Detection-loophole-free nonlocality in the simplest scenario

TL;DR

The paper identifies fundamental efficiency and complexity thresholds for quantum steering in the simplest two-party scenario with one untrusted detector. It proves a minimal detector-efficiency bound of $ε>1/X$ for steering with $X$ one-click measurements and shows this bound is tight for any pure entangled state when $X=2$, enabling detection-loophole-free steering with a single detector above 50% efficiency. The authors derive a family of optimal steering witnesses for the $X=2$ case, validate them analytically via a duality argument, and implement a minimal photonic experiment achieving steering at $ε=(51.6\pm0.4)\%$, demonstrating practical feasibility of the simplest loophole-free steering. They contrast steering with Bell-nonlocality, showing that the Eberhard bound $ε>2/3$ remains universal for Bell tests in the same minimal setup and no-signalling models cannot circumvent it. Overall, the work provides a resource-efficient benchmark for steering-based quantum information protocols and clarifies the distinct roles of entanglement and detector efficiency in steering versus Bell nonlocality.

Abstract

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 $X \geq 2$ 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)\% $.

Figures (4)

• Figure 1: Experimental setup. The source generates 1550 nm polarization-entangled photons via SPDC (spontaneous parametric down-conversion) in a nonlinear crystal (PPKTP) embedded in a Mach-Zehnder interferometer realized by beam displacers (BDs). The entangled photons are sent to Alice and Bob. Alice performs a one-click measurement using a half-waveplate (HWP), a polarizing beam splitter (PBS), and only one detector. Bob's three-outcome POVM is implemented using a partially polarizing beam splitter (PPBS), a HWP, and a PBS with outcomes corresponding to polarizations in the X-Z plane of the Bloch sphere.
• Figure 2: Steering parameters versus Alice's measurement overlap for maximally entangled states. Points in the white region indicate a steering violation. The solid curves show theoretical predictions for an ideal state, Alice's projectors and Bob's POVM, based on the experimentally measured efficiencies; the shaded regions represent the uncertainty for those predictions based on the uncertainty in efficiency. Markers indicate the experimental data, along with the associated error estimated as $\pm1$ standard deviation, obtained by repeating the measurements 10 times (The values of minimum steering parameters and their errors are provided in SM Table \ref{['tab:violation values']}). As the efficiency decreases, Alice's measurements are required to have a high overlap to steer Bob. The inset zooms in on the lowest efficiency curve, with $\epsilon = 0.516 \pm 0.004$.
• Figure 3: White-noise levels ($\eta$) tolerable for the simplest bipartite demonstrations of nonlocality, as a function of the detection efficiency $\epsilon$. Solid curves are for Bell-nonlocality, and dashed curves for EPR-steering. Green is for maximally entangled states and black for optimal states. Red points are experimentally measured white-noise robustness of steering with 1 standard deviation error bars. For details, see text.
• Figure S1: Steering parameters versus Alice's measurement overlap for non-maximally entangled states. Points in the white region indicate a steering violation. The shaded regions of the curves represent theoretical predictions for an ideal state and POVM, based on the experimentally measured efficiency. The band reflects $\pm1$ standard deviation uncertainty in efficiency. Markers represent the experimental data, along with the associated error estimated by repeating the measurements 10 times.

Theorems & Definitions (6)

• Lemma 1: Cutoff efficiency for one-detector steering tests
• proof
• Lemma 2
• proof
• Theorem 3: Most inefficient one-detector steering
• proof