# Two generalized Wigner–Yanase skew information and their uncertainty relations

Two generalized Wigner–Yanase skew information and their uncertainty relations 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 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

# Two generalized Wigner–Yanase skew information and their uncertainty relations

, Volume 15 (12) – Sep 8, 2016
12 pages

/lp/springer_journal/two-generalized-wigner-yanase-skew-information-and-their-uncertainty-t6ZdCmNSxs
Publisher
Springer US
Subject
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-016-1434-5
Publisher site
See Article on Publisher Site

### Abstract

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 .

### Journal

Quantum Information ProcessingSpringer Journals

Published: Sep 8, 2016

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Unlimited reading Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere. ### Stay up to date Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates. ### Organize your research It’s easy to organize your research with our built-in tools. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. ### DeepDyve Freelancer ### DeepDyve Pro Price FREE$49/month

\$360/year
Save searches from