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Journal of Physics A Mathematical and Theoretical

Geometric uncertainty relations on Wigner–Yanase skew information

Bin Chen, Pan Lian

Published Jun 7, 2023• 4 citations
Publisher

Abstract

Abstract We formulate uncertainty relations based on Wigner–Yanase skew information. By using the Kirillov–Kostant–Souriau Kähler structure on the quantum phase space, we present a new geometric uncertainty relation associated to the skew information, which is shown to be tighter than the existing ones. Furthermore, we provide a skew information-based product uncertainty relation, in which the lower bound can also be used to capture the non-commutativity of the observables. We also attempt to generalize the geometric uncertainty inequalities to the case of arbitrary three observables, where the Kähler structure plays a vital role in the proof.

DOI
10.1088/1751-8121/acdc69

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Abstract Uncertainty principle plays a vital role in quantum physics. The Wigner–Yanase skew information characterizes the uncertainty of an observable with respect to the measured state. We generalize the uncertainty relations for two quantum channels to arbitrary N quantum channels based on Wigner–Yanase skew information. We illustrate that these uncertainty inequalities are tighter than the existing ones by detailed examples. Especially, we also discuss the uncertainty relations for N unitary channels, which could be regarded as variance-based sum uncertainty relations with respect to any pure state.
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.
Abstract In this paper, we build some uncertainty relations for arbitrary N quantum channels based on the Wigner–Yanase skew information. It is shown that our lower bounds are stronger than the ones proposed by Zhang et al (2021 Phys. Lett. A 387 127029; 2021 Laser Phys. Lett. 18 095204) respectively in all intervals. In particular, we derive several uncertainty relations for N unitary channels. Moreover, we establish several descending sequences of lower bounds for the uncertainty relations of N quantum channels. Detailed examples are provided.
Abstract We establish tighter uncertainty relations for arbitrary finite observables via ( α , β , γ ) weighted Wigner–Yanase–Dyson (( α , β , γ ) WWYD) skew information. The results are also applicable to the ( α , γ ) weighted Wigner–Yanase–Dyson (( α , γ ) WWYD) skew information and the weighted Wigner–Yanase–Dyson (WWYD) skew information. We also present tighter lower bounds for quantum channels and unitary channels via ( α , β , γ ) modified weighted Wigner–Yanase–Dyson (( α , β , γ ) MWWYD) skew information. Detailed examples are provided to illustrate the tightness of our uncertainty relations.