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Journal of Physics Conference Series

Wigner-Yanase-Dyson skew information and uncertainty relation

Kenjiro Yanagi

Published Feb 15, 2010• 21 citations
Publisher

Abstract

We give a trace inequality related to the uncertainty relation of Wigner-Yanase-Dyson skew information. This inequality corresponds to a generalization of the uncertainty relation derived by S.Luo [8] for the quantum uncertainty quantity excluding the classical mixture. Finally we show a counter-example of the uncertainty relation related to Wigner-Yanase-Dyson skew information given in [6] recently.

DOI
10.1088/1742-6596/201/1/012015

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We give a trace inequality related to the uncertainty relation of Wigner-Yanase-Dyson skew information. This inequality corresponds to a generalization of the uncertainty relation derived by S.Luo [8] for the quantum uncertainty quantity excluding the classical mixture. Finally we show a counter-example of the uncertainty relation related to Wigner-Yanase-Dyson skew information given in [6] recently.
We introduce a generalized Wigner-Yanase skew information and then derive the trace inequality related to the uncertainty relation. This inequality is a non-trivial generalization of the uncertainty relation derived by S.Luo for the quantum uncertainty quantity excluding the classical mixture. And we introduce a generalized Fisher information and then derive a generalized Cramer-Rao inequality. We also give an example for our generalized Fisher information and then derive the uncertainty relation for two observables.
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The variance of an observable in a quantum state is usually used to describe Heisenberg uncertainty relation. For mixed states, the variance includes quantum and classical uncertainties. By means of the skew information and the decomposition of the variance, a stronger uncertainty relation was presented by Luo [Phys. Rev. A 72, 042110 (2005)]. In this paper, by using Wigner-Yanase-Dyson information which is a generalization of the skew information, we propose a general uncertainty relation of mixed states.
A family of skew information quantities is obtained, in which the well-known Wigner-Yanase skew information and quantum Fisher information stand as special cases. A transparent proof of convexity of the generalized skew information is given, implying a simple proof of the Wigner-Yanase-Dyson conjecture. We find in this work an exact skew information inequality for a qubit system, which may be regarded as the information counterpart of the uncertainty relation. A lower bound for the generalized skew information of a pair of incompatible observables in arbitrary dimension and also the upper bound for a qubit system are achieved.