  The uncertainty principle is one of the most important issues that clarify
the distinction between classical and quantum theory. This principle sets a
bound on our ability to predict the measurement outcome of two incompatible
observables precisely. Uncertainty principle can be formulated via Shannon
entropies of the probability distributions of measurement outcome of the two
observables. It has shown that the entopic uncertainty bound can be improved by
considering an additional particle as the quantum memory $B$ which has
correlation with the measured particle $A$. In this work we consider the memory
assisted entropic uncertainty for the case in which the quantum memory and
measured particle are topological qubits. In our scenario the topological
quantum memory $B$, is considered as an open quantum system which interacts
with its surrounding. The motivation for this model is associated with the fact
that the basis of the memory-assisted entropic uncertainty relation is
constructed on the correlation between quantum memory $B$ and measured particle
$A$. In the sense that, Bob who holds the quantum memory $B$ can predict
Alice's measurement results on particle $A$ more accurately, when the amount of
correlation between $A$ and $B$ is great. Here, we want to find the influence
of environmental effects on uncertainty bound while the quantum memory
interacts with its surrounding. In this work we will consider Ohmic-like
Fermionic and Bosonic environment. We have also investigate the effect of the
Fermionic and Bosonic environment on the lower bounds of the amount of the key
that can be extracted per state by Alice and Bob for quantum key distribution
protocols.
