Exploring uncertainty relation and its connection with coherence under the Heisenberg spin model with the Dzyaloshinskii–Moriya interaction

Uncertainty principle is at the heart of quantum physics, taking a fundamental and crucial role in the area of quantum information science, and it provides a remarkable lower bound to quantify our prediction for the measured outcome of two incompatible observables. Herein, the relationship between the lower bound of the measured uncertainty and quantum coherence is investigated under a one-dimensional Heisenberg XXZ spin model with Dzyaloshinskii–Moriya (DM) interactions, and the effect of DM interaction on the entropic uncertainty is also examined in details. We reveal the systematic temperature can give rise to the increase in the measurement uncertainty of interest at thermal equilibrium. By contrast, the stronger coupling strength $\left|J\right|$ or the stronger DM interaction would induce the decrease in the amount with respect to the uncertainty. Moreover, we analyze the dynamical behaviors of quantum coherence and find that the bound of the uncertainty is oppositely correlated with the quantum coherence dramatically. Further, the effects of DM interaction along x-direction (characterized by the parameter D_x) and z-direction (characterized by D_z) on the uncertainty of interest are discussed, respectively. For the antiferromagnetic frame J > 0, it is interesting to obtain that x-direction DM interaction D_x exhibits a more powerful influence on reducing the uncertainty and enhancing the systematic coherence, when comparing with that of D_z. With these in mind, we wish our investigations would better understand the dynamical features of the measured uncertainty in the spin-based solid systems.