Coherent control of the causal order of entanglement distillation

Abstract: Indefinite causal order is an evolving field with potential involvement in quantum technologies. Here we propose and study one possible scenario of practical application in quantum communication: a compound entanglement distillation protocol that features two steps of a basic distillation protocol applied in a coherent superposition of two causal orders. This is achieved by using one faulty entangled pair to control-swap two others before a fourth pair is combined with the two swapped ones consecutively. As a result, the protocol distills the four faulty entangled states into one of a higher fidelity. Our protocol has a higher fidelity of distillation and probability of success for some input faulty pairs than conventional concatenations of the basic protocol that follow a definite distillation order. Our proposal shows the advantage of indefinite causal order in an application setting consistent with the requirements of quantum communication.