A Generalized Skew Information and Uncertainty Relation

Abstract

Abstract-A generalized skew information is defined and a generalized uncertainty relation is established with the help of a trace inequality which was recently proven by Fujii. In addition, we prove the trace inequality conjectured by Luo and Zhang. Finally, we point out that Theorem 1 in S. Luo and Q. Zhang, IEEE Trans. Inf. Theory, vol. 50, pp. 1778-1782, no. 8, Aug. 2004 is incorrect in general, by giving a simple counter-example. Index Terms-Skew information, trace inequalities and uncertainty relation.

Quantum Covariance, Quantum Fisher Information, and the Uncertainty Relations

“…In the case N = 2 the inequality was proved by Luo, Q. Zhang and Z. Zhang [16,17,15], by Kosaki [11] and by Yanagi, Furuichi and Kuriyama [27] for some special functions f. The general case is due to Gibilisco, Imparato and Isola [7,4]. Gibilisco and Isola emphasized the geometric aspects of the inequality (10) and conjectured it for general quantum Fisher information [4].…”

Uncertainty principle and quantum Fisher information

“…13, and of Proposition IV.1 in Ref. 28. Note that, because of Corollary 7.2 the optimal bound previously known was given by f WY , namely, the bound of Wigner-Yanase metric ͑this was due to Kosaki in Ref.…”

“…This conjecture was proved shortly after by Luo and Zhang. 21 The case of WYD metric ͓namely, the metric associated with f ␤ for ␤ ͑0,1/2͔͒ was proved independently by Kosaki 13 and by Yanagi et al 28 In the paper of Gibilisco and Isola, 6 they emphasized the geometric aspects of the question and they succeeded to formulate Eq. ͑1.3͒ for a general quantum Fisher information.…”

Generalized Wigner-Yanase skew information and generalized fisher information

“…We also denote the set of all self-adjoint operators (observables) by L h (H) and the set of all density operators (quantum states) by S(H) on the Hilbert space H. The relation between the Wigner-Yanase skew information and the uncertainty relation was studied in [9]. Moreover the relation between the Wigner-Yanase-Dyson skew information and the uncertainty relation was studied in [6,11]. In our previous paper [11], we defined a generalized skew information and then derived a kind of an uncertainty relation.…”

“…Moreover the relation between the Wigner-Yanase-Dyson skew information and the uncertainty relation was studied in [6,11]. In our previous paper [11], we defined a generalized skew information and then derived a kind of an uncertainty relation. In the section 2, we introduce a new generalized Wigner-Yanase-Dyson skew information.…”