Entanglement-swapping measurements for deterministic entanglement distribution

Abstract

Entanglement swapping is a key primitive for distributing entanglement over quantum networks, but different measurement outcomes can produce end-to-end states with different entanglement, requiring branch-dependent processing or the rejection of unfavorable outcomes. We characterize all projective swapping measurements with full-Schmidt-rank vectors such that, for every pair of pure input links, all outcomes yield the same end-to-end state up to local-unitary corrections. Within this family, the measurements that maximize the average G-concurrence for every input pair are built from complex Hadamard operators, and every outcome individually attains the optimum. Classifying the underlying complex Hadamard operators that preserve optimal deterministic swapping gives one class for $d=2,3$, exactly $72$ classes for $d=5$, and uncountably many whenever $d=4k$. We show further that for $d=2,3$, the corrected end-to-end state in a swapping chain is independent of the swapping order, and discuss noise robustness under depolarizing noise and arbitrary convex input contamination. For pure inputs, these schemes retain every outcome while achieving optimal G-concurrence and therefore eliminate outcome-based postselection.