Detection Efficiency Bounds in (Semi-)Device-Independent Scenarios

TL;DR

This work provides a unified, technically grounded review of detection-efficiency bounds across device-independent and semi-device-independent scenarios, focusing on the detection loophole that can mimic classical correlations. It analyzes the Bell, instrumental, prepare-and-measure, and bilocality frameworks, deriving and compiling threshold efficiencies under various detector models and loss channels, including absorption, extra-output, and state-noise models. A key contribution is the presentation of novel results in the instrumental scenario, demonstrating how different loss treatments and interventional versus observational data shape the achievable quantum violations and efficiency thresholds. The findings have direct implications for designing loophole-free tests and secure quantum protocols in real-world settings, where losses and finite statistics are unavoidable, and they underscore the potential of networked configurations to reduce experimental demands for certifying nonclassicality.

Abstract

This article provides a comprehensive review of the critical role of detection efficiency in demonstrating non-classicality across various device-independent and semi-device-independent scenarios. The central focus is the detection loophole, a challenge in which imperfect detectors can allow classical hidden variable models to mimic quantum correlations, thus masking genuine non-classicality. As a review, the article revisits the paradigmatic Bell scenario, detailing the efficiency requirements for the CHSH inequality, such as the 2/3 threshold for symmetric efficiencies, and traces the historical trajectory toward the first loophole-free tests. The analysis extends to other causal structures to explore how efficiency requirements are affected in different contexts. These include the instrumental scenario, which for binary variables has recently been shown to follow the same inefficiency bounds as the bipartite dichotomic Bell scenario; the prepare-and-measure scenario, where inefficiencies impact the certification of a quantum system's dimension and create security breaches in protocols such as Quantum Key Distribution (QKD); and the bilocality scenario, which exemplifies how employing multiple independent sources can significantly relax the required efficiencies to certify non-classical correlations.

Figures (14)

• Figure 1: Directed acyclic graph representing the causal assumptions of local-realism and measurement independence in the bipartite Bell scenario.
• Figure 2: Violation of the CHSH absorbing the non-detection events in $\hat{a}=1$ and $\hat{b}=0$. In the case where $\hat{a} = 1$ and $\hat{b} = 0$, the figure depicts the boundary curve $\eta_2(\eta_1)$ beyond which a violation of the CHSH inequality is no longer possible. For the symmetric configuration, represented by the green curve, the critical efficiency threshold is $\eta_1 = \eta_2 = 0.67$, corresponding to the intersection point of the two red dashed lines. The green curve itself was obtained with the optimized measurements and states that give the critical efficiencies $\eta_1 = \eta_2 = 0.67$. In the asymmetric regime with one ideal detector ($\eta_1 = 1$), the minimum efficiency for the other detector is $\eta_2 = 0.50$; by symmetry, the same holds if $\eta_2 = 1$ and $\eta_1 = 0.50$. The purple and orange curves are obtained by using the optimized configurations for $\eta_1 = 0.51$ and $\eta_2 = 0.51$, respectively.
• Figure 3: Violation of the CHSH absorbing the non-detection events in $\hat{a}=1$ and $\hat{b}=1$. In the figure we plot the curve $\eta_2(\eta_1)$ below which no violation of the CHSH inequality is possible when no-detection events are absorbed in $\hat{a} = 1$ and $\hat{b} = 1$. In the symmetric case ($\eta_1 = \eta_2$, green curve), the minimum efficiency required is $\eta_1 = \eta_2 = 0.84$, identified by the intersection of the two red dashed curves. In the asymmetric case, with one perfect detector ($\eta_1 = 1$), the minimum occurs at $\eta_2 = 0.50$, and analogously, if $\eta_2 = 1$ is fixed then the minimum over $\eta_1$ is also $0.50$. The orange and purple curves indicate, respectively, the optimized boundary conditions for $\eta_1 = 0.51$ (fixing $\eta_2$) and for $\eta_2 = 0.51$ (fixing $\eta_1$). The curves were obtained with the optimal measurements and state that give the minimum critical efficiency in the symetric and asymmetric case.
• Figure 4: $I_{3}$inequality with qutrits. The non-detection event on side $A/B$ is absorbed in the outcome $2$. This figure shows the values of $\eta_2(\eta_1)$ for which no violation of the $I_3$ inequality occurs. In the symmetric configuration, the minimum efficiency required is $\eta = 0.81$. In the asymmetric case, the critical value of $\eta = 0.68$ applies regardless of which side is perfect ($\eta_1 = 1$ or $\eta_2 = 1$). All curves coincide in the plot. The curves were obtained with the optimal measurements and state that give the minimum critical efficiency in the symetric and asymmetric case.
• Figure 5: DAG depicting the causal structure of a tripartite Bell scenario.