The lower bound of quantum memory-assisted entropic uncertainty and secret rate for two topological qubits under environments

The uncertainty principle sets a bound on our ability to predict the measurement outcome of two incompatible observables precisely. The entropic uncertainty lower bound can be improved by considering an additional particle as the quantum memory B. which has correlation with the measured particle A. Here, we consider the quantum memory-assisted entropic uncertainty for the case in which the quantum memory and the measured particle are topological qubits. In our scenario the topological quantum memory B interacts with environment. We will study Ohmic-like fermionic and bosonic environments. We also investigate the effect of the fermionic and bosonic environments on the lower bounds of the amount of the key which can be extracted per state by Alice and Bob for quantum key distribution protocols.