Loophole-Free Quantum Nonlocality

Updated 8 January 2026

Loophole-free quantum nonlocality is defined by experiments that rigorously close detection, locality, and freedom-of-choice loopholes, ensuring true device independence.
Demonstrations enforce critical thresholds, such as 66.7% efficiency for bipartite CHSH tests and 50% for EPR steering, validating key quantum principles.
These advances support robust applications in quantum cryptography, self-testing, and secure randomness generation through refined Bell inequality protocols.

Quantum nonlocality refers to the ability of quantum correlations, observed via measurements on entangled systems, to violate constraints satisfied by all local hidden-variable (LHV) models. Loophole-free quantum nonlocality is the regime where such violations are demonstrated in experiments with all major “loopholes” rigorously closed: detection loophole, locality/communication loophole, and freedom-of-choice loophole. This ensures the observed violations cannot, even in principle, be simulated by classical models exploiting realistic imperfections. Loophole-free demonstrations are foundational for device-independent cryptography, randomness generation, and for validating quantum theory over classical alternatives.

1. Types of Nonlocality and Hierarchy of Demonstrations

Quantum nonlocality is formally organized as a hierarchy:

Entanglement: Nonseparability under local operations, but may admit local-hidden-state (LHS) models.
Einstein-Podolsky-Rosen (EPR) Steering: Demonstrates remote preparation (steering) of quantum states under the assumption that only one party’s measurements are trusted; rules out all LHS models for the trusted party, but not full LHVs.
Bell Nonlocality: Strongest form—excludes all possible LHV explanations; requires both parties to be untrusted, and strict closure of all loopholes.

Bell nonlocality is certified via violation of Bell inequalities (e.g. CHSH/Clauser–Horne–Shimony–Holt), often requiring high-complexity measurement scenarios and high detection efficiencies. EPR steering can be detected via “steering inequalities,” which require trust in only one party’s quantum description and can be more accommodating to lower detector efficiencies (Wittmann et al., 2011). There is a strict containment: Bell nonlocality ⊂ steering ⊂ entanglement (Chen et al., 2015).

2. Loopholes in Nonlocality Tests

Three principal loopholes must be closed for device-independent certification:

Loophole	Description	Typical Closure Requirements
Detection	Unregistered events bias outcomes; "fair-sampling" invalidates device independence	System efficiency above scenario-specific threshold; all events including “no-clicks” must be incorporated in the analysis (Christensen et al., 2013, Zhao et al., 2024)
Locality	Measurement outcomes/settings influence each other via subluminal signals or shared past	Space-like separation of measurement choices and outcomes (Christensen et al., 2013, Dong et al., 2024)
Freedom-of-choice	Detector or measurement settings correlated with underlying hidden variables	Fast, independent random number generators (“quantum randomness”) and strict timing analysis (Dong et al., 2024, Zhao et al., 2024)

Additional loopholes relevant in advanced or high-dimensional settings include the binarisation loophole (Miao et al., 5 Jan 2026) and the “spoof” loophole for time-coincidence (Kracklauer, 2010).

3. Theoretical Framework and Critical Inequalities

The standard theoretical framework for Bell tests is the violation of a Bell inequality, such as the CHSH parameter:

$S_{\text{CHSH}} = \langle A_0B_0 \rangle + \langle A_0B_1 \rangle + \langle A_1B_0 \rangle - \langle A_1B_1 \rangle \leq 2$

where measurements A and B are performed under settings 0 or 1.

For detection-loophole-free demonstrations, inequalities must incorporate null outcomes (e.g., no-click events) without “fair-sampling.” The Clauser–Horne (CH) inequality, for instance, expresses violations directly in terms of singles and coincidences and is robust to losses:

$p_{12}(a,b)+p_{12}(a,b')+p_{12}(a',b)-p_{12}(a',b') \leq p_1(a)+p_2(b)$

Eberhard’s analysis (Christensen et al., 2013) showed that for bipartite qubit Bell tests with two settings per party, the minimal detection efficiency threshold is $\eta = 2/3$ for suitable nonmaximally entangled states.

Steering inequalities take the form

$S_n = \frac{1}{n} \sum_{k=1}^n \langle A_k \sigma_k^B \rangle \leq C_n(\eta)$

where $C_n(\eta)$ is the efficiency-dependent classical bound (Slussarenko et al., 2020, Wittmann et al., 2011).

Multipartite and network settings demand more complex inequalities (e.g., the Svetlichny and $T_2$ inequalities for genuine tripartite nonlocality), each with scenario-specific critical efficiency thresholds (Ghosh et al., 2024).

4. Experimental Realizations and Loophole Closures

4.1 Loophole-Free Steering and Bell Experiments

Early photonic demonstrations could only close one loophole at a time. Major milestones include:

Loophole-free EPR steering using a three-setting quadratic steering inequality with entangled photons, closing all three major loopholes with fast quantum random number generators, space-like separation, and overall detection arm efficiency $n_A = 38.3\%$ , above the $33.3\%$ threshold (Wittmann et al., 2011).
CHSH Bell tests with local precertification: By integrating superconducting transition-edge sensors (TES) and heralded single-photon sources, detection-loophole-free Bell violations with thresholds as low as $\eta \approx 0.67$ were achieved, allowing future full closure of all loopholes (Cabello et al., 2012).
High-efficiency, time-bin, GHz-repetition demonstrations: Detection-loophole-free steering with overall efficiencies as low as $21\%$ in chip-fiber telecom platforms and ultrafast measurement switching ($1.25$ GHz) (Zeng et al., 2024).

4.2 Critical Efficiency and Measurement Complexity

Table: Minimum detection efficiencies for loophole-free violation in several scenarios.

Scenario	Minimal $\eta$ (symmetric)	Inequality/Test	Reference
bipartite CHSH (nonmax entangled)	$2/3 = 66.7\%$	CH/CHSH	(Christensen et al., 2013, Gigena et al., 2024)
bipartite EPR steering ( $X=2$ )	$1/2 = 50\%$	2-setting steering (one detector)	(Raveendranath et al., 7 Jan 2026)
tripartite T $_2$ nonlocality	$75\%$	Pironio $T_2$ inequality	(Ghosh et al., 2024)
tripartite Svetlichny nonlocality	$88.1\%$	Svetlichny inequality	(Ghosh et al., 2024)

Fundamental thresholds differ between Bell and steering cases, with steering often requiring lower efficiencies and measurement complexity (Chen et al., 2015, Raveendranath et al., 7 Jan 2026).

4.3 Loophole-Free Demonstrations beyond Polarization

Entanglement distribution and nonlocality tests have been extended to orbital angular momentum and time-bin encodings. High-efficiency, binarisation-loophole-free Bell tests with four-dimensional (ququart) path/mode entanglement have ruled out not only all LHV models but also entangled states within Hilbert spaces of dimension below the physical system (Miao et al., 5 Jan 2026).

4.4 Network Nonlocality and Postselection

Recent theory shows that, in multi-source networks built from unreliable sources, inconclusive outcomes (no-clicks) can be safely discarded provided all measurements satisfy a “fair-sampling” property, characterized via completely positive local filters. Saturation of the quantum Finner inequality self-tests the underlying “failing-source” model, guaranteeing loophole-free device-independent nonlocality in complex quantum networks (Boreiri et al., 23 Jan 2025).

5. Advanced Tests and Resource Interplay

Advanced loophole-free experiments now address:

Device-independent lower bounds on entanglement and quantification of local measurement incompatibility via effective overlap, showing the intricate trade-off between incompatibility, entanglement, and maximal attainable nonlocality (Dong et al., 2024).
Hardy-type “all-versus-nothing” loophole-free paradoxes: Using engineered nonmaximally entangled states and properly handling inconclusive outcomes, loophole-free violations with efficiency $82.2\%$ and fidelity $99.10\%$ have been measured, with null-hypothesis rejection at $p < 10^{-16348}$ (Zhao et al., 2024).
Extension to arbitrarily long distance: A protocol randomizing the spatial location of detectors, utilizing already-certified near-source devices to trust distant ones, enables loophole-free nonlocality tests over large distances with arbitrarily small efficiencies at the far end, contingent on near-perfect performance close to the source (Chaturvedi et al., 2022).

6. Methodological Developments and Fundamental Implications

Generalized “tilted” Bell inequalities: Detector losses are optimally incorporated via “doubly-tilted” Bell functionals, whose quantum optima are self-tested via an explicit qubit construction, polynomial equations, and Jordan's lemma; minimal symmetric efficiency is $\eta>2/3$ for CHSH and $1/2$ for one-sided perfect detection (Gigena et al., 2024).
Randomized subset Bell tests: When many-measurement settings are impractical, loophole-free nonlocality can still be certified on a random fraction $f$ of settings, with detection threshold rising as $f$ decreases. Explicit trade-off formulas govern the efficiency/fraction relation (Singh et al., 2023).
Closing the spoof and binarisation loopholes: Strict time-stamping, single-pair emission, and active switching are required to exclude classical “spoof” models based on mispairing or binarisation artifacts. High-dimensional nonlocality now requires truly multi-outcome detectors physically enforcing projective completeness, closing the binarisation loophole (Kracklauer, 2010, Miao et al., 5 Jan 2026).

7. Applications and Outlook

Loophole-free quantum nonlocality provides the gold-standard for:

Device-independent quantum cryptography and randomness, whose security proofs are valid against arbitrary adversaries only if all loopholes are strictly closed (Christensen et al., 2013, Dong et al., 2024, Boreiri et al., 23 Jan 2025).
Self-testing of quantum devices and measurement incompatibility, even in the absence of trust in the hardware realization.
Genuine multipartite and network nonlocality—enabling more complex quantum networks to be certified without detection or post-selection loopholes (Ghosh et al., 2024, Boreiri et al., 23 Jan 2025).
Quantum communication over large distances, made feasible by leveraging trusted near-source devices and trade-offs in spatially distributed Bell scenarios (Chaturvedi et al., 2022).

Future work will address the challenge of simultaneous closure of all loopholes in higher-dimensional systems, large multipartite networks, and hybrid quantum architectures, as well as optimized protocols for randomness generation and QKD at practical rates and device complexities.