Improved tripartite uncertainty relation with quantum memory

The uncertainty principle is a striking and fundamental feature in quantum mechanics, distinguishing it from classical mechanics. It offers an important lower bound to predict outcomes of two arbitrary incompatible observables measured on a particle. In quantum information theory, this uncertainty principle is popularly formulized in terms of entropy. Here, we present an improvement of the tripartite quantum-memory-assisted entropic uncertainty relation. The uncertainty's lower bound is derived by considering mutual information and the Holevo quantity. It shows that the bound derived by this method will be tighter than the lower bound of Renes and Boileau [ Phys. Rev. Lett. 103, 020402 (2009)]. Furthermore, regarding a pair of mutual unbiased bases as the incompatibility, our bound will become extremely tight for the three-qubit $X$-state system, completely coinciding with the entropy-based uncertainty, and can restore Renes and Boileau's bound with respect to arbitrary tripartite pure states. In addition, by applying our lower bound one can attain the tighter bound of the quantum secret key rate, which is of basic importance to enhance the security of quantum key distribution protocols.