Bilocality Inequality in Quantum Networks

• Bilocality inequality is a nonlinear constraint that distinguishes classical and quantum correlations in networks by assuming independent sources for entanglement swapping.
• It is derived using factorization and convexity arguments, leading to the bound √|I| + √|J| ≤ 1 which quantum strategies can violate with maximally entangled states.
• Experimental implementations using Bell-state measurements demonstrate its noise robustness and potential for generalizing to multi-party quantum network protocols.

Bilocality inequalities are nonlinear constraints satisfied by classical models in quantum networks where independent sources distribute quantum systems to multiple parties. Primarily motivated by entanglement swapping scenarios, these inequalities provide a stringent criterion for nonclassical correlations in multipartite systems with source independence. The prototypical bilocality test involves three parties—Alice, Bob, and Charlie—connected via two independent quantum sources. A bilocal hidden-variable model assumes each source emits its own random variable, and the joint distribution of measurement outcomes factorizes with respect to these variables. Violation of the bilocality inequality certifies "nonbilocality," surpassing the standard notion of Bell nonlocality by exploiting the network structure and independence requirements.

1. Bilocal Hidden-Variable Models and Formal Inequality

The bilocality scenario is defined by two independent sources, $S_1$ and $S_2$, distributing systems to the three parties: Alice (A), Bob (B), and Charlie (C). $S_1$ connects $A$ and $B$ via hidden variable $\lambda_1$, and $S_2$ connects $B$ and $C$ via $\lambda_2$. The core assumption is source independence,

$$P(a,b,c|x,y,z) = \int d\lambda_1 d\lambda_2\, \rho(\lambda_1)\rho(\lambda_2)\, P(a|x,\lambda_1)\,P(b|y,\lambda_1,\lambda_2)\,P(c|z,\lambda_2),$$

ensuring that $$\rho(\lambda_1,\lambda_2) = \rho(\lambda_1)\rho(\lambda_2)$$ (Sun et al., 2018, Saunders et al., 2016, Mukherjee et al., 2014, Branciard et al., 2011).

The bilocality inequality, introduced by Branciard et al., bounds certain linear combinations of tripartite correlations. For binary inputs and outputs, one defines: $$I = \frac{1}{4} \sum_{x,z=0}^{1} \langle A_x B_0 C_z \rangle,\qquad J = \frac{1}{4} \sum_{x,z=0}^{1} (-1)^{x+z} \langle A_x B_1 C_z \rangle,$$ with tripartite correlators $$\langle A_x B_y C_z \rangle = \sum_{a,b,c} (-1)^{a+b+c} P(a,b,c|x,y,z)$$. Any bilocal hidden-variable model must satisfy: $B = \sqrt{|I|} + \sqrt{|J|} \leq 1.$ Violation of $B>1$ is incompatible with all bilocal models (Andreoli et al., 2017, Saunders et al., 2016).

2. Physical Interpretation and Derivation

Bilocality inequalities refine the Bell locality concept by enforcing independence of sources. A standard Bell-local model uses a single shared hidden variable $\lambda$; the bilocal model splits $\lambda \mapsto (\lambda_1, \lambda_2)$, imposing that the two sources cannot conspire to jointly influence all parties. The derivation of the bilocality inequality exploits the factorization structure (Cauchy–Schwarz and convexity arguments) and the boundedness of local deterministic strategies. This results in strictly nonlinear constraints—$\sqrt{|I|} + \sqrt{|J|} \leq 1$—which are tighter than any linear Bell inequality on the same system (Branciard et al., 2011, Mukherjee et al., 2014, Andreoli et al., 2017).

Table: Comparison of classical bounds

Inequality type	Scenario	Classical bound	Quantum bound (max violation)
Bell (CHSH)	Bipartite	$S \leq 2$	$S \leq 2\sqrt{2}$
Bilocal (B)	Tripartite	$B \leq 1$	$B \leq \sqrt{2}$

3. Quantum Violations and Explicit Strategies

Quantum resources can violate the bilocality inequality under suitable entanglement swapping protocols. Optimally, sources $S_1$ and $S_2$ emit maximally entangled two-qubit states, and Bob performs a (partial or full) Bell-state measurement. Alice and Charlie select measurements such as $A_0 = C_0 = (\sigma_z + \sigma_x)/\sqrt{2}$, $A_1 = C_1 = (\sigma_z - \sigma_x)/\sqrt{2}$. The maximum attainable value is $B = \sqrt{2}$ for perfect singlet states. Any pair of pure entangled states violates the bilocality inequality, with $B = \sqrt{2}$ for maximal entanglement (Gisin et al., 2017, Andreoli et al., 2017).

In the presence of noise, e.g., when each source emits a Werner state of visibility $v$, the violation persists for $v > 1/\sqrt{2}$ (Mukherjee et al., 2014, Branciard et al., 2011). Notably, the noise threshold for bilocality violation is lower than for standard Bell inequality violation on the swapped pair ($v > 2^{-1/4} \approx 0.841$ for CHSH), indicating superior noise robustness (Saunders et al., 2016).

4. Experimental Realizations and Technical Loophole Closure

Multiple experimental platforms have demonstrated nonbilocal correlations with photonic qubits (Saunders et al., 2016, Sun et al., 2018, Andreoli et al., 2017). Crucial experimental advances include:

• Independent source generation: Each entanglement source is triggered by laser phase randomization, ensuring independence and eliminating the common phase reference loophole (Sun et al., 2018).
• Strict locality constraints: Quantum random number generators, space-like separation of relevant events, and careful timing budgets preclude signaling and measurement independence loopholes.
• Measurement implementation: Full and partial Bell-state measurements, as well as reference-frame-independent protocols, have been realized (Saunders et al., 2016, Andreoli et al., 2017).

Bilinearity violation persists down to detection efficiencies of $\eta = 2/3$ for Alice and Charlie, outperforming the CHSH threshold ($\sim 0.828$), and is more robust against noise (Saunders et al., 2016, Mukherjee et al., 2014).

5. Operational Perspective and the Role of Negative Probabilities

The operational approach recasts bilocality in terms of joint (quasi-)probability distributions and (quasi-)stochastic processes. The bilocal model is manifested as a product of joint distributions for each source and a local processing step at the central node (Onggadinata et al., 2023). Violating the bilocality inequality necessitates negative probabilities in either the sources or the processing step, analogous to the necessity of negative entries in the Bell scenario (Fine's theorem). This negative quasi-probability acts as a "resource" for network nonlocality, with the maximum algebraic violation corresponding to "bi-PR-box" behavior.

Crucially, the set of bilocal correlations forms a strict subset of local correlations: all bilocal correlations are Bell-local, but some Bell-local correlations fail to be bilocal, detectable only by the nonlinear bilocality inequality (Onggadinata et al., 2023).

6. Generalizations: N-Locality, Asymmetry, and Continuous Variables

Bilocality inequalities generalize naturally to networks with more parties and sources ("n-locality"). In a star network of $n$ edge parties and one central node, the relevant inequality becomes

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

with analogous construction for chain topologies (Tavakoli et al., 2014, Mukherjee et al., 2014, Kundu et al., 2020, Andreoli et al., 2017, Sasmal et al., 2023). Asymmetric bilocal scenarios—where edge parties have an unequal number of settings—admit tailored inequalities and sum-of-squares techniques for optimal quantum bounds. Maximum quantum violations can serve as dimension witnesses and are robust to moderate noise (Sasmal et al., 2023).

Continuous-variable extensions employ pseudospin measurements in two-mode squeezed vacuum states and entangled coherent states, reaching maximal bilocality violation ($2\sqrt{2}$) in the infinite-squeezing limit, or via non-Gaussian operations such as photon subtraction even at zero squeezing (Chakrabarty et al., 9 Jun 2025). This demonstrates bilocality violation in both Gaussian and non-Gaussian CV regimes.

7. Entropic Approaches and Information-Theoretic Constraints

Bilocality can also be formulated with Shannon entropies, imposing independence of sources as $H(A_0, A_1, C_0, C_1) = H(A_0, A_1) + H(C_0, C_1)$. Entropic nonsignaling constraints yield linear entropic inequalities such as

$H(A_0,C) \leq H(A_0,B) + H(C|A_1,B),$

which can certify nonbilocality for outcome alphabets beyond the binary case and in network scenarios where correlator-based witnesses are insufficient (Chaves et al., 2016).

Entropic witnesses capture certain forms of nonlocality activation and demonstrate that genuine network effects (such as "activation" via PR-boxes and classical channels) arise only in multipartite scenarios with source independence, often requiring fewer measurement configurations than full probability reconstructions.

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The bilocality inequality is central for identifying network nonlocal correlations fundamentally distinct from standard Bell nonlocality. Its robustness, operational relevance, and generalizability underpin advanced quantum network protocols, providing stringent device-independent certification for entanglement swapping and informing future quantum communication architectures (Sun et al., 2018, Kundu et al., 2020, Andreoli et al., 2017, Chakrabarty et al., 9 Jun 2025).