  In an entanglement distribution network, the function of a quantum switch is
to generate elementary entanglement with its clients followed by entanglement
swapping to distribute end-to-end entanglement of sufficiently high fidelity
between clients. The threshold on entanglement fidelity is any
quality-of-service requirement specified by the clients as dictated by the
application they run on the network. We consider a discrete-time model for a
quantum switch that attempts generation of fresh elementary entanglement with
clients in each time step in the form of maximally entangled qubit pairs, or
Bell pairs, which succeed probabilistically; the successfully generated Bell
pairs are stored in noisy quantum memories until they can be swapped. We focus
on establishing the value of entanglement distillation of the stored Bell pairs
prior to entanglement swapping in presence of their inevitable aging, i.e.,
decoherence: For a simple instance of a switch with two clients, exponential
decay of entanglement fidelity, and a well-known probabilistic but heralded
two-to-one distillation protocol, given a threshold end-to-end entanglement
fidelity, we use the Markov Decision Processes framework to identify the
optimal action policy - to wait, to distill, or to swap that maximizes
throughput. We compare the switch's performance under the optimal
distillation-enabled policy with that excluding distillation. Simulations of
the two policies demonstrate the improvements that are possible in principle
via optimal use of distillation with respect to average throughput, average
fidelity, and jitter of end-to-end entanglement, as functions of fidelity
threshold. Our model thus helps capture the role of entanglement distillation
in mitigating the effects of decoherence in a quantum switch in an entanglement
distribution network, adding to the growing literature on quantum switches.
