Quantitative and Optimal Device-Independent Lower Bounds on Detection Efficiency

TL;DR

This work addresses the problem of certifying the minimum detector efficiency required to observe a Bell violation in a fully device-independent (2,2,2) setup.It combines a numerical search over quantum realizations with convex optimization at the second level of the NPA hierarchy to produce tight lower bounds on the efficiency for a given Eberhard violation; it also provides an analytical bound derived from no-signaling distributions obeying the Tsirelson bound. The results show that the numerical and NPA-based bounds agree to four decimal places, yielding a tight device-independent benchmark, and extend the analysis to include dark counts and general linear noise models. Additionally, the authors derive a closed-form analytical lower bound from a restricted NS/Tsirelson framework, highlighting the gap between analytical and numerically tight bounds and offering a tractable comparison for detector-performance certification in quantum networks.

Abstract

This paper examines a quantitative and optimal lower bound on the detector efficiency in a (2,2,2) Bell experiment within a fully device-independent framework, whereby the detectors used in the experiment are uncharacterized. We provide a tight lower bound on the minimum efficiency required to observe a desired Bell-CHSH violation using the Navascués-Pironio-Acín (NPA) hierarchy, confirming tightness up to four decimal places with numerical optimization over explicit quantum realizations. We then introduce the effect of dark counts and demonstrate how to quantify the minimum required efficiency to observe a desired CHSH violation with an increasing dark count error. Finally, to obtain an analytical closed-form expression of the minimum efficiency, we consider the set of no-signaling behaviors that satisfy the Tsirelson bound, which are easier to characterize than the quantum set. Using such behaviors, we find a simple closed-form expression for a lower bound on the minimum efficiency which is monotonically increasing with the CHSH violation, though the analytically obtained lower bounds are meaningfully below the numerically tight lower bound.

Figures (4)

• Figure 1: A plot of the numerically minimized detector efficiency $\eta^{qr}_{\mathcal{E}_{obs}}$ in the symmetric (2,2,2) setting with respect to the observed Eberhard violation $\mathcal{E}_{obs}$. The black dotted horizontal line represents the critical threshold value $\frac{2}{3}$ below which no Eberhard violation can be observed.
• Figure 2: This figure represents two plots. The blue dotted curve reproduces the numerical minimization curve from Fig. \ref{['numericalplot']}, whereas the yellow dotted curve represents the minimum efficiency curve obtained through Algorithm \ref{['alg2']} which provides the lower bound on detector efficiency $\eta$ over the set $\mathcal{Q}_2$. Notice there is no visible gap between the plots as numerical values of the blue dotted curve have agreement up to four decimal places with the values obtained under $NPA_2$, as can be observed in Table [\ref{['table2']}].
• Figure 3: This figure represents four plots. The blue and yellow dotted curve reproduce the numerical minimization curve and the $NPA_2$ minimization curve from Fig. \ref{['npa2curve']}. The red dotted curve represents the numerical minimization curve of the efficiency for $\xi =0.01$. The pink dotted curve represents the $NPA_2$ minimization curve of the efficiency values for $\xi =0.01$. It is clear from the figure that there is no visible gap between the red and pink dotted curve and the comparisons of the numerical values of the efficiency as described in Table [\ref{['table2']}].
• Figure 4: This figure represents two plots. The blue dotted curve reproduces the numerical plot in Fig. \ref{['numericalplot']} of the minimum detector efficiency $\eta_{\mathcal{E}_{obs}}^{\min}$ (accurate up to $4$ decimals) over all quantum realizations for given Eberhard violation $\mathcal{E}_{obs}>0$. The violet curve represents the analytical lower bound on the detector efficiency $\eta_{ns}$ described in Eq.$~$(\ref{['effns']}), which is an explicit function of the variable $\mathcal{E}_{obs}\in (0, ~\frac{1}{2}(\sqrt{2}-1)]$.

Theorems & Definitions (4)

• Definition 1
• Definition 2
• Proposition 1
• proof