Genuine Tripartite Nonlocality (GTN)

• Genuine Tripartite Nonlocality (GTN) is defined as irreducible three-party quantum correlations that cannot be explained by pairwise resources alone.
• Different models such as Svetlichny, NS2/T2, and LOSR frameworks offer distinct operational criteria and certification routes using specialized inequalities and witnesses.
• Experimental demonstrations in optical, network, and continuous-variable setups highlight GTN's practical significance and pose challenges in addressing noise and decoherence.

Searching arXiv for recent and foundational papers on genuine tripartite nonlocality and related notions. Genuine tripartite nonlocality (GTN) denotes irreducible three-party nonlocal correlation: a tripartite behavior $P(a,b,c\mid x,y,z)$ is said to be genuinely tripartite nonlocal when it cannot be reduced to models in which nonclassicality is confined to fewer than all three parties. In the literature, however, GTN is not a single universally fixed notion. Older work formulates it through Svetlichny-type hybrid decompositions; later work refines this to no-signalling or time-ordered hybrid models; more recent network-causal work defines GTN relative to local operations and shared randomness (LOSR), asking whether any composition of pairwise resources plus global shared randomness can reproduce the observed behavior. The subject is therefore both a classification problem for multipartite correlations and a resource-theoretic question about what kinds of common causes are required to explain them (Bancal et al., 2011, Coiteux-Roy et al., 2021, Coiteux-Roy et al., 2021).

1. Definitions, model classes, and competing notions

The minimal reference point is ordinary tripartite Bell locality. A fully local three-party model has the form

$$P(a,b,c\mid x,y,z)=\sum_\lambda p(\lambda)\,P(a\mid x,\lambda)P(b\mid y,\lambda)P(c\mid z,\lambda).$$

Violation of a standard multipartite Bell inequality excludes only this fully local class; it does not by itself establish that the nonlocality is intrinsically tripartite (Huang et al., 2022).

Svetlichny’s formulation strengthens the foil theory by allowing, in each hidden-variable sector, one bipartition to be nonlocal while the third party remains local. In one common tripartite form,

$$p(a_1,a_2,a_3 \vert x_1,x_2,x_3) = q_{1}p_{a_1 \leftrightarrow a_2}+q_{2}p_{a_1 \leftrightarrow a_3}+q_{3}p_{a_2 \leftrightarrow a_3}, \qquad q_1+q_2+q_3=1,$$

with, for example,

$$p_{a_1 \leftrightarrow a_2} = \sum_{\lambda}p(a_1 \vert x_1,x_2,\lambda)\,p(a_2 \vert x_1,x_2,\lambda)\,p(a_3 \vert x_3,\lambda).$$

Correlations outside this convex hull are Svetlichny-genuinely tripartite nonlocal (Suprano et al., 2022). A related hybrid decomposition used in the no-signalling postselection literature writes absence of GTN as

$$\begin{aligned} P_{abc|xyz} &=\sum_{\lambda_1\in\Lambda_1} P_{\lambda_1}P_{bc|yz\lambda_1}P_{a|x\lambda_1} \ &\quad + \sum_{\lambda_2\in\Lambda_2}P_{\lambda_2}P_{ac|xz\lambda_2}P_{b|y\lambda_2} \ &\quad + \sum_{\lambda_3\in\Lambda_3}P_{\lambda_3}P_{ab|xy\lambda_3}P_{c|z\lambda_3}, \end{aligned}$$

with additional no-signalling restrictions when the Bell inequality under consideration assumes them (Gebhart et al., 2021).

A further refinement distinguishes admissible hidden bipartite resources. The definitions $NS_2$ and $T_2$ introduced for multipartite nonlocality restrict the bipartite sectors respectively to no-signalling resources and to causally time-ordered one-way-signalling resources, and satisfy the strict inclusions

$NS_2 \subset T_2 \subset S_2.$

These inclusions show that “genuine tripartite nonlocality” depends on which hybrid class is being excluded (Bancal et al., 2011).

A conceptually different notion is genuine LOSR tripartite nonlocality, or GTN “in network.” Here the null model is not a Svetlichny convex hull but the full set of behaviors obtainable when each pair of parties may share arbitrary bipartite generalized probabilistic theory resources, all parties may share unlimited classical randomness, and only local operations are allowed at the observed nodes. In this framework, the target question is whether a tripartite correlation can be assembled entirely from pairwise nonclassical resources. The network literature emphasizes that Svetlichny-type GMN can still arise in systems built only from bipartite resources, so Svetlichny’s notion is not aligned with this causal question (Coiteux-Roy et al., 2021, Huang et al., 2022).

Notion	Excluded model class	Representative certification route
Svetlichny GTN	Hybrid bipartition models with one local wing and one nonlocal pair	Svetlichny inequality
$NS_2$/$T_2$	Hybrid models with no-signalling or time-ordered bipartite sectors	Bancal-type facet inequalities
LOSR/network GTN	Arbitrary pairwise GPT resources + local operations + unlimited shared randomness	Inflation-derived Bell-type inequalities

This multiplicity of definitions is not merely terminological. It reflects different operational benchmarks: exclusion of bipartition-local decompositions, exclusion of no-signalling or time-ordered hybrids, or exclusion of all pairwise-resource network explanations.

2. Witnesses, inequalities, and certification mechanisms

The canonical Svetlichny operator used in three-qubit work is

$$P(a,b,c\mid x,y,z)=\sum_\lambda p(\lambda)\,P(a\mid x,\lambda)P(b\mid y,\lambda)P(c\mid z,\lambda).$$0

with the hybrid-model bound

$$P(a,b,c\mid x,y,z)=\sum_\lambda p(\lambda)\,P(a\mid x,\lambda)P(b\mid y,\lambda)P(c\mid z,\lambda).$$1

A state is GTN-detectable in this sense when

$$P(a,b,c\mid x,y,z)=\sum_\lambda p(\lambda)\,P(a\mid x,\lambda)P(b\mid y,\lambda)P(c\mid z,\lambda).$$2

For three-qubit $$P(a,b,c\mid x,y,z)=\sum_\lambda p(\lambda)\,P(a\mid x,\lambda)P(b\mid y,\lambda)P(c\mid z,\lambda).$$3 states, a closed formula exists: $$P(a,b,c\mid x,y,z)=\sum_\lambda p(\lambda)\,P(a\mid x,\lambda)P(b\mid y,\lambda)P(c\mid z,\lambda).$$4 with $$P(a,b,c\mid x,y,z)=\sum_\lambda p(\lambda)\,P(a\mid x,\lambda)P(b\mid y,\lambda)P(c\mid z,\lambda).$$5 a linear combination of diagonal entries. This formula is central in several relativistic and open-system studies because Hawking mixing and decoherence preserve $$P(a,b,c\mid x,y,z)=\sum_\lambda p(\lambda)\,P(a\mid x,\lambda)P(b\mid y,\lambda)P(c\mid z,\lambda).$$6-state structure (Wu et al., 2022, Liu et al., 2024).

Beyond Svetlichny, multipartite Bell theory introduced facet inequalities adapted to $$P(a,b,c\mid x,y,z)=\sum_\lambda p(\lambda)\,P(a\mid x,\lambda)P(b\mid y,\lambda)P(c\mid z,\lambda).$$7 and $$P(a,b,c\mid x,y,z)=\sum_\lambda p(\lambda)\,P(a\mid x,\lambda)P(b\mid y,\lambda)P(c\mid z,\lambda).$$8. A central example is

$$P(a,b,c\mid x,y,z)=\sum_\lambda p(\lambda)\,P(a\mid x,\lambda)P(b\mid y,\lambda)P(c\mid z,\lambda).$$9

whose violation certifies genuine three-way nonlocality under both $$p(a_1,a_2,a_3 \vert x_1,x_2,x_3) = q_{1}p_{a_1 \leftrightarrow a_2}+q_{2}p_{a_1 \leftrightarrow a_3}+q_{3}p_{a_2 \leftrightarrow a_3}, \qquad q_1+q_2+q_3=1,$$0 and $$p(a_1,a_2,a_3 \vert x_1,x_2,x_3) = q_{1}p_{a_1 \leftrightarrow a_2}+q_{2}p_{a_1 \leftrightarrow a_3}+q_{3}p_{a_2 \leftrightarrow a_3}, \qquad q_1+q_2+q_3=1,$$1 definitions (Bancal et al., 2011).

Hardy-type tests provide an inequality-free route. One three-party version requires

$$p(a_1,a_2,a_3 \vert x_1,x_2,x_3) = q_{1}p_{a_1 \leftrightarrow a_2}+q_{2}p_{a_1 \leftrightarrow a_3}+q_{3}p_{a_2 \leftrightarrow a_3}, \qquad q_1+q_2+q_3=1,$$2

All non-signalling hybrid models fail this test, and it is passed by every tripartite entangled pure state after suitable local projection and measurement choice (Yu et al., 2013).

The network/LOSR literature uses different witnesses because the null model is stronger. A prominent tripartite witness combines a conditioned CHSH term,

$$p(a_1,a_2,a_3 \vert x_1,x_2,x_3) = q_{1}p_{a_1 \leftrightarrow a_2}+q_{2}p_{a_1 \leftrightarrow a_3}+q_{3}p_{a_2 \leftrightarrow a_3}, \qquad q_1+q_2+q_3=1,$$3

with a consistency term

$$p(a_1,a_2,a_3 \vert x_1,x_2,x_3) = q_{1}p_{a_1 \leftrightarrow a_2}+q_{2}p_{a_1 \leftrightarrow a_3}+q_{3}p_{a_2 \leftrightarrow a_3}, \qquad q_1+q_2+q_3=1,$$4

to form

$$p(a_1,a_2,a_3 \vert x_1,x_2,x_3) = q_{1}p_{a_1 \leftrightarrow a_2}+q_{2}p_{a_1 \leftrightarrow a_3}+q_{3}p_{a_2 \leftrightarrow a_3}, \qquad q_1+q_2+q_3=1,$$5

The bound $$p(a_1,a_2,a_3 \vert x_1,x_2,x_3) = q_{1}p_{a_1 \leftrightarrow a_2}+q_{2}p_{a_1 \leftrightarrow a_3}+q_{3}p_{a_2 \leftrightarrow a_3}, \qquad q_1+q_2+q_3=1,$$6 holds for all tripartite behaviors obtainable from arbitrary bipartite GPT resources plus shared randomness; hence $$p(a_1,a_2,a_3 \vert x_1,x_2,x_3) = q_{1}p_{a_1 \leftrightarrow a_2}+q_{2}p_{a_1 \leftrightarrow a_3}+q_{3}p_{a_2 \leftrightarrow a_3}, \qquad q_1+q_2+q_3=1,$$7 certifies genuine LOSR tripartite nonlocality (Huang et al., 2022).

A second network witness, used in triangle architectures, sums three parallel bipartite Bell expressions. For chained/BKP inequalities with $$p(a_1,a_2,a_3 \vert x_1,x_2,x_3) = q_{1}p_{a_1 \leftrightarrow a_2}+q_{2}p_{a_1 \leftrightarrow a_3}+q_{3}p_{a_2 \leftrightarrow a_3}, \qquad q_1+q_2+q_3=1,$$8 settings,

$$p(a_1,a_2,a_3 \vert x_1,x_2,x_3) = q_{1}p_{a_1 \leftrightarrow a_2}+q_{2}p_{a_1 \leftrightarrow a_3}+q_{3}p_{a_2 \leftrightarrow a_3}, \qquad q_1+q_2+q_3=1,$$9

and any Svetlichny model on the triangle obeys

$$p_{a_1 \leftrightarrow a_2} = \sum_{\lambda}p(a_1 \vert x_1,x_2,\lambda)\,p(a_2 \vert x_1,x_2,\lambda)\,p(a_3 \vert x_3,\lambda).$$0

Violation of $$p_{a_1 \leftrightarrow a_2} = \sum_{\lambda}p(a_1 \vert x_1,x_2,\lambda)\,p(a_2 \vert x_1,x_2,\lambda)\,p(a_3 \vert x_3,\lambda).$$1 certifies GTN in the Svetlichny sense for that network scenario (Suprano et al., 2022).

3. Relation to genuine tripartite entanglement

GTN is a correlation-level statement, not a state-level one. Genuine tripartite entanglement (GTE) concerns separability structure of $$p_{a_1 \leftrightarrow a_2} = \sum_{\lambda}p(a_1 \vert x_1,x_2,\lambda)\,p(a_2 \vert x_1,x_2,\lambda)\,p(a_3 \vert x_3,\lambda).$$2; GTN concerns whether the observed $$p_{a_1 \leftrightarrow a_2} = \sum_{\lambda}p(a_1 \vert x_1,x_2,\lambda)\,p(a_2 \vert x_1,x_2,\lambda)\,p(a_3 \vert x_3,\lambda).$$3 can be reproduced by a chosen operational class. The distinction is explicit in both experimental and theoretical work: a state can be genuinely tripartite entangled while measured correlations fail to certify GTN because the measurements are suboptimal, the noise is too large, or the targeted notion of GTN is especially demanding (Huang et al., 2022, Wu et al., 2022).

For pure states, the relation can become unusually tight, but only relative to a specified nonlocality notion. One theorem states that every tripartite entangled pure state is genuinely tripartite nonlocal against non-signalling hybrid models, via a three-party Hardy-type test and local projection to a fully entangled three-qubit state (Yu et al., 2013). Independently, numerical evidence for $$p_{a_1 \leftrightarrow a_2} = \sum_{\lambda}p(a_1 \vert x_1,x_2,\lambda)\,p(a_2 \vert x_1,x_2,\lambda)\,p(a_3 \vert x_3,\lambda).$$4 and $$p_{a_1 \leftrightarrow a_2} = \sum_{\lambda}p(a_1 \vert x_1,x_2,\lambda)\,p(a_2 \vert x_1,x_2,\lambda)\,p(a_3 \vert x_3,\lambda).$$5 notions suggests that all pure genuinely tripartite entangled three-qubit states can produce three-way nonlocal correlations, even though Svetlichny’s criterion may fail on some of them (Bancal et al., 2011).

The gap between entanglement and Svetlichny GTN is especially clear for Gaussian states. In the three-mode Gaussian theory, fully symmetric pure states are fully inseparable for every $$p_{a_1 \leftrightarrow a_2} = \sum_{\lambda}p(a_1 \vert x_1,x_2,\lambda)\,p(a_2 \vert x_1,x_2,\lambda)\,p(a_3 \vert x_3,\lambda).$$6, yet Svetlichny violation occurs only for

$$p_{a_1 \leftrightarrow a_2} = \sum_{\lambda}p(a_1 \vert x_1,x_2,\lambda)\,p(a_2 \vert x_1,x_2,\lambda)\,p(a_3 \vert x_3,\lambda).$$7

For mixed three-mode Gaussian states, purity imposes a sharp obstruction: $$p_{a_1 \leftrightarrow a_2} = \sum_{\lambda}p(a_1 \vert x_1,x_2,\lambda)\,p(a_2 \vert x_1,x_2,\lambda)\,p(a_3 \vert x_3,\lambda).$$8 At fixed residual Rényi-2 tripartite entanglement, the maximal Svetlichny value admits tight upper and lower bounds; the lower bound is saturated by fully symmetric states, and all pure states with

$$p_{a_1 \leftrightarrow a_2} = \sum_{\lambda}p(a_1 \vert x_1,x_2,\lambda)\,p(a_2 \vert x_1,x_2,\lambda)\,p(a_3 \vert x_3,\lambda).$$9

necessarily violate Svetlichny (Adesso et al., 2013).

The same separation appears in optical GHZ-network experiments. A high GHZ fidelity supports the plausibility of a GTN violation but does not constitute GTN by itself. In the locality-constrained photonic network test, the measured postselected GHZ fidelity was $$\begin{aligned} P_{abc|xyz} &=\sum_{\lambda_1\in\Lambda_1} P_{\lambda_1}P_{bc|yz\lambda_1}P_{a|x\lambda_1} \ &\quad + \sum_{\lambda_2\in\Lambda_2}P_{\lambda_2}P_{ac|xz\lambda_2}P_{b|y\lambda_2} \ &\quad + \sum_{\lambda_3\in\Lambda_3}P_{\lambda_3}P_{ab|xy\lambda_3}P_{c|z\lambda_3}, \end{aligned}$$0, just above the theoretical threshold needed for the LOSR-network witness, yet the GTN claim rests on the observed Bell-type inequality violation rather than on the fidelity witness (Huang et al., 2022).

4. Experimental demonstrations and network realizations

Two 2022 photonic experiments established distinct experimental meanings of tripartite irreducibility. In a strict-locality photonic triangular network, a postselected three-photon GHZ state was tested against the LOSR/network witness

$$\begin{aligned} P_{abc|xyz} &=\sum_{\lambda_1\in\Lambda_1} P_{\lambda_1}P_{bc|yz\lambda_1}P_{a|x\lambda_1} \ &\quad + \sum_{\lambda_2\in\Lambda_2}P_{\lambda_2}P_{ac|xz\lambda_2}P_{b|y\lambda_2} \ &\quad + \sum_{\lambda_3\in\Lambda_3}P_{\lambda_3}P_{ab|xy\lambda_3}P_{c|z\lambda_3}, \end{aligned}$$1

The measured value

$$\begin{aligned} P_{abc|xyz} &=\sum_{\lambda_1\in\Lambda_1} P_{\lambda_1}P_{bc|yz\lambda_1}P_{a|x\lambda_1} \ &\quad + \sum_{\lambda_2\in\Lambda_2}P_{\lambda_2}P_{ac|xz\lambda_2}P_{b|y\lambda_2} \ &\quad + \sum_{\lambda_3\in\Lambda_3}P_{\lambda_3}P_{ab|xy\lambda_3}P_{c|z\lambda_3}, \end{aligned}$$2

exceeded the universal bipartite-resource bound by $$\begin{aligned} P_{abc|xyz} &=\sum_{\lambda_1\in\Lambda_1} P_{\lambda_1}P_{bc|yz\lambda_1}P_{a|x\lambda_1} \ &\quad + \sum_{\lambda_2\in\Lambda_2}P_{\lambda_2}P_{ac|xz\lambda_2}P_{b|y\lambda_2} \ &\quad + \sum_{\lambda_3\in\Lambda_3}P_{\lambda_3}P_{ab|xy\lambda_3}P_{c|z\lambda_3}, \end{aligned}$$3 standard deviations. The setup spacelike separated all relevant basis-choice and measurement events, used fast QRNGs and high-speed polarization analyzers, and realized a triggered postselected GHZ source from two independent Bell-pair sources rather than a direct three-photon source. The experiment closed the locality loophole among the three parties, but retained fair-sampling, detection, and postselection assumptions (Huang et al., 2022).

A second experiment demonstrated GTN in a quantum triangle network consisting only of three independent bipartite entangled sources and local separable measurements. Here the target notion was explicitly Svetlichny-esque. Using parallel BKP inequalities, the total triangle scores were

$$\begin{aligned} P_{abc|xyz} &=\sum_{\lambda_1\in\Lambda_1} P_{\lambda_1}P_{bc|yz\lambda_1}P_{a|x\lambda_1} \ &\quad + \sum_{\lambda_2\in\Lambda_2}P_{\lambda_2}P_{ac|xz\lambda_2}P_{b|y\lambda_2} \ &\quad + \sum_{\lambda_3\in\Lambda_3}P_{\lambda_3}P_{ab|xy\lambda_3}P_{c|z\lambda_3}, \end{aligned}$$4

violating the corresponding Svetlichny bounds $$\begin{aligned} P_{abc|xyz} &=\sum_{\lambda_1\in\Lambda_1} P_{\lambda_1}P_{bc|yz\lambda_1}P_{a|x\lambda_1} \ &\quad + \sum_{\lambda_2\in\Lambda_2}P_{\lambda_2}P_{ac|xz\lambda_2}P_{b|y\lambda_2} \ &\quad + \sum_{\lambda_3\in\Lambda_3}P_{\lambda_3}P_{ab|xy\lambda_3}P_{c|z\lambda_3}, \end{aligned}$$5, $$\begin{aligned} P_{abc|xyz} &=\sum_{\lambda_1\in\Lambda_1} P_{\lambda_1}P_{bc|yz\lambda_1}P_{a|x\lambda_1} \ &\quad + \sum_{\lambda_2\in\Lambda_2}P_{\lambda_2}P_{ac|xz\lambda_2}P_{b|y\lambda_2} \ &\quad + \sum_{\lambda_3\in\Lambda_3}P_{\lambda_3}P_{ab|xy\lambda_3}P_{c|z\lambda_3}, \end{aligned}$$6, and $$\begin{aligned} P_{abc|xyz} &=\sum_{\lambda_1\in\Lambda_1} P_{\lambda_1}P_{bc|yz\lambda_1}P_{a|x\lambda_1} \ &\quad + \sum_{\lambda_2\in\Lambda_2}P_{\lambda_2}P_{ac|xz\lambda_2}P_{b|y\lambda_2} \ &\quad + \sum_{\lambda_3\in\Lambda_3}P_{\lambda_3}P_{ab|xy\lambda_3}P_{c|z\lambda_3}, \end{aligned}$$7. The strongest normalized ratio occurred at $$\begin{aligned} P_{abc|xyz} &=\sum_{\lambda_1\in\Lambda_1} P_{\lambda_1}P_{bc|yz\lambda_1}P_{a|x\lambda_1} \ &\quad + \sum_{\lambda_2\in\Lambda_2}P_{\lambda_2}P_{ac|xz\lambda_2}P_{b|y\lambda_2} \ &\quad + \sum_{\lambda_3\in\Lambda_3}P_{\lambda_3}P_{ab|xy\lambda_3}P_{c|z\lambda_3}, \end{aligned}$$8, as predicted by the robustness analysis, and the protocol used only separable local measurements, with no Bell-state measurement or entangled joint measurement at any node (Suprano et al., 2022).

These demonstrations sharpened an existing conceptual divide. The triangle-network experiment showed that Svetlichny-type GTN can emerge from a structure built entirely from pairwise entangled sources, whereas the strict-locality GHZ-network experiment targeted a stronger causal statement: that no model based only on bipartite resources plus shared randomness can reproduce the observed statistics. The two results therefore certify different operational statements about “genuineness” (Suprano et al., 2022, Huang et al., 2022).

Finite-data statistical analysis has also been revisited. For LOSR-GMNL experiments, the prediction-based ratio (PBR) protocol yields valid $$\begin{aligned} P_{abc|xyz} &=\sum_{\lambda_1\in\Lambda_1} P_{\lambda_1}P_{bc|yz\lambda_1}P_{a|x\lambda_1} \ &\quad + \sum_{\lambda_2\in\Lambda_2}P_{\lambda_2}P_{ac|xz\lambda_2}P_{b|y\lambda_2} \ &\quad + \sum_{\lambda_3\in\Lambda_3}P_{\lambda_3}P_{ab|xy\lambda_3}P_{c|z\lambda_3}, \end{aligned}$$9-values against the null hypothesis of local operations on bipartite resources plus unlimited shared randomness, even under memory effects. Applied to one of the 2022 datasets, the PBR analysis returned

$NS_2$0

while an efficient optimization over a Bell-saturating approximating polytope reduced the number of constraints by more than an order of magnitude without sacrificing statistical validity (Patra et al., 2024).

5. Sequential, continuous-variable, and many-body generalizations

Sequential one-sided scenarios reveal that GTN is markedly less shareable than weaker multipartite nonlocality notions. In the original unsharp-measurement GHZ protocol, at most two Charlies could simultaneously demonstrate genuine tripartite nonlocality with a single Alice and Bob, while up to six Charlies could simultaneously demonstrate standard tripartite nonlocality via Mermin inequality violation (Saha et al., 2018). Later work showed that arbitrarily many sequential Charlies can share standard tripartite nonlocality and also genuinely non-signalling tripartite nonlocality, but not Svetlichny-type GTN; the authors explicitly noted that their method does not extend to the Svetlichny setting, where prior work limited sharing to at most two Charlies (Fu et al., 2022). A subsequent projective-measurement-only protocol sharpened the distinction further: with unbiased projective choices, at most one Charlie can share GTN, whereas biased projective choices plus shared classical randomness can raise this to two Charlies, still with no route beyond two in that setting (Xu et al., 2024).

In continuous-variable systems, GTN persists despite the positivity of Gaussian Wigner functions, provided the measurements are non-Gaussian. Three-mode Gaussian theory formulates Svetlichny and $NS_2$1 tests using displaced parity measurements, exploiting

$NS_2$2

For pure states, the maximal Svetlichny violation can be optimized over a reduced set of phase-space displacements; for fully symmetric pure states the violation saturates asymptotically at

$NS_2$3

below the general quantum maximum $NS_2$4. Although Svetlichny does not detect all fully inseparable pure Gaussian states, weaker $NS_2$5 inequalities were found numerically to be violated for all $NS_2$6 tested fully inseparable pure states (Adesso et al., 2013).

A concrete many-body realization arises in Bose–Einstein condensates undergoing collective atomic recoil in a ring cavity. The resulting three-mode Gaussian state of two atomic sidebands plus the scattered cavity field exhibits genuine tripartite entanglement that increases unboundedly with evolution time and exceeds any reduced bipartite entanglement. Under displaced parity measurements, the optimized Svetlichny parameter violates the classical bound in a broad parameter region, providing a continuous-variable realization of GTN in a nonsymmetric Gaussian state generated by collective atomic recoil instability (Piano et al., 2013).

6. Postselection, curved spacetime, and open-system dynamics

Postselection is pervasive in multipartite optical schemes, but it is not automatically legitimate for GTN certification. A causal-diagram analysis established sufficient conditions under which postselected data still support a valid genuine multipartite nonlocality claim. In the tripartite case, the key theorem states that a postselection which can be equivalently decided by any two of the three parties is valid for verification of genuine three-partite nonlocality. This “all-but-one” principle was applied to the Yurke–Stoler independent-source scheme, yielding a valid postselected demonstration of maximal Svetlichny violation

$NS_2$7

from three independent particle sources under the theorem’s assumptions (Gebhart et al., 2021).

Relativistic quantum information studies use the Svetlichny criterion to track GTN across horizons. For a GHZ-like Dirac-field state in Schwarzschild spacetime, one analysis found that physically accessible GTN in the exterior subsystem $NS_2$8 decreases with Hawking temperature and exhibits sudden death at

$NS_2$9

while no physically inaccessible GTN is generated in any reduced subsystem involving interior modes. In that setting, GTN does not pass through the event horizon, although genuine tripartite entanglement can be redistributed there (Wu et al., 2022).

A later study showed that this conclusion is not universal for mixed initial states. For

$T_2$0

with two accelerated detectors, Hawking effect can generate physically inaccessible GTN in states such as $T_2$1. The sudden-birth condition can be written as

$T_2$2

which requires

$T_2$3

Thus, for suitable mixed fermionic states, GTN can be redistributed across the horizon in a sense not present for pure GHZ input or for $T_2$4 (Zhang et al., 2022).

Decoherence further modifies the picture. In an open Dirac-field model with generalized amplitude damping on Alice and local filtering,

$T_2$5

physically accessible GTN can be completely destroyed by decoherence, yet local filtering can restore or generate accessible GTN within a finite Hawking-temperature window. In the same model, physically inaccessible GTN is never generated, even though inaccessible genuine tripartite entanglement can be produced and enhanced (Liu et al., 2024). More generally, numerical optimization of the maximal Svetlichny value has been used to quantify GTN for arbitrary three-qubit states in dissipative systems, verify complementarity between tripartite and bipartite nonlocality, and show that the quantum Zeno effect can protect both GTN and GTE under decoherence (Xiong et al., 2024).

Taken together, these results show that GTN is not a monolithic property. Its content depends on the excluded causal model, its witnesses range from Hardy paradoxes to inflation-derived network inequalities, its experimental status depends sensitively on locality, sampling, and postselection assumptions, and its dynamical behavior differs sharply from that of entanglement. What remains constant across these frameworks is the central operational claim: some tripartite correlations are irreducibly three-party, in the sense relevant to the underlying notion, and cannot be reduced to purely pairwise nonclassical structure.