  The uncertainty principle determines the distinction between the classical
and quantum worlds. This principle states that it is not possible to measure
two incompatible observables with the desired accuracy simultaneously. In
quantum information theory, Shannon entropy has been used as an appropriate
measure to express the uncertainty relation. According to the applications of
entropic uncertainty relation, studying and trying to improve the bound of this
relation is of great importance. Uncertainty bound can be altered by
considering an extra quantum system as the quantum memory $B$ which is
correlated with the measured quantum system $A$. One can extend the bipartite
quantum memory assisted entropic uncertainty relation to tripartite quantum
memory assisted entropic uncertainty relation in which the memory is split into
two parts. In this work, we obtain a lower bound for the tripartite quantum
memory assisted entropic uncertainty relation. Our lower bound has two
additional terms compared to the lower bound in [Phys. Rev. Lett. 103, 020402
(2009)] which depending on the conditional von Neumann entropy, the Holevo
quantity and mutual information. It is shown that the bound obtained in this
work is more tighter than other bounds. In addition, using our lower bound, a
lower bound for the quantum secret key rate has been obtained. The lower bound
is also used to obtain the states for which the strong subadditivity inequality
and Koashi-Winter inequality is satisfied with equality.
