In this paper, we first define two generalized Wigner–Yanase skew information $$|K_{\rho ,\alpha }|(A)$$ | K ρ , α | ( A ) and $$|L_{\rho ,\alpha }|(A)$$ | L ρ , α | ( A ) for any non-Hermitian Hilbert–Schmidt operator A and a density operator $$\rho $$ ρ on a Hilbert space H and discuss some properties of them, respectively. We also introduce two related quantities $$|S_{\rho ,\alpha }|(A)$$ | S ρ , α | ( A ) and $$|T_{\rho ,\alpha }|(A)$$ | T ρ , α | ( A ) . Then, we establish two uncertainty relations in terms of $$|W_{\rho ,\alpha }|(A)$$ | W ρ , α | ( A ) and $$|\widetilde{W}_{\rho ,\alpha }|(A)$$ | W ~ ρ , α | ( A ) , which read $$\begin{aligned}&|W_{\rho ,\alpha }|(A)|W_{\rho ,\alpha }|(B)\ge \frac{1}{4}\left| \mathrm {tr}\left( \left[ \frac{\rho ^{\alpha }+\rho ^{1-\alpha }}{2} \right] ^{2}[A,B]^{0}\right) \right| ^{2},\\&\sqrt{|\widetilde{W}_{\rho ,\alpha }|(A)| \widetilde{W}_{\rho ,\alpha }|(B)}\ge \frac{1}{4} \left| \mathrm {tr}\left( \rho ^{2\alpha }[A,B]^{0}\right) \mathrm {tr} \left( \rho ^{2(1-\alpha )}[A,B]^{0}\right) \right| . \end{aligned}$$ | W ρ , α | ( A ) | W ρ , α | ( B ) ≥ 1 4 tr ρ α + ρ 1 - α 2 2 [ A , B ] 0 2 , | W ~ ρ , α | ( A ) | W ~ ρ , α | ( B ) ≥ 1 4 tr ρ 2 α [ A , B ] 0 tr ρ 2 ( 1 - α ) [ A , B ] 0 .
Quantum Information Processing – Springer Journals
Published: Sep 8, 2016
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