Metric adjusted skew information, Metric adjusted correlation measure and Uncertainty relations

Inspired by the recent results in [4] and the concept of metric adjusted skew information introduced by Hansen in [6], we here give a further generalization for Schrödinger-type uncertainty relation applying metric adjusted correlation measure introduced in [6]. We firstly give some notations according to those in [4]. Let Mn(C) and Mn,sa(C) be the set of all n × n complex matrices and all n × n self-adjoint matrices, equipped with the Hilbert-Schmidt scalar product ⟨A,B⟩ = Tr[A∗B], respectively. Let Mn,+(C) be the set of all positive definite matrices of Mn,sa(C) and Mn,+,1(C) be the set of all density matrices, that is Mn,+,1(C) ≡ {ρ ∈ Mn,sa(C)|Trρ = 1, ρ > 0} ⊂ Mn,+(C). Here X ∈ Mn,+(C) means we have ⟨φ|X|φ⟩ ≥ 0 for any vector |φ⟩ ∈ Cn. In the study of quantum physics, we usually use a positive semidefinite matrix with a unit trace as a density operator ρ. In this section, we assume the invertibility of ρ. A function f : (0,+∞) → R is said operator monotone if the inequalities 0 ≤ f(A) ≤ f(B) hold for any A,B ∈ Mn,sa(C) such that 0 ≤ A ≤ B. An operator monotone function f : (0,+∞) → (0,+∞) is said symmetric if f(x) = xf(x−1) and normalized if f(1) = 1. We represents the set of all symmetric normalized operator monotone functions by Fop. We have the following examples as elements of Fop: