A volume inequality for quantum Fisher information and the uncertainty principle

Let A (1),...,A (N) be complex self-adjoint matrices and let rho be a density matrix. The Robertson uncertainty principle [GRAPHICS] 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 <.,.>(rho,) (f) be the associated quantum Fisher information. Based on previous results of several authors, we propose here as a conjecture the inequality [GRAPHICS] whose validity would give a non-trivial bound for any N epsilon N using the commutators i[rho, A(h) ].