Bilocality in Quantum Networks

Updated 7 January 2026

Bilocality is a framework in quantum networks that enforces independent hidden variables from separate sources, yielding stricter constraints than standard Bell-local models.
The bilocality inequality uses nonlinear witnesses to distinguish quantum violations from classical correlations, with measured parameters exceeding classical bounds.
Experiments confirm bilocality via entanglement swapping and source independence, achieving high-precision violations and addressing key loopholes in quantum network tests.

Bilocality is a foundational concept in quantum network theory that refines the standard notion of locality by imposing independence on multiple sources in multipartite networks. The bilocality scenario is central for certifying nonclassical features in entanglement-swapping networks and establishing device-independent quantum network security.

1. Bilocality: Definition and Structure

The bilocality scenario consists of three parties, conventionally Alice (A), Bob (B), and Charlie (C), arranged linearly and connected by two independent sources S₁ and S₂. S₁ emits one particle to A and one to B; S₂ emits one to B and one to C. The essential classical description is the bilocal hidden-variable (BLHV) model, where each source emits an independent hidden variable, λ₁ and λ₂. The joint probability for measurement outcomes $a, b, c$ given settings $x, y, z$ is then

$P(a,b,c|x,y,z) = \sum_{\lambda_1, \lambda_2} p(\lambda_1)p(\lambda_2) P(a|x,\lambda_1)P(b|y,\lambda_1, \lambda_2)P(c|z,\lambda_2)$

This independence, $p(\lambda_1, \lambda_2) = p(\lambda_1)p(\lambda_2)$ , strengthens the constraints compared to standard Bell-local models, making the set of bilocal correlations strictly smaller than the local polytope (Branciard et al., 2011, Andreoli et al., 2017, Gisin et al., 2017, Mukherjee et al., 2014, Tavakoli et al., 2014).

2. Bilocality Inequalities: Nonlinear Witnesses

To distinguish bilocal from genuinely quantum or post-classical correlations, the central tool is the bilocality inequality. For the canonical binary-input, binary-output case, Branciard et al. derived the fundamental nonlinear witness:

$\mathcal{B} = \sqrt{|I|} + \sqrt{|J|} \leq 1$

where

$I = \frac{1}{4} \sum_{x, z=0,1} \langle A_x B_0 C_z \rangle, \quad J = \frac{1}{4} \sum_{x, z=0,1} (-1)^{x+z} \langle A_x B_1 C_z \rangle$

and

$\langle A_x B_y C_z \rangle = \sum_{a,b,c} (-1)^{a+b+c} P(a,b,c|x,y,z)$

Violations, i.e., $\mathcal{B}>1$ , certify non-bilocal (network-nonlocal) correlations. Unlike linear Bell inequalities, the root structure in bilocality inequalities arises directly from the independence between hidden variables as enforced by Hölder or Cauchy–Schwarz bounds (Branciard et al., 2011, Andreoli et al., 2017, Gisin et al., 2017). Experimental demonstration of violations with strict source independence and space-like separation has been achieved, with measured bilocal parameters as high as $1.181\pm0.004$ , representing $45\sigma$ deviation from the BLHV bound (Sun et al., 2018).

3. Quantum Violations and Entanglement Swapping

Quantum mechanics enables violation of bilocality inequalities via entanglement swapping, even when each link is classically local. The paradigm involves two independent entangled pairs (e.g., singlets) generated at S₁ and S₂. Bob performs a Bell-state measurement (BSM) on his two qubits. Conditionally, Alice and Charlie’s qubits become entangled, enabling quantum non-bilocal correlations:

Optimal local measurement settings for A and C are typically $(\sigma_z \pm \sigma_x)/\sqrt{2}$ .
In the ideal case, quantum theory yields $I=J=1/(2\sqrt{2})$ , so that $\mathcal{B}=\sqrt{2} \approx 1.414>1$ , certifying non-bilocality (Andreoli et al., 2017, Branciard et al., 2011, Gisin et al., 2017).
All pairs of entangled pure two-qubit states yield a violation. Precise necessary and sufficient conditions for mixed states involve the singular values $\xi_1, \xi_2, \zeta_1, \zeta_2$ of the Bloch correlation matrices, with violation occurring if $\xi_1\zeta_1 + \xi_2\zeta_2 > 1$ (Gisin et al., 2017).
Bilocality violations can occur even in cases where neither link individually violates CHSH; conversely, some CHSH-nonlocal links do not produce bilocal violations (Andreoli et al., 2017).
Device-independent, reference-frame-independent tests have been developed to eliminate loopholes associated with shared randomness or calibration (Andreoli et al., 2017).

4. Extensions: Star Networks and n-Locality

Bilocality generalizes to n-locality in star-shaped and linear networks:

Star network: One central node (Bob) shares independent sources with $n$ edge nodes (Alices). The n-locality assumption decomposes the joint probability into $n$ independent hidden variables, with all edge parties responding locally and the center depending on all variables (Tavakoli et al., 2014, Andreoli et al., 2017).
n-Locality inequality: In star networks with dichotomic settings,

$|I|^{1/n} + |J|^{1/n} \leq 1$

where $I, J$ are generalized correlator sums. Quantum violation reaches $\sqrt{2}$ for $n=2$ (bilocality), and is robust for arbitrary $n$ (with threshold visibilities $v > 1/\sqrt{2}$ for Werner sources) (Tavakoli et al., 2014, Andreoli et al., 2017, Mukherjee et al., 2014). The robustness does not scale in linear chains but increases exponentially for star topologies.

Network Bell inequalities beyond the standard form have been constructed, e.g. via Elegant Joint Measurement (EJM) bases, yielding noise-tolerant quantum violations unattainable by local or even bilocal models (Tavakoli et al., 2020).

5. Classical, Quantum, and Operational Characterizations

Bilocality has been explored in several structural frameworks:

Correlation Tensors (CT) and Probability Tensors (PT): The bilocality condition admits equivalent continuous (C-LHVM) and discrete (D-LHVM) formulations, with bilocal tensors forming a compact, path-connected, but nonconvex set in probability space; star-convexity applies to slices with fixed marginals (Xiao et al., 2022).
Entropic approach: Bilocality constraints induce linear independence relations at the level of Shannon entropies, allowing construction of tight entropic bilocality inequalities—52 in the simplest setting. Quantum states do not violate these entropic cones, but super-quantum (no-signalling) boxes do (Chaves et al., 2012).
Quasi-probabilities: Violations of the bilocality inequality require negative entries in the underlying joint quasi-distributions (JQDs) or quasi-stochastic maps; positive-definite JPDs cannot exceed the bilocal bound. This framework aligns bilocality with the operational foundations of Bell and network nonlocality (Onggadinata et al., 2023).

6. Experimental Implementation and Loophole Closure

State-of-the-art experiments have addressed all major loopholes:

Source Independence: Achieved via phase-randomized, gain-switched lasers; no fixed phase or classical correlation can pass between S₁ and S₂ (Sun et al., 2018).
Locality and Measurement Independence: Realized using space-like separation protocols with precise spatial and temporal separation of emissions, QRNG events, and detections (Sun et al., 2018).
Reference Frame Loophole: Violations robust even with arbitrary local rotations, leveraging multiple orthogonal measurement bases at each node (Andreoli et al., 2017).
Detection and Noise Robustness: Bilocality tests require lower visibility thresholds than standard Bell tests ( $\sim$ 50% versus $1/\sqrt{2}$ for ideal sources; similarly for detection efficiencies), making them advantageous for practical quantum network certification (Branciard et al., 2011, Mukherjee et al., 2014).

7. Generalizations and Theoretical Developments

The bilocality scenario underpins ongoing theoretical innovation:

Continuous Variables: Bilocality violations persist for all nonzero squeezing in two-mode squeezed vacuum states; non-Gaussianity such as photon subtraction enhances the violation, even achieving maximality at zero squeezing (Chakrabarty et al., 9 Jun 2025).
Minimal Network Nonclassicality: The emergence of correlations exhibiting nonclassicality genuinely irreducible to Bell-type scenarios has been formalized via minimal network nonclassicality (MNN), distinguishing novel quantum effects from those "in the shadow of Bell’s theorem" (Ciudad-Alañón et al., 2024).
Contextuality and Causal Compatibility: Certain network tasks (e.g., conclusive exclusion as in the PBR construction) yield causal compatibility inequalities violated in quantum theory by purely possibilistic means—thus providing new kinds of quantum-classical gaps in the bilocality scenario (Yīng et al., 3 Dec 2025).
Combinatorial and Topological Features: Bilocal and n-local sets exhibit nontrivial geometry: compactness, path-connectedness, and star-convexity, supporting both discrete optimization and continuous-variable analyses (Xiao et al., 2022).

Bilocality thus offers a refined framework for probing and certifying network quantum nonlocality, underpinning both foundational studies and practical device-independent quantum networking. Analytical, operational, and experimental results establish that the independence of sources, enforced through the bilocal model, yields nonconvex, nonlinear constraints and admits violations unattainable by standard local hidden-variable or Bell-local models.