Uncertainty Relation on Generalized Skew Information with aMonotone Pair

In this paper, we first define a generalized (f,g)-skew information \(\left |I_{ \rho }^{(f, g)}\right |(A)\) and two related quantity \(\left |J_{ \rho }^{(f, g)}\right |(A)\) and \(\left |U_{ \rho }^{(f, g)}\right |(A)\) for any non-Hermitian Hilbert-Schmidt operator A and a density operator ρ on a Hilbert space H and discuss some properties of them. And then, we obtain the following uncertainty relation in terms of \(\left |U_{ \rho }^{(f, g)}\right |(A)\):

$$\begin{array}{@{}rcl@{}} \left|U_{ \rho}^{(f, g)}\right|(A)\left|U_{ \rho}^{(f, g)}\right|(B)\geq \beta_{(f, g)}\left|Tr\left( f(\rho)g(\rho)A, B]^{0}\right)\right|^{2}, \end{array} $$

which is a generalization of a known uncertainty relation in Ko and Yoo (J. Math. Anal. Appl. 383, 208–214, 11).