the associated quantum Fisher information. We have conjectured the inequalityimagethat gives a non-trivial bound for any image using the commutators [ρ,Ah]. In the present paper the conjecture is proved by mean of the Kubo–Ando mean inequality.