We present an eliminative analysis of no-signaling equalities in the NIST loophole-free Bell test (Shalm et al., 2015).Standard analyses examine outcome correlations to compute CHSH S values. We instead examine whether one party'sdetection rate depends on the other party's setting choice. Under quantum mechanics, these marginal probabilities must beindependent of the remote setting. Using a six-gate streaming pipeline on 356 million event-level trials, we apply theexperiment's own timing calibration (phase filter: pk=90, radius=4) to select genuine entangled photon detections inPockels cell slots 5-7. In the phase-filtered data, Alice's marginal detection rate for setting a=2 shows a stationary deviation:z = 2.41, permutation p = 0.017, 95% bootstrap CI [1.44e-6, 1.31e-5] excluding zero. The combined chi-squared is 9.40 (4df), one tick below the p < 0.05 threshold of 9.49. The deviation is temporally stationary across 100 time windows spanningthe full experimental run, with near-zero autocorrelation and near-zero linear trend. Without phase filtering, all trialsproduce a null result (p = 0.40), consistent with noise dilution from background detector clicks. A second NIST run using adifferent random number generator (ClassicalRNG XOR) lacks detection-level data and cannot be phase-filtered,precluding direct replication with matching detection definitions.
We present an eliminative analysis of no-signaling equalities in the NIST loophole-free Bell test (Shalm et al., 2015).Standard analyses examine outcome correlations to compute CHSH S values. We instead examine whether one party'sdetection rate depends on the other party's setting choice. Under quantum mechanics, these marginal probabilities must beindependent of the remote setting. Using a six-gate streaming pipeline on 356 million event-level trials, we apply theexperiment's own timing calibration (phase filter: pk=90, radius=4) to select genuine entangled photon detections inPockels cell slots 5-7. In the phase-filtered data, Alice's marginal detection rate for setting a=2 shows a stationary deviation:z = 2.41, permutation p = 0.017, 95% bootstrap CI [1.44e-6, 1.31e-5] excluding zero. The combined chi-squared is 9.40 (4df), one tick below the p < 0.05 threshold of 9.49. The deviation is temporally stationary across 100 time windows spanningthe full experimental run, with near-zero autocorrelation and near-zero linear trend. Without phase filtering, all trialsproduce a null result (p = 0.40), consistent with noise dilution from background detector clicks. A second NIST run using adifferent random number generator (ClassicalRNG XOR) lacks detection-level data and cannot be phase-filtered,precluding direct replication with matching detection definitions.
We analyze measurement-setting sequences from the 2015 loophole-free Bell tests: Hensen et al. (Delft, two runs, 8,664 trials) and Shalm et al. (NIST, main blind and classical-RNG XOR, 535 million settings). We search for signatures predicted by the minimum-MI class of superdeterministic models. Under a Markov persistence model, the bound I(X_i;X_{i+1})<3×10^{-11} bits constrains λ persistence to q<2×10^{-4}, excluding any persistent-λ model at >1000σ. The NIST blind dataset provides the most stringent limits: |P(X_i=X_{i-1})−0.5|<4×10^{-5} at 95% CL and consecutive-setting MI below 3×10^{-11} bits. Cross-source comparison of quantum and classical RNGs reveals no structure. Delft 2 pre-upgrade data shows mild non-uniformity (χ²=10.84, p=0.013) traceable to a known phase-diffusion artifact; all other Delft 2 tests are null. These results constrain any superdeterministic model with memory across the full complexity hierarchy.
Let us imagine an experimentalist, who has maximally entangled photons and wants to violate Bell’s inequality in his laboratory. Among many obstacles he has to overcome, two of them are the most significant. Firstly, he has to avoid exchanging information about the measurement settings between the particles. Secondly, he has to detect particles with a high enough probability, because if the detection efficiency is too low, one can reproduce quantum correlations using a LHV (Local Hidden Variables) model. These two problems are known in the literature, respectively, as the locality loophole [1] and the detection loophole [2, 3]. Fortunately, it is possible to close the locality loophole using photons in the experiment [4, 5, 6, 7, 8]. However, closing the detection loophole is still an object of study. The authors of [9] showed that for the CHSH inequality, the lower bound for detector efficiency needed to close the loophole is 82.8% for maximally entangled states. This bound was lowered to 66.7% by Eberhard [10]. Furthermore, the detector efficiency has been studied for other Bell inequalities [11, 12, 13, 14, 15, 16]. In our research, we are trying to answer the following question: If we prepare a Bell experiment and we have two detectors, having different efficiencies both smaller than 100%, which detector should each party receive? And also: Which representation of the chosen Bell inequality should we consider? These are very crucial questions, which arise in laboratories. Hence, we hope that our results will be useful for experimentalists. We focus on the I3322 inequality [17], which is a natural candidate for testing, because of its asymmetry [14, 18]. We consider the case of a post-selection strategy, which consists in rejecting all the rounds in the Bell experiment when one of the detectors does not click. We used numerical optimisation methods to find both the best classical strategy for Alice and Bob and the best detector clicking strategy. Thus, we have found minimal efficiencies of the detectors of Alice and Bob needed to violate I3322. We made the same studies also for symmetric representations of I3322 [19]. Our results show that the minimal efficiencies of Alice and Bob’s detectors of the asymmetric representation of I3322 is lower than in case of the symmetric one [20]. It is work in progress, and we are examining other post-selection methods as well as other Bell inequalities. We want to consider also the case when non-maximally entangled states are used in Bell experiments.
The recently reported violation of a Bell inequality using entangled electronic spins in diamonds (Hensen et al., Nature 526, 682-686) provided the first loophole-free evidence against local-realist theories of nature. Here we report on data from a second Bell experiment using the same experimental setup with minor modifications. We find a violation of the CHSH-Bell inequality of 2.35 ± 0.18, in agreement with the first run, yielding an overall value of S = 2.38 ± 0.14. We calculate the resulting P-values of the second experiment and of the combined Bell tests. We provide an additional analysis of the distribution of settings choices recorded during the two tests, finding that the observed distributions are consistent with uniform settings for both tests. Finally, we analytically study the effect of particular models of random number generator (RNG) imperfection on our hypothesis test. We find that the winning probability per trial in the CHSH game can be bounded knowing only the mean of the RNG bias. This implies that our experimental result is robust for any model underlying the estimated average RNG bias, for random bits produced up to 690 ns too early by the random number generator.
Loophole-free quantum nonlocality often demands experiments with high complexity (defined by all parties' settings and outcomes) and multiple efficient detectors. Here, we identify the fundamental efficiency and complexity thresholds for quantum steering using two-qubit entangled states. Remarkably, it requires only one photon detector on the untrusted side, with efficiency ε>1/X, where X≥2 is the number of settings on that side. This threshold applies to all pure entangled states, in contrast to analogous Bell-nonlocality tests, which require almost unentangled states to be loss-tolerant. We confirm these predictions in a minimal-complexity (X=2 for the untrusted party and a single three-outcome measurement for the trusted party), detection-loophole-free photonic experiment with $ε= (51.6 \pm 0.4)\% $.
The reported violations of Bell inequalities are routinely read as an experimental refutation of local realism. This preprint asks how firmly the record supports that reading. We catalogue the classical routes by which a strictly local apparatus can report |S| > 2 - from outcome-dependent loss and coincidence-window selection (demonstrated in hardware) to synchronisation, electronics-level click shifts and analysis-parameter selection - and show that below the Eberhard efficiency the exact quantum correlation curve certifies nothing, being reproduced exactly by a known local detection model. We then define the first family-level synchronisation audit (plateau membership, bias index, sub-bound fraction), validate it on synthetic data with known ground truth, and execute it under preregistered protocols on three public archives. The raw Innsbruck record passes: across the full longdist series, every standard-configuration violation is a property of the raw data, not of the calibration. On the Delft event-ready data, a 50,000-member heralding-acceptance family answers the standing postselection critique: the violation is generic, not an acceptance choice. On the NIST 2015 record, an outsider reconstruction of the certified pipeline recovers the Bell geometry and gate optima but stalls at undisclosed cleaning parameters - a documented limit of third-party reproducibility. The programme distils into seven cleanliness criteria (C1–C7) for any dataset and eight provenance requirements (M1–M8) for future experiments. All figures and numbers are reproducible from the supplementary scripts (deterministic, fixed seeds, minutes on a laptop); raw data are fetched from the official public archives.
Loophole-free Bell tests for quantum nonlocality and long-distance secure communication require photodetection efficiencies beyond a threshold ${\ensuremath{\eta}}_{\mathrm{crit}}$ that depends on the Bell inequality and the noise affecting the entangled state received by the distant parties. Most calculations of ${\ensuremath{\eta}}_{\mathrm{crit}}$ assume that the noise is random and can be modeled as white noise. However, most sources suffer from colored noise. Indeed, since entangled states are usually created as a superposition of two possible deexcitation paths, a partial distinguishability between the two processes leads to the appearance of colored noise in the generated state. Recently, there was a proposal for a loophole-free Bell test [A. Cabello and F. Sciarrino, Phys. Rev. X 2, 021010 (2012)], where a specific colored noise appears as a consequence of the precertification of the photon's presence through single-photon spontaneous parametric down-conversion. Here we obtain ${\ensuremath{\eta}}_{\mathrm{crit}}$, the optimal quantum states, and the local settings for a loophole-free Bell test as a function of the amount of colored noise. We consider three bipartite Bell inequalities with n dichotomic settings: Clauser-Horne-Shimony-Holt (n=2), I3322 (n=3), and A5 (n=4), both for the case of symmetric efficiencies, corresponding to photon-photon Bell tests, and for the totally asymmetric case, corresponding to atom-photon Bell tests. Remarkably, in all these cases, ${\ensuremath{\eta}}_{\mathrm{crit}}$ is robust against the colored noise. The present analysis can find application in any test of Bell inequalities in which the dominant noise is of the colored type.