Fig. 2. Experimental setup.

Pairs of entangled photons from the source S₀, in modes a and b, respectively, are distributed to Alice’s and Bob’s friend, who locally measure their respective photon in the {∣h〉, ∣v〉}-basis using entangled sources S_A, S_B and type I fusion gates. These use nonclassical interference on a polarizing beam splitter (PBS) together with a set of half-wave (HWP) and quarter-wave plates (QWP). The photons in modes α′ and β′ are detected using superconducting nanowire single-photon detectors (SNSPDs) to herald the successful measurement, while the photons in modes α and β record the friends’ measurement results. Alice (Bob) then either performs a Bell-state measurement via nonclassical interference on a 50/50 beam splitter (BS) on modes a and α (b and β) to measure A₁ (B₁) and establish her (his) own fact or removes the BS to measure A₀ (B₀) to infer the fact recorded by their respective friend (see the Supplementary Materials for details).