Loophole-Free Photonic Experiment

• The paper demonstrates that threshold-dependent post-selection can mimic quantum violations, challenging the validity of nonlocality certification.
• It employs discrete-event simulations and detailed voltage trace analyses to reveal how setting-dependent thresholds undermine fair sampling in photonic tests.
• The findings have significant implications for quantum cryptography and randomness generation, urging the development of detection protocols that ensure complete data inclusion.

A detection-loophole-free photonic experiment is an experimental protocol designed to rigorously test quantum nonlocality, EPR steering, or foundational postulates by ruling out explanations based on detection inefficiencies or local post-selection. The detection loophole arises when detected events represent only a subset of the ensemble, potentially allowing violations of local realism or quantum bounds to be explained by outcome-dependent losses rather than genuine quantum effects. Photonic platforms are especially vulnerable due to intrinsic inefficiencies in transmission, coupling, and single-photon detection. Achieving a truly detection-loophole-free experiment therefore requires achieving efficiencies and analysis protocols that preclude any advantage to a local hidden-variable (LHV) model exploiting selective detection or thresholding.

1. Definition of the Detection (Photon-Identification) Loophole

In standard photonic Bell–EPRB experiments, not every raw detector event is classified as a valid photon detection. Instead, detection events (typically negative voltage traces from single-photon detectors) are compared to software-set photon-identification thresholds. Only if the voltage $v \leq \mathcal{V}$ does the event contribute to data analysis, with the identification flag $$w(a, \varphi, \hat{r}) = \Theta(\mathcal{V} - v(a, \varphi, \hat{r}))$$, where $\Theta$ is the Heaviside function, $a$ is the local analyzer setting, $\varphi$ a polarization or hidden variable, and $\hat{r}$ a local random number (Raedt et al., 2017). This thresholding process, if not strictly independent of measurement settings, can violate the fair-sampling assumption, thus opening the detection loophole, because the selected sample may not fairly represent the whole. The detection efficiency $\eta$ is defined as the ratio of identified pairs $N'$ to total emitted pairs $N_{\text{max}}$, $\delta = N'/N_{\text{max}} \leq 1$.

2. Experimental Protocols and the Role of Thresholding

Prominent experiments, such as those by Giustina et al. and Shalm et al. (2015), introduced high-efficiency superconducting detectors and advanced data acquisition to claim "loophole-free" Bell violations. However, both crucially rely on local, software-defined voltage thresholding to classify detection events. After the experimental run, a photon-identification threshold $\mathcal{V}$ is chosen (typically post hoc to maximize Bell violation), and only events with $v \leq \mathcal{V}$ are accepted (Raedt et al., 2017). This post-selection discards most raw detection events, locally at each station, prior to Bell parameter calculation. While measurement hardware may be Einstein-local, this thresholding constitutes a local selection based on both measurement setting and detector outcome, violating the core fair-sampling premise and enabling the photon-identification loophole.

3. Simulation Models Demonstrating the Loophole

De Raedt, Michielsen, and Hess constructed a discrete-event Monte Carlo model replicating this protocol: each simulated detection implements the local thresholding rule and outputs only when both sides have $w=1$. The essential simulation steps are:

• For each photon, hidden variables ($\varphi$) and local settings are randomly generated.
• The local outcome $x_i = \mathrm{sign}(1 + c_i - 2 r_i)$, with $c_i = \cos 2(a_i - \varphi_i)$, $r_i$ uniform in $[0,1]$.
• The simulated voltage $$v_i = \hat{r_i} |s_i|^{d}(V_{\text{max}} - V_{\text{min}}) - V_{\text{max}}$$ with $s_i = \sin 2(a_i - \varphi_i)$, $\hat{r_i}$ uniform, and $d$ a parameter.
• A pair contributes to the correlation analysis only if both $v_1, v_2$ satisfy $v_i \leq \mathcal{V}$.

Using this model, manifestly local-hidden-variable correlations can match quantum singlet predictions—including the critical CHSH value $S_{\text{max}} \approx 2\sqrt{2}$—because the Bell bound is modified from $|S| \leq 2$ to $|S| \leq 4 - 2\delta$; for $\delta<1$, "violations" up to the quantum limit become possible even in local models (Raedt et al., 2017).

4. Impact on Bell and EPRB Inequality Derivations

The loophole exposes a failure of Bell's theorem under these experimental conditions:

• Locality: All operations are performed locally; the model is Einstein-local.
• Counterfactual definiteness (CFD): When all events are kept ($w \equiv 1$), standard Bell inequalities are recovered.
• Fair sampling: This assumption critically fails because thresholding discards a non-representative subset, which may depend on settings and outcomes. The detection efficiency becomes a function not only of instrument physics but also of software thresholding, which is not part of Bell's original theoretical framework.

Consequently, the standard derivation of the Bell–CHSH inequality (or its variants) does not hold, as the subset of detected events can be locally engineered to mimic quantum correlations, precluding any inference about genuine nonlocality.

5. Experimental Strategies to Close the Detection Loophole

There are only two rigorous approaches to closing the detection (photon-identification) loophole:

1. Perfect/near-perfect detectors: All physical detection events, regardless of voltage trace, are unconditionally accepted as valid, with no software or hardware selection, i.e., $\eta = 1$. This is technologically infeasible with current single-photon detectors.
2. Distributional analysis of voltage traces: Record the full distribution of $v$ and associated data, and demonstrate empirically that the detection probability $P(v \leq \mathcal{V} | a)$ is strictly independent of $a$ (i.e., of the local setting). If this is convincingly shown, the fair-sampling assumption is effectively restored (Raedt et al., 2017).
3. Alternative detection paradigms: Event-ready or heralded schemes can decouple detection efficiency from transmission/channel losses by heralding only on successful photonic input or interaction events (Cabello et al., 2012, Niemietz et al., 2021). However, without ensuring threshold independence, selectivity remains a vulnerability.

Absent these, any claimed detection-loophole-free photonic test that employs local thresholding retains the possibility of an alternative local–realist explanation via the photon-identification loophole.

6. Implications for Certification of Nonlocality and Quantum Information

The photon-identification loophole has direct implications for the certification of nonlocality, device-independent cryptography, and quantum random number generation. If threshold-dependent post-selection is present, no claim of device-independent security, nonlocal randomness expansion, or refutation of local realism is enforcible, since a local model exploiting the same thresholds can achieve the observed levels of Bell inequality violation (Raedt et al., 2017). Thus, the experimental demonstration of quantum nonlocality demands either elimination or stringent control and characterization of event selection mechanisms, with detection and analysis protocols verifiably immune to setting- or outcome-dependent bias.

7. Summary and Outlook

Detection-loophole-free photonic experiments are only achieved when every photodetection—irrespective of raw detector readout or software threshold—is unconditionally included in the analysis, or when thresholding is shown to be strictly independent of all local settings and outcomes. Otherwise, the photon-identification loophole persists, and the experimental results cannot be uniquely attributed to nonlocal quantum correlations. The work of De Raedt, Michielsen, and Hess rigorously establishes that high-efficiency, setting-dependent thresholding enables classical models to perfectly reproduce quantum correlations, invalidating claims of loophole-free Bell violations in such scenarios (Raedt et al., 2017). Future photonic tests must eliminate or fully characterize local selection effects to guarantee genuine detection-loophole-free status.