A Volume Inequality for Quantum Fisher Information and the Uncertainty Principle

Abstract Let A1,…,AN be complex self-adjoint matrices and let ρ be a density matrix. The Robertson uncertainty principle $$\det\{\mathop{Cov}_{\rho}(A_{h},A_{j})\}\geq \det \biggl\{-\frac{i}{2}\mathop{Tr}(\rho [A_{h},A_{j}])\biggr\}$$ gives a bound for the quantum generalized covariance 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 〈⋅,⋅〉ρ,f be the associated quantum Fisher information. Based on previous results of several authors, we propose here as a conjecture the inequality $$\det\{\mathop{Cov}_{\rho}(A_{h},A_{j})\}\geq \det \biggl\{\frac{f(0)}{2}\langle i[\rho,A_{h}],i[\rho,A_{j}]\rangle_{\rho,f}\biggr\}$$ whose validity would give a non-trivial bound for any N∈ℕ using the commutators i[ρ,Ah].