PHYSICAL REVIEW A 104, 052414 (2021) Tighter sum uncertainty relations based on metric-adjusted skew information Ruonan Ren , 1 Ping Li , 2 Mingfei Ye, 1 and Yongming Li 1, * 1 School of Mathematics and Statistics, Shaanxi Normal University, Xi’an 710062, China 2 College of Computer Science, Shaanxi Normal University, Xi’an 710062, China (Received 20 April 2021; accepted 25 October 2021; published 12 November 2021) 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 I. INTRODUCTION Uncertainty principle is one of the essential building blocks of quantum mechanics and plays an important role in quantum information processing. Uncertainty principle was put for- ward by Heisenberg [1] in 1927 through the analysis of ideal experiments, which indicates that the results of two incom- patible measurements cannot be accurately predicted at the same time. Uncertainty relations have very useful and wide applications from the foundations of physics to technological applications, they are important for formulating quantum me- chanics [2] (e.g., to prove the complex structure of the Hilbert space [3] or as a basic foundation of quantum mechanics and quantum gravity [4]) for entanglement detection [5,6] for the security analysis of quantum key distribution [7] in quantum cryptography and so on. As an important resource, quantum coherence is a fun- damental aspect of quantum theory, felicitating the defining properties from superposition principle to quantum correla- tions. With the development of quantum coherence framework [8], quantifying quantum coherence has become one of the most active research fields. Quantum coherence can be ap- plied to many fields, such as quantum information, quantum computing [9], quantum biology [10], quantum metrology [11], and quantum phase transition [12]. Luo et al. [13] elab- orate on the idea that coherence and quantum uncertainty are dual viewpoints of the same quantum substrate and address coherence quantification by identifying coherence of a state (with respect to a measurement) with quantum uncertainty of * liyongm@snnu.edu.cn a measurement (with respect to a state). Consequently, coher- ence measures may be set into correspondence with measures of quantum uncertainty. Entropy is another well-known way to express the uncer- tainty principle. Deutsch [14] was the first to establish general entropic uncertainty relations for two finite-dimensional ob- servables with nondegenerate spectra. Later it was improved by Maassen and Uffink [15]. In addition, various uncer- tainty relations relating to different entropies were proposed [16–23]. These entropic uncertainty relations are shown to be useful in various quantum information and computation tasks [24]. Recently, the study of the uncertainty relation of the skew information has attracted more and more attention. Luo [25] reveals that the skew information is a new notion to quantify the Heisenberg uncertainty principle. Wigner-Yanase skew information was introduced in Ref. [26], and it was defined as the following: I ρ (A):=− 1 2 Tr[ √ ρ, A] 2 . The skew information as a measure of quantum uncertainty is used to characterize the intrinsic features of the state and the observable. The skew information was later generalized by Dyson to I α ρ (A):=− 1 2 Tr[ρ α , A][ρ 1−α , A], 0 <α< 1, which is called Wigner-Yanase-Dyson skew information, and its convexity was solved by Lieb and Ruskai [27], Lieb [28] and Wehrl [29]. First of all, the skew information as a measure of quantum uncertainty is comparable and sometimes superior to the usual variance. For the pure state, the skew information and the 2469-9926/2021/104(5)/052414(8) 052414-1 ©2021 American Physical Society