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Indian Journal of Advanced Physics

Trade-off Relations in Genuine Four-qubit Nonlocality Systems

Sk Sahadat Hossain

Published Oct 27, 2026• 0 citations
Publisher

Abstract

We investigate trade-off relations governing genuine four-qubit nonlocality in multipartite quantum systems. By analyzing the reduced four-qubit density operators obtained from five-qubit generalized GHZ and MS states, we establish nontrivial constraints on the expectation values of the SeevinckSvetlichny operators associated with these subsystems. In particular, we demonstrate that genuine four-qubit nonlocal correlations across different four-qubit reductions are not mutually independent. For both classes of five-qubit states considered, we show that the sum of the maximal squared expectation values of the Seevinck-Svetlichny operators over all four-qubit reductions is subject to a state-dependent upper bound. Furthermore, this collective bound is shown to be tighter than the bounds obtained by independently imposing the maximal allowed expectation value of the Svetlichny operator on each four-qubit subsystem. These results reveal a monogamy-like trade-off structure for genuine four-qubit nonlocal correlations and provide tighter constraints on their simultaneous manifestation in multipartite quantum systems.

DOI
10.54105/ijap.b1079.06021026

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We investigate trade-off relations governing genuine four-qubit nonlocality in multipartite quantum systems. By analyzing the reduced four-qubit density operators obtained from five-qubit generalized GHZ and MS states, we establish nontrivial constraints on the expectation values of the SeevinckSvetlichny operators associated with these subsystems. In particular, we demonstrate that genuine four-qubit nonlocal correlations across different four-qubit reductions are not mutually independent. For both classes of five-qubit states considered, we show that the sum of the maximal squared expectation values of the Seevinck-Svetlichny operators over all four-qubit reductions is subject to a state-dependent upper bound. Furthermore, this collective bound is shown to be tighter than the bounds obtained by independently imposing the maximal allowed expectation value of the Svetlichny operator on each four-qubit subsystem. These results reveal a monogamy-like trade-off structure for genuine four-qubit nonlocal correlations and provide tighter constraints on their simultaneous manifestation in multipartite quantum systems.
We study the trade-off relations satisfied by the genuine tripartite nonlocality in multipartite quantum systems. From the reduced three-qubit density matrices of the four-qubit generalized Greenberger-Horne-Zeilinger (GHZ) states and W states (4-qubit entangled state), we find that there exists a trade-off relation among the mean values of the Svetlichny operators associated with these reduced states. Namely, the genuine three-qubit nonlocalities are not independent. For four-qubit generalized GHZ states and W states, the summation of all their three-qubit maximal (squared) mean values of the Svetlichny operator has an upper bound. This bound is better than the one derived from the upper bounds of individual three-qubit mean values of the Svetlichny operator. Detailed examples are presented to illustrate the trade-off relation among the three-qubit nonlocalities.
Identifying the nonlocality of a multiparty quantum state is an important task in quantum mechanics. Seevinck and Svetlichny [Phys. Rev. Lett. 89, 060401 (2002)], and independently, Collins and co-workers [Phys. Rev. Lett. 88, 170405 (2002)] have generalized the tripartite notion of Svetlichny nonlocality to n-parties. Here we have developed a tight upper bound for genuine four party Svetlichny type nonlocality. The constraints on the quantum states for the tightness of the bound are also presented. The method enables us to provide necessary and sufficient conditions for violating the four qubit Svetlichny type inequality for several quantum states. The relations between the genuine multipartite entanglement and the maximal quantum value of the Seevinck and Svetlichny operators for pure four qubit states are also discussed. Consequently, we have exhibited genuine four qubit hidden nonlocality under local filtering. Our result provides an effective and operational method for further study of multipartite quantum nonlocality.
We introduce a feature of no-signaling (Bell) nonlocal theories: namely, when a system of multiple parties manifests genuine nonlocal correlation, then there cannot be arbitrarily high nonlocal correlation among any subset of the parties. We call this feature complementarity of genuine multipartite nonlocality. We use Svetlichny's criterion for genuine multipartite nonlocality and nonlocal games to derive the complementarity relations under no-signaling constraints. We find that the complementarity relations are tightened for the much stricter quantum constraints. We compare this notion with the well-known notion of monogamy of nonlocality. As a consequence, we obtain tighter nontrivial monogamy relations that take into account genuine multipartite nonlocality. Furthermore, we provide numerical evidence showcasing this feature using a bipartite measure and several other well-known tripartite measures of nonlocality.
The violation of the Svetlichny's inequality (SI) [Phys. Rev. D 35, 3066 (1987)] is sufficient but not necessary for genuine tripartite nonlocal correlations. Here we quantify the relationship between tripartite entanglement and the maximum expectation value of the Svetlichny operator (which is bounded from above by the inequality) for the two inequivalent subclasses of pure three-qubit states: the Greenberger-Horne-Zeilinger (GHZ) class and the WWW class. We show that the maximum for the GHZ-class states reduces to Mermin's inequality [Phys. Rev. Lett. 65, 1838 (1990)] modulo a constant factor, and although it is a function of the three tangle and the residual concurrence, large numbers of states do not violate the inequality. We further show that by design SI is more suitable as a measure of genuine tripartite nonlocality between the three qubits in the WWW-class states, and the maximum is a certain function of the bipartite entanglement (the concurrence) of the three reduced states, and only when their sum attains a certain threshold value do they violate the inequality.
It is a well-known fact that both quantum entanglement and nonlocality (implied by the violation of Bell inequalities) constitute quantum correlations that cannot be arbitrarily shared among subsystems. They are both monogamous, albeit in a different fashion. In the present contribution we focus on nonlocality monogamy relations such as the Toner-Verstraete, the Seevinck, and a derived monogamy inequality for three parties and compare them with multipartite nonlocality measures for the whole set of pure states distributed according to the Haar measure. In this numerical endeavor, we also see that, although monogamy relations for nonlocality cannot exist for more than three parties, in practice the exploration of the whole set of states for different numbers of qubits will return effective bounds on the maximum value of all bipartite Bell violations among subsystems. Hence, we shed light on the effective nonlocality monogamy bounds in the multiqubit case.
Two overlapping bipartite binary Bell inequalities cannot be simultaneously violated as this would contradict the usual no-signaling principle. This property is known as monogamy of Bell inequality violations and generally Bell monogamy relations refer to trade-offs between simultaneous violations of multiple inequalities. It turns out that multipartite Bell inequalities admit weaker forms of monogamies that allow for violations of a few inequalities at once. Here we systematically study monogamy relations between correlation Bell inequalities both within quantum theory and under the sole assumption of no signaling. We first investigate the trade-offs in Bell violations arising from the uncertainty relation for complementary binary observables, and exhibit several network configurations in which a tight trade-off arises in this fashion. We then derive a tight trade-off relation which cannot be obtained from the uncertainty relation showing that it does not capture monogamy entirely. The results are extended to Bell inequalities involving different numbers of parties and find applications in device-independent secret sharing and device-independent randomness extraction. Although two multipartite Bell inequalities may be violated simultaneously, we show that genuine multiparty nonlocality, as evidenced by a generalized Svetlichny inequality, does exhibit monogamy property. Finally, using the relations derived we reveal the existence of flat regions in the set of quantum correlations.