A volume inequality for quantum Fisher information and the uncertainty principle

Let be complex self-adjoint matrices and let be a density matrix. The Robertson uncertainty principle det(Cov_\rho(A_h,A_j)) \geq det(- \frac{i}{2} Tr(\rho [A_h,A_j])) gives a bound for the quantum generalized covariance in terms of the commutators . The right side matrix is antisymmetric and therefore the bound is trivial (equal to zero) in the odd case . Let be an arbitrary normalized symmetric operator monotone function and let be the associated quantum Fisher information. In this paper we conjecture the inequality det (Cov_\rho(A_h,A_j)) \geq det (\frac{f(0)}{2} < i[\rho, A_h],i[\rho,A_j] >_{\rho,f}) that gives a non-trivial bound for any natural number using the commutators . The inequality has been proved in the cases by the joint efforts of many authors. In this paper we prove the case N=3 for real matrices.
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