Trilocal Inequality in Quantum Networks

• Trilocal inequality is a Bell-type constraint for networks with three independent sources, leading to nonlinear and nonconvex classical correlation sets.
• Different network topologies (linear chains, triangles, stars) yield distinct trilocal formulations with root-based bounds that certify nontrilocality.
• Derivation methods use hidden-variable factorization, Hölder-type bounds, and sum-of-squares techniques, with quantum violations demonstrating robustness under noise.

Searching arXiv for the cited papers to ground the article in current literature. A trilocal inequality is a Bell-type constraint derived for a network with three independent sources, where the hidden-variable distribution factorizes across those sources. In contrast with standard Bell locality, the source-independence assumption generally makes the classical correlation set non-convex, so trilocal constraints are typically nonlinear and depend strongly on network topology, measurement structure, and symmetry reductions. The term is therefore used for a family of related inequalities: for linear four-party chains, five-party multipartite-source networks, triangle networks with fixed local measurements, star-shaped four-party networks, asymmetric multisource configurations, and the fully symmetric no-input binary-output triangle network. Across these settings, violation of the relevant inequality certifies nontrilocality, i.e. network nonclassicality that cannot be explained by local response functions fed by three independent hidden variables (Mukherjee et al., 2014, Mukherjee et al., 2017, Mukherjee, 2023, Silva et al., 20 Mar 2025).

1. Source independence and the meaning of trilocality

The common structural assumption is factorization of hidden variables. In the five-party scenario of Mukherjee et al., the trilocal model is

$$P(a,b,c,d,e|x,y,z,w,u) = \int d\lambda_1 d\lambda_2 d\lambda_3\, \rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3) \,P(a|x,\lambda_1)P(b|y,\lambda_1,\lambda_2,\lambda_3) P(c|z,\lambda_1,\lambda_2,\lambda_3)P(d|w,\lambda_2)P(e|u,\lambda_3),$$

with the strict independence condition

$$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$

(Mukherjee et al., 2017).

In the linear $n$-local framework specialized to $n=3$, the four-party conditional probability likewise factorizes over three sources,

$$P(a_1,a_2,a_3,a_4|x_1,x_2,x_3,x_4) =\int d\lambda_1 d\lambda_2 d\lambda_3\, \rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)\, P(a_1|x_1,\lambda_1)P(a_2|x_2,\lambda_1,\lambda_2) P(a_3|x_3,\lambda_2,\lambda_3)P(a_4|x_4,\lambda_3)$$

(Mukherjee et al., 2014).

In triangle-network work, the same principle appears in a no-input form. For the fully symmetric binary-output triangle network, a distribution is triangle-local, or “trilocal,” iff

$$p(a,b,c)=\sum_{\alpha,\beta,\gamma} q(\alpha)r(\beta)s(\gamma)\, A(a|\beta,\gamma)\,B(b|\gamma,\alpha)\,C(c|\alpha,\beta),$$

with independent sources $q,r,s$ and deterministic response functions $A,B,C\in\{0,1\}$ (Silva et al., 20 Mar 2025).

A central conceptual consequence is that the relevant classical sets differ from ordinary Bell-local polytopes. In standard Bell scenarios the local set is convex, whereas in network scenarios the trilocal set is generally non-convex, and the corresponding witnesses become nonlinear or algebraic rather than linear (Silva et al., 20 Mar 2025).

2. Canonical formulations across network topologies

The phrase “trilocal inequality” does not denote a single universal formula. It denotes the inequality appropriate to a specified three-source network.

Scenario	Hidden-variable structure	Representative trilocal inequality
Linear four-party chain (Mukherjee et al., 2014)	$S_1,S_2,S_3$ in a line	$\sqrt{|I|}+\sqrt{|J|}\le 1$
Five-party multipartite-source network (Mukherjee et al., 2017)	three independent tripartite sources	$$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$0
Triangle with fixed local measurements (Mukherjee, 2023)	three pairwise sources, no inputs, four outcomes	$$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$1
Four-party star network (Mukherjee, 20 Mar 2026)	central node and three edge links	$$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$2
Asymmetric four-party networks (Sasmal et al., 2023)	central Bob and three edge parties	$$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$3, $$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$4
Fully symmetric no-input triangle (Silva et al., 20 Mar 2025)	three pairwise sources, binary outputs	six boundary inequalities $$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$5 plus bit-flips

In the linear four-party chain, the correlators are

$$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$6

and one defines

$$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$7

$$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$8

after which the trilocal bound is

$$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$9

(Mukherjee et al., 2014).

In the five-party network of Mukherjee et al., one introduces eight grouped correlators

$n$0

and for all $n$1 obtains

$n$2

The summary states that this yields $n$3 independent inequalities (Mukherjee et al., 2017).

In the triangle-network criteria of Ghosh et al., each party performs a fixed local measurement with four possible outcomes, encoded by bit pairs. The resulting theorem is

$n$4

where $n$5 are linear combinations of correlators built from observables $n$6 (Mukherjee, 2023).

In the symmetric no-input triangle network, the distribution is reduced to the symmetrized correlators

$n$7

with

$n$8

subject to the positivity tetrahedron. The boundary then breaks into GHZ, $n$9, and $n=3$0 lobes, and the local region is tested by a family of algebraic inequalities rather than by a single root inequality (Silva et al., 20 Mar 2025).

3. Derivation methods

A common derivational pattern is to start from the trilocal factorization, convert the observable of interest into products of single-source or single-party expectations, and then apply absolute-value bounds together with Hölder-type or related inequalities.

For the linear $n=3$1 chain, one inserts the factorized model into $n=3$2 and $n=3$3, integrates out the middle source contributions using $n=3$4, and then uses a two-term Hölder/triangle argument to obtain $n=3$5. The same work explicitly contrasts this with the ordinary Bell-local bound

$n=3$6

which no longer uses source factorization (Mukherjee et al., 2014).

For the five-party network, the derivation proceeds by introducing local expectations such as $n=3$7, bounding grouped correlators by successive applications of $n=3$8, and then applying the inequality

$n=3$9

together with the dichotomic constraint $$P(a_1,a_2,a_3,a_4|x_1,x_2,x_3,x_4) =\int d\lambda_1 d\lambda_2 d\lambda_3\, \rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)\, P(a_1|x_1,\lambda_1)P(a_2|x_2,\lambda_1,\lambda_2) P(a_3|x_3,\lambda_2,\lambda_3)P(a_4|x_4,\lambda_3)$$0. This yields the cube-root trilocal inequality (Mukherjee et al., 2017).

For the triangle-network inequalities with four outcomes and no inputs, the proof outline uses factorization at the $$P(a_1,a_2,a_3,a_4|x_1,x_2,x_3,x_4) =\int d\lambda_1 d\lambda_2 d\lambda_3\, \rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)\, P(a_1|x_1,\lambda_1)P(a_2|x_2,\lambda_1,\lambda_2) P(a_3|x_3,\lambda_2,\lambda_3)P(a_4|x_4,\lambda_3)$$1-level and then the elementary inequality

$$P(a_1,a_2,a_3,a_4|x_1,x_2,x_3,x_4) =\int d\lambda_1 d\lambda_2 d\lambda_3\, \rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)\, P(a_1|x_1,\lambda_1)P(a_2|x_2,\lambda_1,\lambda_2) P(a_3|x_3,\lambda_2,\lambda_3)P(a_4|x_4,\lambda_3)$$2

to derive

$$P(a_1,a_2,a_3,a_4|x_1,x_2,x_3,x_4) =\int d\lambda_1 d\lambda_2 d\lambda_3\, \rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)\, P(a_1|x_1,\lambda_1)P(a_2|x_2,\lambda_1,\lambda_2) P(a_3|x_3,\lambda_2,\lambda_3)P(a_4|x_4,\lambda_3)$$3

(Mukherjee, 2023).

The derivation in asymmetric four-party trilocal scenarios is more elaborate. Under the factorized hidden-variable model, the correlators are bounded by repeated use of a product-sum inequality, and the optimal quantum values are obtained by a sum-of-squares construction. The paper constructs a positive-semidefinite operator

$$P(a_1,a_2,a_3,a_4|x_1,x_2,x_3,x_4) =\int d\lambda_1 d\lambda_2 d\lambda_3\, \rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)\, P(a_1|x_1,\lambda_1)P(a_2|x_2,\lambda_1,\lambda_2) P(a_3|x_3,\lambda_2,\lambda_3)P(a_4|x_4,\lambda_3)$$4

with $$P(a_1,a_2,a_3,a_4|x_1,x_2,x_3,x_4) =\int d\lambda_1 d\lambda_2 d\lambda_3\, \rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)\, P(a_1|x_1,\lambda_1)P(a_2|x_2,\lambda_1,\lambda_2) P(a_3|x_3,\lambda_2,\lambda_3)P(a_4|x_4,\lambda_3)$$5 for $$P(a_1,a_2,a_3,a_4|x_1,x_2,x_3,x_4) =\int d\lambda_1 d\lambda_2 d\lambda_3\, \rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)\, P(a_1|x_1,\lambda_1)P(a_2|x_2,\lambda_1,\lambda_2) P(a_3|x_3,\lambda_2,\lambda_3)P(a_4|x_4,\lambda_3)$$6 or $$P(a_1,a_2,a_3,a_4|x_1,x_2,x_3,x_4) =\int d\lambda_1 d\lambda_2 d\lambda_3\, \rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)\, P(a_1|x_1,\lambda_1)P(a_2|x_2,\lambda_1,\lambda_2) P(a_3|x_3,\lambda_2,\lambda_3)P(a_4|x_4,\lambda_3)$$7 for $$P(a_1,a_2,a_3,a_4|x_1,x_2,x_3,x_4) =\int d\lambda_1 d\lambda_2 d\lambda_3\, \rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)\, P(a_1|x_1,\lambda_1)P(a_2|x_2,\lambda_1,\lambda_2) P(a_3|x_3,\lambda_2,\lambda_3)P(a_4|x_4,\lambda_3)$$8, and chooses $$P(a_1,a_2,a_3,a_4|x_1,x_2,x_3,x_4) =\int d\lambda_1 d\lambda_2 d\lambda_3\, \rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)\, P(a_1|x_1,\lambda_1)P(a_2|x_2,\lambda_1,\lambda_2) P(a_3|x_3,\lambda_2,\lambda_3)P(a_4|x_4,\lambda_3)$$9 so that $$p(a,b,c)=\sum_{\alpha,\beta,\gamma} q(\alpha)r(\beta)s(\gamma)\, A(a|\beta,\gamma)\,B(b|\gamma,\alpha)\,C(c|\alpha,\beta),$$0. The SOS method then gives the optimal quantum bounds without assuming the system dimension (Sasmal et al., 2023).

The symmetric binary-output triangle case follows a different route. Da Silva et al. numerically search for local models in carefully chosen two-dimensional affine planes, observe that optimal boundary models collapse to small two-parameter families with hidden-variable cardinalities $$p(a,b,c)=\sum_{\alpha,\beta,\gamma} q(\alpha)r(\beta)s(\gamma)\, A(a|\beta,\gamma)\,B(b|\gamma,\alpha)\,C(c|\alpha,\beta),$$1 in the symmetric case, and then eliminate the parameters using resultant or Gröbner-basis methods. The resulting master polynomial relations in $$p(a,b,c)=\sum_{\alpha,\beta,\gamma} q(\alpha)r(\beta)s(\gamma)\, A(a|\beta,\gamma)\,B(b|\gamma,\alpha)\,C(c|\alpha,\beta),$$2 define the conjectured analytic boundary surfaces (Silva et al., 20 Mar 2025).

4. Quantum violations, thresholds, and robustness

Quantum violations are known in several trilocal scenarios. In the linear four-party chain, an entanglement-swapping chain of three singlets with Bell-basis measurements at the two middle parties and observables

$$p(a,b,c)=\sum_{\alpha,\beta,\gamma} q(\alpha)r(\beta)s(\gamma)\, A(a|\beta,\gamma)\,B(b|\gamma,\alpha)\,C(c|\alpha,\beta),$$3

at the ends yields

$$p(a,b,c)=\sum_{\alpha,\beta,\gamma} q(\alpha)r(\beta)s(\gamma)\, A(a|\beta,\gamma)\,B(b|\gamma,\alpha)\,C(c|\alpha,\beta),$$4

hence

$$p(a,b,c)=\sum_{\alpha,\beta,\gamma} q(\alpha)r(\beta)s(\gamma)\, A(a|\beta,\gamma)\,B(b|\gamma,\alpha)\,C(c|\alpha,\beta),$$5

For Werner sources $$p(a,b,c)=\sum_{\alpha,\beta,\gamma} q(\alpha)r(\beta)s(\gamma)\, A(a|\beta,\gamma)\,B(b|\gamma,\alpha)\,C(c|\alpha,\beta),$$6, the summary gives $$p(a,b,c)=\sum_{\alpha,\beta,\gamma} q(\alpha)r(\beta)s(\gamma)\, A(a|\beta,\gamma)\,B(b|\gamma,\alpha)\,C(c|\alpha,\beta),$$7, whereas the usual CHSH threshold is $$p(a,b,c)=\sum_{\alpha,\beta,\gamma} q(\alpha)r(\beta)s(\gamma)\, A(a|\beta,\gamma)\,B(b|\gamma,\alpha)\,C(c|\alpha,\beta),$$8. Thus the interval $$p(a,b,c)=\sum_{\alpha,\beta,\gamma} q(\alpha)r(\beta)s(\gamma)\, A(a|\beta,\gamma)\,B(b|\gamma,\alpha)\,C(c|\alpha,\beta),$$9 is non-trilocal but Bell-local (Mukherjee et al., 2014).

In the five-party multipartite-source scenario, the GGHZ family

$q,r,s$0

leads to the maximal violation

$q,r,s$1

Therefore any GGHZ state with $q,r,s$2 is nontrilocal, and the perfect GHZ state gives $q,r,s$3. The same summary states that genuine entanglement content is not an essential requirement: a biseparable state can also display nontrilocality, and for noisy GHZ sources the condition is $q,r,s$4, corresponding to a symmetric threshold $q,r,s$5 (Mukherjee et al., 2017).

In the four-outcome triangle-network criteria, explicit quantum violations are given for three Bell states with an entangled measurement basis, where

$q,r,s$6

and this exceeds $q,r,s$7 for $q,r,s$8. The same work reports that the product basis $q,r,s$9 yields $A,B,C\in\{0,1\}$0, $A,B,C\in\{0,1\}$1, hence $A,B,C\in\{0,1\}$2. For detector inefficiency modeled by POVM elements $A,B,C\in\{0,1\}$3, violation holds when $A,B,C\in\{0,1\}$4 exceeds approximately $A,B,C\in\{0,1\}$5 for an optimal $A,B,C\in\{0,1\}$6 (Mukherjee, 2023).

In the star-shaped four-party setting used for full-network-nonlocality-based QKD, if each source distributes a two-qubit state with diagonalized correlation tensor $A,B,C\in\{0,1\}$7, then the relevant expression is

$A,B,C\in\{0,1\}$8

and any trilocal model must satisfy $A,B,C\in\{0,1\}$9. For three singlets, $S_1,S_2,S_3$0, and the summary states that no known measurement gives a larger value than $S_1,S_2,S_3$1 in that symmetric three-singlet scenario (Mukherjee, 20 Mar 2026).

For asymmetric trilocal scenarios, the optimal quantum bounds are

$S_1,S_2,S_3$2

to be compared with trilocal bounds $S_1,S_2,S_3$3 and $S_1,S_2,S_3$4, respectively. Under white-noise admixture, the critical visibility is reported as $S_1,S_2,S_3$5 for Scenario I and $S_1,S_2,S_3$6 for Scenario II (Sasmal et al., 2023).

5. The symmetric triangle network and analytic boundary inequalities

The most detailed recent use of the term concerns the minimal triangle network with no inputs and binary outcomes. In the fully permutation-symmetric sector, the observed distribution is encoded by the three correlators $S_1,S_2,S_3$7, and the triangle-local set splits into three boundary lobes centered on the archetypal GHZ, $S_1,S_2,S_3$8, and $S_1,S_2,S_3$9 points (Silva et al., 20 Mar 2025).

The GHZ lobe is described by a two-parameter family of local models of cardinality $\sqrt{|I|}+\sqrt{|J|}\le 1$0, leading to an inequality of the form

$\sqrt{|I|}+\sqrt{|J|}\le 1$1

where, defining

$\sqrt{|I|}+\sqrt{|J|}\le 1$2

one has

$\sqrt{|I|}+\sqrt{|J|}\le 1$3

$\sqrt{|I|}+\sqrt{|J|}\le 1$4

Its bit-flipped copy is

$\sqrt{|I|}+\sqrt{|J|}\le 1$5

The summary notes that the square root requires $\sqrt{|I|}+\sqrt{|J|}\le 1$6 to be real; if this fails, the GHZ-lobe test is not applied, and one resorts to the $\sqrt{|I|}+\sqrt{|J|}\le 1$7-type tests (Silva et al., 20 Mar 2025).

The $\sqrt{|I|}+\sqrt{|J|}\le 1$8 lobe is covered by five distinct two-parameter local-model families. Their inequalities are:

$\sqrt{|I|}+\sqrt{|J|}\le 1$9

$$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$00

$$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$01

provided $$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$02,

$$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$03

$$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$04

subject to three auxiliary polynomial domain constraints,

$$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$05

$$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$06

where $$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$07 is defined implicitly as the minimal $$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$08 compatible with the symmetry equations and positivity, equivalently by solving a two-variable Lagrange-multiplier system (Silva et al., 20 Mar 2025).

For triangle-locality, the classical bound is the same structural rule throughout: the displayed expression is nonnegative, and equality defines the analytic boundary. The bit-flipped counterparts $$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$09, $$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$10 cover the $$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$11 lobe. The summary further states that all five $$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$12 must be satisfied by any triangle-local point in the $$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$13 lobe, and that violation of all five signals $$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$14-type nonlocality (Silva et al., 20 Mar 2025).

A significant open point is quantum realizability. The current status reported there is that no explicit symmetric quantum realization is known to violate these exact inequalities, although inflation-based outer approximations of the no-signaling-plus-source-independence region leave small wedges where quantum nonlocality may yet appear (Silva et al., 20 Mar 2025).

6. Relation to Bell nonlocality, generalizations, and applications

Trilocal inequalities differ from ordinary Bell inequalities in both structure and operational meaning. In the four-party linear chain, ordinary locality yields the linear constraint $$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$15, whereas trilocality yields the nonlinear bound $$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$16 (Mukherjee et al., 2014). In the five-party scenario, the summary explicitly states that without source independence one recovers only linear five-party Bell inequalities, which are far less noise-tolerant (Mukherjee et al., 2017).

The triangle-network criteria sharpen this distinction further. Ghosh et al. emphasize that their inequalities can be violated even when each source state is individually CHSH-local; for Bell-diagonal states of the form specified in their Eq. (31), the individual pairs are CHSH-local yet the triangle-network criterion $$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$17 can still be violated for suitable $$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$18 (Mukherjee, 2023). This isolates a form of network nonlocality not reducible to standard Bell-CHSH nonlocality.

The literature also distinguishes full network nonlocality from pairwise Bell nonlocality. In the star-network QKD setting, violation of the trilocal inequality cannot be obtained if any single source is classical, even if the other two are maximally nonlocal. The summary therefore presents the test as a genuine four-party nonlocality witness requiring all three sources (Mukherjee, 20 Mar 2026). This notion underlies the security analysis of the protocol $$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$19, where a subset of runs is used to test the trilocal inequality and the remaining runs determine the quantum bit error rate $$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$20. For three identical two-qubit states, the reported threshold is

$$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$21

while for three non-identical states the strongest bound is $$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$22. A CHSH-based comparison protocol gives $$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$23 (Mukherjee, 20 Mar 2026).

Several generalizations are explicit. Mukherjee et al. state that their proof extends to an $$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$24-local star-chain with $$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$25 sources and $$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$26 parties, replacing cube roots by $$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$27-th roots and yielding $$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$28 inequalities (Mukherjee et al., 2017). Ghosh et al. generalize the fixed-measurement triangle-network construction to any $$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$29-sided polygon, introducing inequalities

$$\rho(\lambda_1,\lambda_2,\lambda_3)=\rho_1(\lambda_1)\rho_2(\lambda_2)\rho_3(\lambda_3)$$30

for cycle networks (Mukherjee, 2023). Choudhary et al. develop asymmetric trilocal scenarios with unequal numbers of settings on the edge parties and central node, and use SOS methods to compute optimal quantum violations in a dimension-independent way (Sasmal et al., 2023).

Taken together, these results show that “trilocal inequality” has evolved into a broad technical category: nonlinear constraints enforcing three-source independence, tailored to specific network geometries, and used both as foundational diagnostics of network nonclassicality and as operational witnesses in tasks such as entanglement witnessing, noise-threshold analysis, and network-based quantum key distribution (Mukherjee et al., 2014, Mukherjee et al., 2017, Mukherjee, 2023, Mukherjee, 20 Mar 2026).