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We now describe in detail the measurement procedure sketched in Fig. 2. Source S0 and the HWP on its right output arm produce an entangled pair of photons in the state of Eq. 3. This photon pair is distributed to the laboratories of Alice’s friend and Bob’s friend, who measure their photon using type-I fusion gates (17). Each fusion gate is implemented with a polarizing beam splitter (PBS), where horizontally and vertically polarized photons are transmitted and reflected, respectively (by convention collecting a phase i for the latter). Two photons entering the PBS from two different inputs with opposite polarization, ∣h〉 ∣v〉 or ∣v〉 ∣ h〉, will exit from the same output port and will therefore not lead to coincident detection. Only the coincident ∣h〉 ∣h〉 and ∣v〉 ∣v〉 components will be recorded in post-selection. For these post-selected photons, the fusion gate induces the following transformationshhPBShhQ/HWPhh+iv2,vvPBSvvQ/HWPvhiv2(5)where Q/HWP refers to the combination of a quarter-wave plate at π/4 and a half-wave plate at π/8 behind the PBS (see Fig. 2). The second (heralding) photon in the above equation is then projected onto the state ∣h〉 via another PBS. The type-I fusion gate thus implements the operationFGI=12(hhhvvv)(6)where the factor 12 indicates the success probability of the gate of 12.

To use the fusion gate to measure photon a (see Fig. 2) nondestructively, Alice’s friend uses an ancilla from the entangled pair created by SA, prepared as ∣Ψα′α. Depending on the state of the incoming photon, the operation performed by Alice’s friend transforms the overall state ashaΨαα=12(hahαvαhavαhα)FGI12havα,vaΨαα=12(vahαvαvavαhα)FGI12vahα(7)Hence, the state ∣ha or ∣va of the external photon in mode a is copied, after being flipped (hv), onto Alice’s friend’s photon in mode α. This corresponds to a measurement of the incoming photon in the {∣h〉, ∣v〉}-basis, with the outcome being recorded in the state of photon α such that we can writephoton is hα=vα,photon is vα=hα(8)The amplitudes 12 in Eq. 7 indicate the total success probability of 14 for this procedure.

Consider now the central source S0 together with Alice’s and Bob’s friend laboratories. According to Eq. 3, the state generated by S0 is, after the unitary U7π16Ψ˜ab=12cosπ8 (havb+vahb)+12sinπ8 (hahbvavb)(9)

The transformations induced by Alice’s and Bob’s friend are then, according to Eq. 7Ψ˜abΨααΨββFGI214Ψ˜aαbβ(10)with a global success probability of 116. The stateΨ˜aαbβ=12cosπ8 (hvaαvhbβ+vhaαhvbβ)+12sinπ8 (hvaαhvbβvhaαvhbβ)(11)is the four-photon state shared by Alice and Bob when both fusion gates are successful.

Recalling from Eq. 8 how the friends’ measurement results are encoded in their polarization states, the observables of Eq. 4 to be measured on Ψ˜aαbβ areA0=B0=1(v vh h),A1=B1=Ψ+ Ψ+Ψ Ψ(12)with Ψ±=12(hv±vh). To obtain 〈AxBy〉, we projected these states onto all combinations of eigenstates of Ax and By individually and recorded six-photon coincidence events for a fixed duration. More specifically, to measure A0 (similarly B0), we projected onto ∣hvaα and ∣vvaα (eigenvalue +1), and ∣hhaα and ∣vhaα (eigenvalue −1) using QWP and HWP to implement local rotations before the final PBS, not using the beam splitter (BS) in Fig. 2. Note that A0 cannot be simply measured by ignoring photon a due to the probabilistic nature of the photon source. Hence, this photon has to be measured in a polarization-insensitive way, which, due to the polarization-sensitive nature of the photon-detectors, is best achieved by summing over the projections onto both orthogonal polarizations. To measure A1 (B1), we used a 50/50 BS followed by projection onto ∣vh〉. Because of nonclassical interference in the BS, this implements a projection onto the singlet state ∣Ψaα with success probability 12. Using quantum measurement tomography, we verified this Bell-state measurement with a fidelity of Fbsm=96.840.05+0.05. Projections on the other Bell states are possible via local rotations using the same QWP and HWP as before. Here, ∣Ψ+aα takes eigenvalue +1, ∣Ψaα takes eigenvalue −1, and Φ±aα=12(hh±vv)aα takes eigenvalue 0. Probabilities are obtained from normalizing the measured counts with respect to the total of the 16 measurements for each pair of observables (see fig. S2). The theoretically expected values for the various probabilities are 14(1+12)0.427, 14(1+12)0.073, or 0. In addition to this result, an alternative measurement protocol for A0 and B0 is presented in the Supplementary Materials.

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