A Volume Inequality for Quantum Fisher Information and the Uncertainty Principle

Abstract

Let A ₁,…,A _N 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 [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 〈⋅,⋅〉 _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,A _h ].