Quantum Fisher Information and Uncertainty Principle

Recently S. Luo and Q. Zhang proved a different kind of uncertainty principle (see [3], Theorem 2), in the SchrNodinger form, where the lower bound appears because the variables A,B are not commuting with the state ρ (in contrast with standard uncertainty principle where the bound depends on the commutator [A,B]). This result was conjectured by S. Luo himself and Z. Zhang in a previous paper [4]. The inequality by Luo and Zhang has been recently generalized by Kosaki [2] and Yanagi-FuruichiKuriyama [5] and the final result is Varρ(A)·Varρ(B)−|Re{Covρ(A,B)}| ≥ Iρ,α(A)Iρ,α(B)−|Re{Corrρ,α(A, B)}|2 ∀α ∈ (0, 1) where I and Corr are related to the Wigner-Yanase-Dyson skew information of parameter α. The purpose of this talk is to put the above inequality in a more geometric form by means of quantum Fisher information (namely the monotone metrics classified by Petz). In this way the lower bound will appear as a simple function of the area spanned by the commutators i[A, ρ], i[B, ρ] in the tangent space to the state ρ, provided the state space is equipped with a suitable monotone metric. At this point it is natural to ask whether such an inequality holds for every quantum Fisher information. The answer turns out to be negative and a counterexample will be given (see [1]). This is a joint work with T.Isola.