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arXiv:2511.19302 · Physical Review A

Quantitative and Optimal Device-Independent Lower Bounds on Detection Efficiency

Arkaprabha Ghosal, Soumyadip Patra, Peter Bierhorst

quant-phSubmitted Nov 24, 2025• Announced Nov 25, 2025• 0 citations• Revised Jun 11, 2026 (v2)
v1 · Nov 24, 2025v2 · Jun 11, 2026
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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.

Journal reference
Phys. Rev. A 114, 012404 (2026)
DOI
10.1103/3xdr-w3z4
Comments
22 pages, 4 figures

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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.
Bell inequalities are an important tool for studying nonlocality, however, they quickly become computationally intractable as the system size grows. We consider a method for finding an upper bound for the quantum violation of such inequalities by combining the Navascu\'es-Pironio-Ac\'{\i}n (NPA) hierarchy, the method of alternating projections, and a standard memory-efficient optimization algorithm. While our method may not give the tightest upper bound possible, it often does so several orders of magnitude faster than state-of-the-art solvers, with minimal memory usage, thus allowing solutions to problems that would otherwise be intractable. We benchmark using the well-studied I3322{I}_{3322}I3322​ inequality as well as a more general large-scale randomized inequality Rxx22{R}_{xx22}Rxx22​. For randomized inequalities with 130 inputs either side (a first-level moment matrix of size $261\ifmmode\times\else\texttimes\fi{}261$), our method is $\ensuremath{\sim}100\ifmmode\times\else\texttimes\fi{}$ faster than both mosek and SCS while giving a bound only $\ensuremath{\sim}2%$ above the optimum.
The rates of several device-independent (DI) protocols, including quantum key-distribution (QKD) and randomness expansion (RE), can be computed via an optimization of the conditional von Neumann entropy over a particular class of quantum states. In this work we introduce a numerical method to compute lower bounds on such rates. We derive a sequence of optimization problems that converge to the conditional von Neumann entropy of systems defined on general separable Hilbert spaces. Using the Navascués-Pironio-Acín hierarchy we can then relax these problems to semidefinite programs, giving a computationally tractable method to compute lower bounds on the rates of DI protocols. Applying our method to compute the rates of DI-RE and DI-QKD protocols we find substantial improvements over all previous numerical techniques, demonstrating significantly higher rates for both DI-RE and DI-QKD. In particular, for DI-QKD we show a minimal detection efficiency threshold which is within the realm of current capabilities. Moreover, we demonstrate that our method is capable of converging rapidly by recovering all known tight analytical bounds up to several decimal places. Finally, we note that our method is compatible with the entropy accumulation theorem and can thus be used to compute rates of finite round protocols and subsequently prove their security.
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The Navascués--Pironio--Acín (NPA) hierarchy gives the standard semidefinite outer approximations to quantum behaviors. Whether \emph{any} finite level can already equal the quantum set has remained open even in the bipartite scenario with two binary measurements per party. We demonstrate that \emph{no finite level} is exact. For the symmetric doubly tilted CHSH functional hα=A0B0+A0B1+A1B0−A1B1+α(A0+B0)h_α=A_0B_0+A_0B_1+A_1B_0-A_1B_1+α(A_0+B_0)hα​=A0​B0​+A0​B1​+A1​B0​−A1​B1​+α(A0​+B0​), set T=1−αT=1-αT=1−α. Its quantum maximum satisfies [ωQ(1−T)−(4−2T)]/T3→4/3[ω_{\rm Q}(1-T)-(4-2T)]/T^3\to4/3[ωQ​(1−T)−(4−2T)]/T3→4/3, whereas every fixed NPA level satisfies [ωL(1−T)−ωQ(1−T)]/T3→+∞[ω_L(1-T)-ω_{\rm Q}(1-T)]/T^3\to+\infty[ωL​(1−T)−ωQ​(1−T)]/T3→+∞. Under the corresponding boundary rescaling, an explicit expectation of the positive operator ωQ(1−t2)I−Htω_{\rm Q}(1-t^2)I-H_tωQ​(1−t2)I−Ht​ converges to the Motzkin polynomial. A bounded fixed-level error would therefore make the Motzkin polynomial plus a nonnegative constant a sum of squares, which is impossible. Consequently, every standard NPA relaxation based on a fixed finite list of words in the measurement projectors strictly contains the complete quantum set, and its nonquantum behaviors accumulate at a local deterministic behavior. Thus, the finite-level exactness of CHSH and all one-sided tilted CHSH maxima does not extend to an exact finite-level description of the complete quantum set in the minimal scenario.
The rates of several device-independent (DI) protocols, including quantum key-distribution (QKD) and randomness expansion (RE), can be computed via an optimization of the conditional von Neumann entropy over a particular class of quantum states. In this work we introduce a numerical method to compute lower bounds on such rates. We derive a sequence of optimization problems that converge to the conditional von Neumann entropy of systems defined on general separable Hilbert spaces. Using the Navascués-Pironio-Acín hierarchy we can then relax these problems to semidefinite programs, giving a computationally tractable method to compute lower bounds on the rates of DI protocols. Applying our method to compute the rates of DI-RE and DI-QKD protocols we find substantial improvements over all previous numerical techniques, demonstrating significantly higher rates for both DI-RE and DI-QKD. In particular, for DI-QKD we show a minimal detection efficiency threshold which is within the realm of current capabilities. Moreover, we demonstrate that our method is capable of converging rapidly by recovering all known tight analytical bounds up to several decimal places. Finally, we note that our method is compatible with the entropy accumulation theorem and can thus be used to compute rates of finite round protocols and subsequently prove their security.
This work derives an exact and operational tradeoff between Bell–CHSH violation and measurement dependence in local hidden-variable models. Measurement independence is relaxed via a single parameter MMM, defined as the maximum total-variation distance between hidden-variable distributions conditioned on different setting pairs. Under locality, we prove the sharp inequality ∣S∣≤min⁡(2+6M,4)|S| \le \min(2 + 6M, 4)∣S∣≤min(2+6M,4), valid for all admissible MMM, and construct an explicit two-state local model saturating the bound for every M∈[0,1/3]M \in [0, 1/3]M∈[0,1/3]. The result is invertible: any observed CHSH value ∣S∣=2+Δ|S| = 2 + \Delta∣S∣=2+Δ certifies a minimum required measurement dependence M≥Δ/6M \ge \Delta/6M≥Δ/6. The parameter MMM has a direct hypothesis-testing interpretation, placing Bell violations and setting predictability in the same operational units used in statistical distinguishability and cryptography. The paper also provides a linear-programming template that reproduces the analytic frontier and can be adapted to other Bell functionals.