  The uncertainty principle is deemed as one of cornerstones in quantum
mechanics, and exploring its lower limit of uncertainty will be helpful to
understand the principle's nature. In this study, we propose a generalized
entropic uncertainty relation for arbitrary multiple-observable in multipartite
system, and further derive a tighter lower bound by considering Holevo quality
and mutual information. Importantly, we detailedly discuss the proposed
uncertainty relations and quantum coherence in the context of Schwarzschild
space-time. It is interesting to find that Hawking radiation will damage the
coherence of the physically accessible region and increase the uncertainty.
Furthermore, we argue that the properties of the uncertainty in Schwarzschild
space-time can be explained from the systems' purity and the information
redistribution of the different regions. Therefore, it is believed that our
findings provide the generalized entropic uncertainty relations in multipartite
systems, which may facilitate us deeper understanding of quantumness and
information paradox of the black holes.
