Full Network Nonlocality

• Full network nonlocality is defined as correlations that remain nonclassical even when any one source is modeled as a classical link, ensuring all links contribute to the nonlocal behavior.
• It extends standard network nonlocality by excluding hybrid models where a single nonclassical source could be responsible for observed correlations, thereby imposing a stricter validation of nonlocality.
• Witness inequalities and experimental implementations in varied network topologies confirm full network nonlocality, paving the way for robust cryptographic protocols and noise-tolerant quantum networks.

Searching arXiv for recent and foundational papers on full network nonlocality to ground the article in the literature. arxiv.search query="full network nonlocality" max_results=10 Full network nonlocality is a strengthening of network nonlocality for multipartite correlations generated by several independent sources. In a network-local model, each source emits an independent hidden variable and each party responds only to the variables carried by the adjacent links. Full network nonlocality requires more: the observed correlation must remain incompatible with every hybrid model in which any one source is replaced by a classical link while the remaining sources are allowed to be arbitrary no-signaling resources. It therefore certifies that nonclassicality is distributed across every link of the network, rather than being attributable to a single nonclassical source embedded in an otherwise classical architecture (Pozas-Kerstjens et al., 2021, Wang et al., 2022).

1. Formal definition and causal structure

A network with sources $S_1,\dots,S_m$ and parties $P_1,\dots,P_n$ is network-local when its input-output distribution admits the factorized hidden-variable decomposition

$$p(\mathbf{a}|\mathbf{x}) =\int d\lambda_1\,\mu_1(\lambda_1)\cdots \int d\lambda_m\,\mu_m(\lambda_m) \prod_{k=1}^n p(a_k|x_k,\bar\lambda_k),$$

where each $\lambda_j$ is emitted independently by source $S_j$, and $\bar\lambda_k$ is the subset available to party $P_k$. Violation of this structure is ordinary network nonlocality (Pozas-Kerstjens et al., 2021).

Full network nonlocality replaces the benchmark. A correlation is fully network nonlocal iff, for every choice of one source singled out as classical, there is no hybrid decomposition in which that source is modeled by a classical hidden variable while all remaining sources are promoted to arbitrary no-signaling resources. In the bilocal chain, this means excluding both the “Classical–NS” and “NS–Classical” models

$$p(a,b,c|x,z)=\int d\lambda\,\rho(\lambda)\,p(a|x,\lambda)\,p(b,c|\lambda,z),$$

and its source-swapped analogue; in the three-star network, if the $A^{(1)}\!-\!B$ link is classical, admissible correlations must take the form

$$p(a_1,a_2,a_3,b|x_1,x_2,x_3) =\sum_\lambda p(\lambda)\,p(a_1|x_1,\lambda)\,p_{\mathrm{NSI}}(a_2,a_3,b|x_2,x_3,\lambda),$$

with $P_1,\dots,P_n$0 no-signaling and the $P_1,\dots,P_n$1 marginal factorized after tracing out $P_1,\dots,P_n$2 (Gu et al., 2023, Wang et al., 2022).

This definition is explicitly theory-independent on the nonclassical side: the nonclassical sources need not be quantum, only no-signaling. The certification therefore does not assume quantum mechanics for the nonclassical resources. In the experimental star-network realization, this feature is described as certification “without assuming quantum mechanics” for the sources (Wang et al., 2022).

2. Relation to Bell nonlocality, bilocality, and ordinary network nonlocality

Full network nonlocality is strictly stronger than standard network nonlocality. All fully network nonlocal correlations are network nonlocal, but the converse fails: an ordinary network Bell violation may be explained by a single nonclassical source embedded in a larger network of classical links (Pozas-Kerstjens et al., 2021).

This distinction is particularly sharp in the bilocal entanglement-swapping scenario. The Branciard–Gisin–Pironio inequality,

$P_1,\dots,P_n$3

witnesses network nonlocality, yet it does not witness full network nonlocality. A no-go result shows that every violation $P_1,\dots,P_n$4, including the maximal quantum value $P_1,\dots,P_n$5, can be reproduced by a model with one classical source and one arbitrary nonlocal resource. In the same setting, the full-network-nonlocality-admissible region in the $P_1,\dots,P_n$6-plane is exactly

$P_1,\dots,P_n$7

which is weaker than the bilocal bound $P_1,\dots,P_n$8 (Pozas-Kerstjens et al., 2021).

The same misconception appears in experimental practice. Violations of bilocality inequalities used in earlier entanglement-swapping experiments certify that two classical sources cannot explain the data, but they do not certify the nonclassicality of each source separately. This limitation is emphasized in both the photonic bilocal experiment and the later loophole-constrained realization, where full network nonlocality is introduced precisely to rule out any model with one classical source and one no-signaling source (Håkansson et al., 2022, Gu et al., 2023).

A related but distinct operational notion is genuine network quantum nonlocality. In that framework, the excluded models are “quantum-wirable” simulations built from bipartite quantum boxes plus arbitrary classical wirings. In the bilocal network, entanglement swapping with Bell-state measurement yields correlations that are non-bilocal and not quantum-wirable, even though the tripartite distribution can remain Bell-local in the usual single-source sense (Šupić et al., 2021). This suggests that full network nonlocality and genuine network quantum nonlocality probe different obstructions: one excludes classical-source hybrids against no-signaling adversaries, the other excludes reductions to standard Bell nonlocal resources plus wirings.

3. Witness inequalities and certification methods

The first systematic formulation of full network nonlocality also provided constructive witness methods. In the three-branch star network, the linear $P_1,\dots,P_n$9-star Bell inequality can be tightened under the assumption that one source is classical. For $$p(\mathbf{a}|\mathbf{x}) =\int d\lambda_1\,\mu_1(\lambda_1)\cdots \int d\lambda_m\,\mu_m(\lambda_m) \prod_{k=1}^n p(a_k|x_k,\bar\lambda_k),$$0, any correlation with one classical source and two no-signaling sources satisfies

$$p(\mathbf{a}|\mathbf{x}) =\int d\lambda_1\,\mu_1(\lambda_1)\cdots \int d\lambda_m\,\mu_m(\lambda_m) \prod_{k=1}^n p(a_k|x_k,\bar\lambda_k),$$1

whereas the quantum strategy reaches $$p(\mathbf{a}|\mathbf{x}) =\int d\lambda_1\,\mu_1(\lambda_1)\cdots \int d\lambda_m\,\mu_m(\lambda_m) \prod_{k=1}^n p(a_k|x_k,\bar\lambda_k),$$2. Thus $$p(\mathbf{a}|\mathbf{x}) =\int d\lambda_1\,\mu_1(\lambda_1)\cdots \int d\lambda_m\,\mu_m(\lambda_m) \prod_{k=1}^n p(a_k|x_k,\bar\lambda_k),$$3 certifies full network nonlocality in the trilocal star (Pozas-Kerstjens et al., 2021).

More generally, hybrid inflation was introduced as a certification framework. In the three-star network, inflating the network and solving linear programs yields explicit inequalities $$p(\mathbf{a}|\mathbf{x}) =\int d\lambda_1\,\mu_1(\lambda_1)\cdots \int d\lambda_m\,\mu_m(\lambda_m) \prod_{k=1}^n p(a_k|x_k,\bar\lambda_k),$$4, $$p(\mathbf{a}|\mathbf{x}) =\int d\lambda_1\,\mu_1(\lambda_1)\cdots \int d\lambda_m\,\mu_m(\lambda_m) \prod_{k=1}^n p(a_k|x_k,\bar\lambda_k),$$5, and $$p(\mathbf{a}|\mathbf{x}) =\int d\lambda_1\,\mu_1(\lambda_1)\cdots \int d\lambda_m\,\mu_m(\lambda_m) \prod_{k=1}^n p(a_k|x_k,\bar\lambda_k),$$6, one for each choice of classical branch. Full network nonlocality is certified iff all three are violated simultaneously. The witness $$p(\mathbf{a}|\mathbf{x}) =\int d\lambda_1\,\mu_1(\lambda_1)\cdots \int d\lambda_m\,\mu_m(\lambda_m) \prod_{k=1}^n p(a_k|x_k,\bar\lambda_k),$$7 is a linear combination of four-body, three-body, two-body, and one-body correlators such as $$p(\mathbf{a}|\mathbf{x}) =\int d\lambda_1\,\mu_1(\lambda_1)\cdots \int d\lambda_m\,\mu_m(\lambda_m) \prod_{k=1}^n p(a_k|x_k,\bar\lambda_k),$$8, with the explicit correlator convention

$$p(\mathbf{a}|\mathbf{x}) =\int d\lambda_1\,\mu_1(\lambda_1)\cdots \int d\lambda_m\,\mu_m(\lambda_m) \prod_{k=1}^n p(a_k|x_k,\bar\lambda_k),$$9

The same inflation logic also underlies bilocal polynomial witnesses (Wang et al., 2022).

In the bilocal scenario, Pozas-Kerstjens and coauthors derived two polynomial witnesses, commonly written as $\lambda_j$0 and $\lambda_j$1. Each single violation rules out one designated classical source, and simultaneous violation excludes both hybrid decompositions. The derivation clones the classical-source outputs in an inflated model while respecting no-cloning of the no-signaling resource, then imposes positivity and no-signaling constraints to obtain the bounds (Gu et al., 2023).

Subsequent work extended this program in two directions. First, a single inequality $\lambda_j$2 and its summed form $\lambda_j$3 were shown to rule out a hybrid model in which a tripartite source is only biseparable while a bipartite source is arbitrary no-signaling; the same inequality also witnesses genuine multipartite nonlocality of the tripartite source (Ning-Ning et al., 2024). Second, arbitrary-party and unbounded-input families of inequalities were introduced for star and chain networks: $\lambda_j$4 with optimal quantum values

$\lambda_j$5

Their local–nonlocal hybrid bounds are $\lambda_j$6 and $\lambda_j$7, respectively, creating analytic full-network-nonlocality gaps without dimension assumptions (Munshi et al., 2024).

4. Quantum realizations and experimental demonstrations

The standard quantum architecture for full network nonlocality uses independent entangled sources and a central entanglement-swapping measurement. In the bilocal photonic implementation, two independent polarization-entangled photon sources feed a partial Bell-state measurement at Bob, while Alice and Charlie measure $\lambda_j$8, $\lambda_j$9, $S_j$0, and $S_j$1. The ideal quantum prediction is

$S_j$2

and the experiment reported

$S_j$3

both above the classical bound $S_j$4 by more than $S_j$5 (Håkansson et al., 2022).

A later loophole-constrained realization imposed strict source independence, measurement independence, and locality. Two type-0 SPDC sources in PPMgLN crystals inside Sagnac loops were pumped by a 250 MHz pulse-pattern generator; phase randomization erased pulse-to-pulse coherence, fast QRNGs generated the settings, and the space-time analysis established inter-node separations $S_j$6 m. In the symmetric case $S_j$7, the measured values were

$S_j$8

both $S_j$9 by more than $\bar\lambda_k$0 (Gu et al., 2023).

The first experimental demonstration beyond the bilocal scenario used a star-shaped photonic network with three independent sources of maximally entangled qubit pairs $\bar\lambda_k$1, branch observables

$\bar\lambda_k$2

and a three-qubit GHZ projection at the center. Optimizing over $\bar\lambda_k$3 yields a maximal quantum violation $\bar\lambda_k$4 at $\bar\lambda_k$5, $\bar\lambda_k$6, and full network nonlocality persists for source visibility $\bar\lambda_k$7. Experimentally, 155 019 six-fold coincidences gave $\bar\lambda_k$8 and

$\bar\lambda_k$9

each exceeding the bound $P_k$0 by $P_k$1 (Wang et al., 2022).

A further photonic experiment combined a tripartite and a bipartite source, with Charlie performing a partial Bell-state measurement and the remaining parties using Pauli measurements. Over eight settings with approximately 5 300 six-fold coincidences each, it obtained

$P_k$2

and therefore

$P_k$3

by $P_k$4, simultaneously certifying genuine multipartite nonlocality and a strengthened form of full network nonlocality (Ning-Ning et al., 2024).

Scenario	Witness	Reported result
Bilocal photonic network	$P_k$5	$P_k$6 (Håkansson et al., 2022)
Bilocal with strict locality constraints	$P_k$7	$P_k$8 (Gu et al., 2023)
Three-source star network	$P_k$9	$$p(a,b,c|x,z)=\int d\lambda\,\rho(\lambda)\,p(a|x,\lambda)\,p(b,c|\lambda,z),$$0 (Wang et al., 2022)
Hybrid tripartite–bipartite network	$$p(a,b,c|x,z)=\int d\lambda\,\rho(\lambda)\,p(a|x,\lambda)\,p(b,c|\lambda,z),$$1	$$p(a,b,c|x,z)=\int d\lambda\,\rho(\lambda)\,p(a|x,\lambda)\,p(b,c|\lambda,z),$$2 (Ning-Ning et al., 2024)

5. Generalizations and variants

Several extensions show that full network nonlocality is not confined to the original fixed-input photonic architectures. In continuous-variable networks, one-way network nonlocality was defined for a two-source chain in which Bob may send classical information to Alice and Charlie. The witness

$$p(a,b,c|x,z)=\int d\lambda\,\rho(\lambda)\,p(a|x,\lambda)\,p(b,c|\lambda,z),$$3

is violated by an all-optical continuous-variable entanglement-swapping scheme based on two-mode squeezed vacuum states, a high-gain parametric amplifier, and sign-binned homodyne measurements. The same work explicitly states that a genuine full-network certification in longer chains or star networks would require nonlinear Bell-type inequalities or hierarchies of chain tests (Jiang et al., 26 Jun 2025).

A stronger hybrid benchmark has also been proposed. In the four-party network with a tripartite GHZ source and a bipartite EPR source, the excluded model allows the tripartite source to be merely biseparable while the bipartite source may be arbitrary no-signaling. This “FNN♯” condition is stronger than the original full network nonlocality benchmark and is witnessed by the same single inequality that detects genuine multipartite nonlocality (Ning-Ning et al., 2024).

Another generalization concerns measurement structure. Earlier full-network-nonlocality demonstrations in the bilocal scenario relied on entangled measurements at the central node, such as partial Bell-state measurements or Elegant Joint Measurements. A later construction showed that entangled measurements are not necessary: separable product measurements augmented with bidirectional classical feedforward suffice. In that protocol, the bilocal FNN witnesses

$$p(a,b,c|x,z)=\int d\lambda\,\rho(\lambda)\,p(a|x,\lambda)\,p(b,c|\lambda,z),$$4

reach $$p(a,b,c|x,z)=\int d\lambda\,\rho(\lambda)\,p(a|x,\lambda)\,p(b,c|\lambda,z),$$5 for Werner-state visibility $$p(a,b,c|x,z)=\int d\lambda\,\rho(\lambda)\,p(a|x,\lambda)\,p(b,c|\lambda,z),$$6, whereas the entangled-measurement strategy yields $$p(a,b,c|x,z)=\int d\lambda\,\rho(\lambda)\,p(a|x,\lambda)\,p(b,c|\lambda,z),$$7 (Polino et al., 13 Apr 2026). The same architecture also realizes minimal network nonclassicality, a different notion in which the distribution lies outside the fully classical bilocal set but inside both single-source-classical hybrid sets.

Topological robustness provides a further perspective. In large ring networks, token-counting correlations generate a classical inequality $$p(a,b,c|x,z)=\int d\lambda\,\rho(\lambda)\,p(a|x,\lambda)\,p(b,c|\lambda,z),$$8 while the honest quantum strategy gives $$p(a,b,c|x,z)=\int d\lambda\,\rho(\lambda)\,p(a|x,\lambda)\,p(b,c|\lambda,z),$$9. An outlined extension shows that trusting only a small subgraph of two or three neighboring parties can suffice to certify full-network-nonlocality-type behavior across the whole ring by conditioning the rest of the network into an effective triangle (Boreiri et al., 2024).

6. Noise, resource theory, and applications

The robustness of full network nonlocality is currently understood only in restricted families. In the three-source star experiment, source noise modeled as

$A^{(1)}\!-\!B$0

still permits full network nonlocality for $A^{(1)}\!-\!B$1 (Wang et al., 2022). In the simultaneous GMN–FNN♯ test, a total Werner-state visibility $A^{(1)}\!-\!B$2 is sufficient to violate the same inequality (Ning-Ning et al., 2024).

Noise can also be treated at the channel level. In a trilocal star network, the full-network bound can be written as

$A^{(1)}\!-\!B$3

A single-qubit depolarizing channel becomes $A^{(1)}\!-\!B$4-use full-network-nonlocality breaking when

$A^{(1)}\!-\!B$5

and for $A^{(1)}\!-\!B$6 this holds for all $A^{(1)}\!-\!B$7, so five channel uses suffice to destroy any full network nonlocality in that scenario. By contrast, the dephasing channel considered there does not satisfy the paper’s unital-breaking criterion and therefore does not break full network nonlocality in that setting (Mukherjee et al., 30 Oct 2025).

Resource-theoretic questions include recyclability and measurement dependence. In an extended bilocal network with weak measurements, passive full-network-nonlocality sharing is impossible, while active sharing is possible: numerical optimization yields simultaneous violations for $A^{(1)}\!-\!B$8, with maximal common value $A^{(1)}\!-\!B$9 at $$p(a_1,a_2,a_3,b|x_1,x_2,x_3) =\sum_\lambda p(\lambda)\,p(a_1|x_1,\lambda)\,p_{\mathrm{NSI}}(a_2,a_3,b|x_2,x_3,\lambda),$$0; however, active sharing requires $$p(a_1,a_2,a_3,b|x_1,x_2,x_3) =\sum_\lambda p(\lambda)\,p(a_1|x_1,\lambda)\,p_{\mathrm{NSI}}(a_2,a_3,b|x_2,x_3,\lambda),$$1 under Werner noise, indicating strong fragility (Cai et al., 2022). Under one-sided measurement dependence, classical no-signaling models can reproduce the maximal quantum network violations once the measurement-dependence parameter reaches $$p(a_1,a_2,a_3,b|x_1,x_2,x_3) =\sum_\lambda p(\lambda)\,p(a_1|x_1,\lambda)\,p_{\mathrm{NSI}}(a_2,a_3,b|x_2,x_3,\lambda),$$2 in the bilocal case and $$p(a_1,a_2,a_3,b|x_1,x_2,x_3) =\sum_\lambda p(\lambda)\,p(a_1|x_1,\lambda)\,p_{\mathrm{NSI}}(a_2,a_3,b|x_2,x_3,\lambda),$$3 in the 3-local star case, showing that freedom-of-choice assumptions remain operationally relevant for both standard and full network nonlocality (Kundu et al., 2024).

Security-motivated applications have begun to appear. A four-party entanglement-assisted QKD protocol uses violation of a trilocal full-network-nonlocality inequality as a security check; compared with a CHSH-based construction, it is reported to be more secure, with the quantum bit error rate reducible below $$p(a_1,a_2,a_3,b|x_1,x_2,x_3) =\sum_\lambda p(\lambda)\,p(a_1|x_1,\lambda)\,p_{\mathrm{NSI}}(a_2,a_3,b|x_2,x_3,\lambda),$$4 using full network nonlocality, compared with $$p(a_1,a_2,a_3,b|x_1,x_2,x_3) =\sum_\lambda p(\lambda)\,p(a_1|x_1,\lambda)\,p_{\mathrm{NSI}}(a_2,a_3,b|x_2,x_3,\lambda),$$5 when exploiting Bell-CHSH nonlocality (Mukherjee, 20 Mar 2026). A plausible implication is that full network nonlocality is not only a foundational refinement of network Bell theory, but also a certification primitive for distributed cryptographic tasks in which every link must be intrinsically quantum.

The field remains structurally open. Existing work identifies several unresolved directions: closing the detection and memory loopholes in photonic implementations, deriving noise-tolerant rigidity statements, extending certification to higher-dimensional and more complex topologies, and obtaining genuinely device-independent self-testing in arbitrary-party networks using experimentally simpler two-output measurements (Gu et al., 2023, Renou et al., 2022, Munshi et al., 2024).