Abstract Bell experiment in a network gives rise to a form of quantum nonlocality which is conceptually different from traditional multipartite Bell nonlocality. In this work, the star‐network configuration involving arbitrary n independent sources and parties, including n edge parties and one central party is considered. Each of the n edge parties shares a physical system with the central party. Each edge party receives number of inputs, and the central party receives an arbitrary m number of inputs. The conditional dependence on the inputs of each edge party is imposed so that the local probabilities satisfy a set of constraints. A family of generalized n‐locality inequalities is proposed in the arbitrary input scenario by imposing the set of constraints on inputs. The optimal quantum violation of the inequalities is derived by using an elegant sum‐of‐squares approach without specifying the dimension of the quantum system. Notably, the optimal quantum value is achieved only when the set of linear constraints on inputs is satisfied, which, in turn, self‐tests the observables required for each edge party. It shows that while conditional dependence on inputs significantly reduces the n‐local bound of the inequalities, the optimal quantum violation remains invariant. It argues that this implies a more robust test of network non‐locality, which can be revealed for smaller visibility parameters of the corresponding state. Further, the network nonlocality is characterized and examine its correspondence with suitably derived standard Bell nonlocality.
Network Bell experiments reveal a form of nonlocality conceptually different from standard Bell nonlocality. Standard multiparty Bell experiments involve a single source shared by a set of observers. In contrast network Bell experiments feature multiple independent sources and each of them may distribute physical systems to a set of observers who perform randomly chosen measurements. The
n-locality scenario in star-network configuration is an instance of network Bell experiment. Such a scenario involves
n number of edge observers (Alices) a central observer (Bob) and
n number of independent sources having no prior correlation. Each Alice shares an independent state with the central observer Bob. Usually in network Bell experiments one considers that each party measures only two observables. In this work we propose a nontrivial generalization of
n-locality scenario in star-network configuration where each Alice performs some integer
m number of binary-outcome measurements and the central party Bob performs ${2}^{m\ensuremath{-}1}$ measurements. We derive a family of generalized
n-locality inequalities for any arbitrary
m. Using an elegant sum-of-squares approach we derive the optimal quantum violation of the aforementioned inequalities when each and every Alice measures
m number of mutually anticommuting observables. For
m=2 and 3 one obtains the optimal quantum value with a two-qubit entangled state shared between each Alice and Bob. We further demonstrate that the optimal quantum violation of
n-locality inequality for any arbitrary
m can be obtained when every Alice shares $\ensuremath{\lfloor}m/2\ensuremath{\rfloor}$ copies of two-qubit maximally entangled state with the central party Bob. We also argue that for
m>3 a single copy of a two-qubit-entangled state may not be enough to exhibit the violation of
n-locality inequality but multiple copies of it can activate the quantum violation. We discuss the implications of our study and raise some open questions.
The nonlocality revealed in a multiparty multisource network Bell experiment is conceptually different than the standard multiparty Bell nonlocality involving a single common source. Here, by introducing variants of asymmetric bilocal as well as trilocal network scenarios, we go beyond the typical bilocal network scenario where both the edge parties have an equal number of measurement settings. We first introduce an asymmetric bilocal network where one of the edge parties (say, Alice) receives ${2}^{n\ensuremath{-}1}$ inputs and the other edge party (say, Charlie) receives
n inputs. We derive two variants of asymmetric bilocality inequalities and demonstrate their optimal quantum violations. Further, we explore two types of asymmetric trilocal scenarios: (i) when two edge parties receive ${2}^{n\ensuremath{-}1}$ inputs each and the other edge party receives
n inputs, and (ii) when one edge party receives ${2}^{n\ensuremath{-}1}$ inputs and the other two edge parties have
n inputs each. We use an elegant sum-of-squares technique that enables us to evaluate the quantum optimal values of the proposed network inequalities without assuming the dimension of the systems for both the asymmetric bilocal as well as the trilocal scenarios. Further, we demonstrate the robustness of the quantum violations of the proposed inequalities in the presence of white noise.
The multipartite quantum networks feature multiple independent sources, in contrast to the conventional multipartite Bell experiment involving a single source. Despite the initial independence of resources, the multiple observers in the network can suitably choose measurements on their local subsystems and generate a form of quantum nonlocality across the network. So far, network nonlocality has been explored when each source produces a two-qubit entangled state. In this work we demonstrate the network nonlocality when each party performs a black-box measurement, and the dimension of the system remains unspecified. In an interesting work, by considering each source produces two-qubit entangled states in the conventional bilocal scenario, Gisin et al. [Phys. Rev. A 96, 020304 (2017)] demonstrated a correspondence between the violations of bipartite Clauser-Horne-Shimony-Halt inequality and the bilocality inequality. We introduce a variant of the sum-of-squares approach to reproduce their results without assuming the dimension of the system. We then generalize the argument for network nonlocality in star-network topology. Furthermore, we propose a set of
n-locality inequalities in a star-network configuration where each of the
n parties performs an arbitrary number of dichotomic measurements and demonstrates the above correspondence between the quantum violations of the
n-locality inequalities and the chained Bell inequalities. A similar correspondence is demonstrated based on a recently formulated family of
n-locality inequalities whose optimal quantum violation cannot be obtained when each source emits a two-qubit entangled state and requires multiple copies of two-qubit entangled states. Throughout this paper, each party in the network performs black-box measurements, and the dimension of the system remains unspecified.
The nonlocality arising in a multiparty network involving multiple independent sources radically differs from the standard multipartite Bell nonlocality involving a single source. The notion of the full network nonlocality (FNN) [A. Pozas-Kerstjens et al., Full network nonlocality, Phys. Rev. Lett. 128, 010403 (2022)PRLTAO0031-900710.1103/PhysRevLett.128.010403] characterizes the quantum correlations that cannot be reproduced by a local-nonlocal model featuring one local source and the rest of nonlocal no-signaling sources. However, the demonstration of FNN was limited to bilocal and trilocal star-shaped network scenarios involving three or two dichotomic measurements for edge parties. In this Letter, we first demonstrate that a large class of prevailing network inequalities does not exhibit FNN. We then introduce an elegant set of arbitrary-party and unbounded-input network inequalities in star-shaped and linear-chain networks whose optimal quantum violation exhibits FNN, certifying that the nonlocality is genuinely distributed to the entire network. Contrasting to existing demonstrations of FNN that inevitably require fixed-input and four-output elegant joint measurements for the central party, our generalized inequalities are more experimentally friendly, requiring only two output measurements. Moreover, our derivation of optimal quantum violation is fully analytic and devoid of assuming the dimension of the quantum system, thereby showcasing its potential for device-independent self-testing.
Quantum nonlocality in networks featuring multiple independent sources underpins large-scale quantum communication and poses fundamental challenges for its characterization. In this work, we construct a family of explicit nonlinear Bell inequalities to verify the nonlocality across the general multi-input quantum networks. The construction of these inequalities relies on the number of leaf nodes, a network parameter that can be identified by a linear-time algorithm. Our approach establishes a structural connection between bipartite full-correlation Bell inequalities and network Bell inequalities, enabling the analytical derivation of optimal quantum violations and the conditions under which they occur. We further quantify the upper bound on maximal violations achievable by arbitrary two-qubit mixed states in such networks, under separable measurements, and evaluate the noise robustness of the proposed inequalities via the visibilities of noisy states. Finally, we demonstrate that these inequalities can, in a device-independent manner, distinguish between network topologies of equal size that differ in the number of leaf nodes.