  Quantum entanglement is the key to quantum communications over considerable
distances. The first step for entanglement distribution among quantum
communication nodes is to generate link-level Einstein-Podolsky-Rosen (EPR)
pairs between adjacent communication nodes. EPR pairs may be continuously
generated and stored in a few quantum memories to be ready for utilization by
quantum applications. A major challenge is that qubits suffer from unavoidable
noise due to their interaction with the environment, which is called
decoherence. This decoherence results in the known exponential decay model of
the fidelity of the qubits with time, thus, limiting the lifetime of a qubit in
a quantum memory and the performance of quantum applications.
  In this paper, we evaluate the fidelity of the stored EPR pairs under two
opposite dynamical and probabilistic phenomena, first, the aforementioned
decoherence and second purification, i.e. an operation to improve the fidelity
of an EPR pair at the expense of sacrificing another EPR pair. Instead of
applying the purification as soon as two EPR pairs are generated, we introduce
a Purification scheme Beyond the Generation time (PBG) of two EPR pairs. We
analytically show the probability distribution of the fidelity of stored
link-level EPR pairs in a system with two quantum memories at each node
allowing a maximum of two stored EPR pairs. In addition, we apply a PBG scheme
that purifies the two stored EPR pairs upon the generation of an additional
one. We finally provide numerical evaluations of the analytical approach and
show the fidelity-rate trade-off of the considered purification scheme.
