Tightening the tripartite quantum-memory-assisted entropic uncertainty relation

The uncertainty principle determines the distinction between the classical and the quantum world. This principle states that it is not possible to measure two incompatible observables with the desired accuracy simultaneously. In quantum information theory, the Shannon entropy is used as an appropriate measure to express the uncertainty relation. Improving the bound of the entropic uncertainty relation is of great importance. The bound can be varied by considering an extra quantum system as the quantum memory which is correlated with the measured quantum system. One can extend the bipartite quantum-memory-assisted entropic uncertainty relation to the tripartite one in which the memory is split into two parts. Here, a lower bound is obtained for the tripartite quantum-memory-assisted entropic uncertainty relation. This lower bound has two extra terms compared with the lower bound in Renes and Boileau [J. M. Renes and J.-C. Boileau, Phys. Rev. Lett. 110, 20402 (2013)] which depends on the conditional von Neumann entropy, the Holevo quantity, and the mutual information. It is shown that the bound is tighter than other bounds derived earlier. It also leads to a lower bound for the quantum secret key rate. In addition, it is applied to obtain the states for which both the strong subadditivity and the Koashi-Winter inequalities turn into equalities.