Two generalized Wigner–Yanase skew information and their uncertainty relations

Abstract

In this paper, we first define two generalized Wigner–Yanase skew information $|K_{\rho ,\alpha }|(A)$ and $|L_{\rho ,\alpha }|(A)$ for any non-Hermitian Hilbert–Schmidt operator A and a density operator $\rho$ on a Hilbert space H and discuss some properties of them, respectively. We also introduce two related quantities $|S_{\rho ,\alpha }|(A)$ and $|T_{\rho ,\alpha }|(A)$. Then, we establish two uncertainty relations in terms of $|W_{\rho ,\alpha }|(A)$ and $|\widetilde{W}_{\rho ,\alpha }|(A)$, which read

$$\begin{aligned}&|W_{\rho ,\alpha }|(A)|W_{\rho ,\alpha }|(B)\ge \frac{1}{4}\left| \mathrm {tr}\left( \left[ \frac{\rho ^{\alpha }+\rho ^{1-\alpha }}{2} \right] ^{2}[A,B]^{0}\right) \right| ^{2},\\&\sqrt{|\widetilde{W}_{\rho ,\alpha }|(A)| \widetilde{W}_{\rho ,\alpha }|(B)}\ge \frac{1}{4} \left| \mathrm {tr}\left( \rho ^{2\alpha }[A,B]^{0}\right) \mathrm {tr} \left( \rho ^{2(1-\alpha )}[A,B]^{0}\right) \right| . \end{aligned}$$