Entanglement-swapping measurements for deterministic entanglement distribution

Abstract: Entanglement swapping is a key primitive for distributing entanglement across nodes in quantum networks. In standard protocols, the outcome of the intermediate measurement determines the resulting state, making the process inherently probabilistic and requiring postselection. In this work, we fully characterize those measurements under which entanglement swapping becomes deterministic: for arbitrary pure inputs, every measurement outcome produces local-unitarily equivalent states. We also show that an optimal measurement, maximizing a concurrence-type entanglement measure, is built from complex Hadamard matrices. For this optimal protocol, we provide a complete, dimension-dependent classification of deterministic entanglement-swapping measurements: unique in dimensions $d=2,3$, infinite for $d=4$, and comprising $72$ inequivalent classes for $d=5$. We further consider a general network with multiple swapping nodes and show that, for $d=2,3$ the resulting end-to-end state is independent of the order in which the repeaters perform the optimal measurements. Our results establish optimal entanglement-swapping schemes that are post-selection free, in the sense that they distribute entanglement across generic quantum network architectures without unfavorable measurement outcomes.