Sum uncertainty relations based on $(\hat{I}\pm ,{\hat{I}}^2,{\hat{I}}^3)$ weighted Wigner-Yanase-Dyson skew information

Abstract

We introduce ($\hat{I}\pm ,{\hat{I}}^2,{\hat{I}}^3$) weighted Wigner-Yanase-Dyson (($\hat{I}\pm ,{\hat{I}}^2,{\hat{I}}^3$) WWYD) skew information and ($\hat{I}\pm ,{\hat{I}}^2,{\hat{I}}^3$) modified weighted Wigner-Yanase-Dyson (($\hat{I}\pm ,{\hat{I}}^2,{\hat{I}}^3$) MWWYD) skew information. We explore the sum uncertainty relations for arbitrary $N$ mutually noncommutative observables based on ($\hat{I}\pm ,{\hat{I}}^2,{\hat{I}}^3$) WWYD skew information. A series of uncertainty inequalities are derived. We show by detailed example that our results cover and improve the previous ones based on the original Wigner-Yanase (WY) skew information. Finally, we establish new sum uncertainty relations in terms of the ($\hat{I}\pm ,{\hat{I}}^2,{\hat{I}}^3$) MWWYD skew information for arbitrary $N$ quantum channels.

19 pages, 1 figure