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Physical Review A

Tighter sum uncertainty relations based on metric-adjusted skew information

Ruonan Ren, Ping Li, Mingfei Ye, Yongming Li

Published Nov 12, 2021• 26 citations
Publisher

Abstract

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 $n$ observables based on metric-adjusted skew information, and we give two new uncertainty relations for arbitrary finite $N$ 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.

DOI
10.1103/physreva.104.052414

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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.
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.
Tighter sum uncertainty relations via (α,β,γ)(α,β,γ)(α,β,γ) weighted Wigner-Yanase-Dyson skew information
We establish tighter uncertainty relations for arbitrary finite observables via (α,β,γ)(α,β,γ)(α,β,γ) weighted Wigner-Yanase-Dyson ((α,β,γ)(α,β,γ)(α,β,γ)WWYD) skew information. The results are also applicable to the (α,γ)(α,γ)(α,γ) weighted Wigner-Yanase-Dyson ((α,γ)(α,γ)(α,γ)WWYD) skew information and the weighted Wigner-Yanase-Dyson (WWYD) skew information. We also present tighter lower bounds of quantum channels and unitary channels via (α,β,γ)(α,β,γ)(α,β,γ) modified weighted Wigner-Yanase-Dyson ((α,β,γ)(α,β,γ)(α,β,γ)MWWYD) skew information. Detailed examples are provided to illustrate tightness of our uncertainty relations.
Abstract We use a novel formation to illustrate the ( α , β , γ ) modified weighted Wigner–Yanase–Dyson (( α , β , γ ) MWWYD) skew information of quantum channels. By using operator norm inequalities, we explore the sum uncertainty relations for arbitrary N quantum channels and for unitary channels. These uncertainty inequalities are shown to be tighter than the existing ones by a detailed example. Our results are also applicable to the MWWYD skew information and the ( α , γ ) modified weighted Wigner–Yanase–Dyson (( α , γ ) MWWYD) skew information of quantum channels as special cases.
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.