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Tohoku Mathematical Journal

Schrödinger uncertainty relation and convexity for the monotone pair skew information

Chul Ki Ko, Hyun Jae Yoo

Published Jan 1, 2014• 2 citations
Publisher

Abstract

Furuichi and Yanagi showed a Schrödinger uncertainty relation for the Wigner-Yanase-Dyson skew information, which is a special monotone pair skew information. In this paper, we give a Schrödinger uncertainty relation based on a monotone pair skew information, and extend the result of Furuichi and Yanagi. Moreover, we show that some monotone pair skew information becomes a metric adjusted skew information and therefore the convexity of it follows from known results.

DOI
10.2748/tmj/1396875665

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Furuichi and Yanagi showed a Schrödinger uncertainty relation for the Wigner-Yanase-Dyson skew information, which is a special monotone pair skew information. In this paper, we give a Schrödinger uncertainty relation based on a monotone pair skew information, and extend the result of Furuichi and Yanagi. Moreover, we show that some monotone pair skew information becomes a metric adjusted skew information and therefore the convexity of it follows from known results.
Uncertainty principle is one of the most essential features of quantum mechanics, and it reveals the intrinsic difference that distinguishes the quantum world from the classical world. In this paper, we focus on the uncertainty relations based on metric-adjusted skew information. By making full use of the norm property, we establish an uncertainty relation for arbitrary finite nnn observables based on metric-adjusted skew information, and we give two new uncertainty relations for arbitrary finite NNN quantum channels based on metric-adjusted skew information. For arbitrary two observables and two channels, the uncertainty relations we give are not only better than the uncertainty relations detailed in [Quantum Inf. Process. 20, 72 (2021)], but also are the equations. The equation of the uncertainty relation is more accurate than the inequation of the usual uncertainty relation, which has important advantages in the application of quantum information technology, such as quantum communication and the quantum precision measurement. Meanwhile, our results are suitable to Wigner-Yanase-Dyson skew information that is a special metric-adjusted skew information and Wigner-Yanase skew information that is a special Wigner-Yanase-Dyson skew information. Some examples about Wigner-Yanase-Dyson skew information are given and show that the new lower bounds are tighter. The results play an important role in quantum information processing in this paper.
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.