Enumerating all bilocal Clifford distillation protocols through symmetry reduction

Entanglement distillation is an essential building block in quantum communication protocols. Here, we study the class of near-term implementable protocols that use bilocal Clifford operations followed by a single round communication. We introduce tools to enumerate and optimise over all for up $n=5$ (not necessarily equal) Bell-diagonal states using commodity desktop computer. Furthermore, exploiting symmetries input states, find $n=8$ copies Werner state. For latter case, present circuits achieve highest fidelity with perfect no decoherence. These have modest depth number two-qubit gates. Our results are based on correspondence between double cosets symplectic group, improve previously known