Experimental test of local observer independence

In the quantum domain, there may not be an objective story to tell.


Supplementary Text
Towards a loophole-free "Bell-Wigner" test. Since our experiment relies on the same assumptions as traditional Bell tests, it is subject to the same conceptual and technical loopholes: locality, freedom of choice and the detection loophole. Due to the increased complexity of our experiment, compared to a standard Bell test, the practical requirements for closing these loopholes are significantly more challenging. We now briefly discuss how these loopholes could be closed in the future.
The configuration of our experiment makes it analogous to an "event-ready" Bell test, where the detection of the ancilla photons in the fusion gates heralds which events should be kept for the Bell-Wigner test. In such a configuration, closing the locality and freedom of choice loopholes requires the heralding events to be space-like separated from Alice and Bob's setting choices, which should each be space-like separated from the measurement outcome of the other party. This imposes stringent space-time location requirements for a Bell-Wigner test closing these loopholes.
The detection loophole arises because only a fraction of all created photons is detected. In our "event-ready" configuration, the limited success probability of the fusion gates is not an issue: only heralded events will contribute to the Bell-Wigner test. Nevertheless, to ensure that the fusion gates are indeed event-ready, the ancilla detectors should be photon-numberresolving.
To measure the observables A x , B y , we chose to project the photon states onto their different eigenstates separately. To close the detection loophole one cannot follow such an approach: the measurement protocol should be able to project the states onto any of the eigenstates in any run of the experiment.
To measure A 0 /B 0 from Eq.(4), one could pass the friends' photon through a PBS, with detectors at both outputs. As for A 1 /B 1 , a full Bell-state measurement (which is impossible with linear quantum optics ( )) is not required: it suffices to distinguish |Ψ + , |Ψ − , and have a third outcome for |Φ ± (see Eq. (12)). This can be realised with a small modification to our setup, with detectors added on the second outputs of Alice and Bob's PBS ( ). An even simpler measurement would discriminate e.g. |Ψ − from the other three Bell states, thus measuring the observables A 1 = B 1 = 1 − 2|Ψ − Ψ − |; this would not change anything in an ideal implementation, but simplifies the analysis with detection inefficiencies below.
Even the best photon detectors aren't 100% efficient and optical loss is unavoidable. Assuming a symmetric combined detection efficiency per photon of η, the measurement of A 0 /B 0 requires one detector to click and would succeed with probability η, while the measurement of A 1 /B 1 requires two detectors to fire and would work as expected with probability η 2 . When a detector fails to click, a simple strategy is to output a fixed pre-defined value for the measurement outcome, e.g. +1. Then, for Eq.(11) the average values A x B y are theoretically expected to be With these values, the minimal required detection efficiency to violate inequality (2) with (unrealistically) perfect quantum states and measurements 29 30 is η > 2 3(1 − 1 √ 2 ) − 1 0.875. This is a more stringent requirement than for a standard test of the CHSH inequality, for which a similar analysis for maximally entangled states yields η > 2 √ 2 − 2 0.828. To relax this requirement, one might attempt similar tricks as for standard Bell tests, e.g. to use non-maximally entangled states ( ), although this will come at the cost of a reduced violation of the inequality.
Note, finally, that in the conclusions we draw from the violation of inequality (2), we need to trust that A 0 and B 0 indeed directly measure the memory of Alice and Bob's friends, so as to unveil their respective facts. A new loophole may be open, now specific to Bell-Wigner tests, if such an interpretation cannot be maintained. To address this loophole with a setup like ours, one should use measurement devices for A 0 and B 0 that clearly separate the initial systems and the memories of each friend, and only "looks" at the memory photons, rather than at the system photon + memory photon together; we also leave this possibility as a challenge for future Bell-Wigner experimental tests.

Alternative observables
which have a slightly different physical interpretation. The observables used in the main text and defined in Eq.(4) directly measure the facts established by the friend, as recorded in their memory. In contrast, the observables in Eq.(S1) can be understood as not only a measurement of the friend's record (to establish a "fact for the friend"), but also of the original photon measured by the friend, as a consistency check: if the state of the photon is found to be inconsistent with the friend's record, the definition above assigns a value 0 for the measurement result.
Our experiment also allows us to test inequality (2) using this alternative definition of A 0 , B 0 . Indeed, from the experimental data shown in ig S2, it suffices (according to Eq. (S1) and recalling Eq.(8)) to assign the eigenstate/eigenvalue according to |hv → +1, |vh → −1 and |hh , |vv → 0 in the calculation of the average values A x B y . We thus obtain the three average values A 0 B 0 = 0.662 +0.033 −0.033 , A 0 B 1 = 0.573 +0.039 −0.039 and A 1 B 0 = 0.600 +0.040 −0.040 with A 1 B 1 unchanged. With these values, we have S exp = 2.407 +0.073 −0.073 , again violating inequality (2) by more than 5 standard deviations. As in the main text, errors are computed assuming Poissonian photon counting statistics, see below for details.
Alternative measurement protocol for A 0 , B 0 . Recall that in order to measure A 0 (similarly B 0 ), the beam splitter for Alice in Fig.(2) has to be removed compared to the measurement of A 1 . A less invasive method (which does not compromise the alignment of our optical elements) is to introduce linear polarisers in modes a(b) and α(β). This effectively measures the photons before the BS, preventing interference.
We implemented this procedure for the alternative definition of A 0 and B 0 in Eq. (S1). Since this approach leads to a reduced success probability of the measurement of A 0 (B 0 ) by a factor f 1/4, we measured all 16 eigenvectors only for A 1 B 1 . For the other observables we measured the eigenvectors with non-zero eigenvalues and normalised all data with respect to the total counts for A 1 B 1 , ig. S3. This slightly increases experimental uncertainties, which we have taken into account in our error analysis. The expectation values so obtained are A 0 B 0 = 0.609 +0.048 −0.048 , A 0 B 1 = 0.577 +0.049 −0.049 and A 1 B 0 = 0.588 +0.049 −0.049 with A 1 B 1 unchanged, and S exp = 2.346 +0.110 −0.110 , violating the Bell-Wigner inequality by more than 3 standard deviations. We note that the violation observed with this method is somewhat reduced because of ∼ 4.83 ± 0.97% loss that is introduced by the polarisers. This effectively reduces the number of counts that are observed in the settings A 0 and B 0 compared to the normalisation used, and thereby reduces the expectation values A 0 B 1 and A 1 B 0 , and A 0 B 0 , leading to a reduced violation. Detailed experimental setup. The Ti:sapphire laser beam is protected from backreflections by a Faraday isolator and spatially filtered using a short single-mode fibre (not shown). The laser beam is then temporally multiplexed to effectively quadruple the pulse rate. The pump is then delivered to three Sagnac-interferometer sources to create polarisation entangled photon pairs. The outputs of each source are coupled to single-mode fibres and delivered to the measurement stages. Fibre polarisation controllers are used to maintain the polarisation states of the photons during transport. The three entangled pairs are then subject to two fusion gates, where temporal mode matching is achieved by employing physical delays as indicated. One photon at each measurement stage acts as a heralding signal for the success of the fusion gate, while the other two are subject to a Bell-state measurement on a 50/50 beam splitter, or to a direct measurement without the BS (for A 0 , B 0 ), followed by projection onto orthogonal polarisations. Finally, all six photons are fibre-coupled and detected by the SNSPDs whose detection is processed by a classical computer to find 6-photon coincidence events.