Abstract The interest in quantum network correlations has surged, as Bell inequalities originating from a single source fall short of capturing the intricate many-body correlations within generic quantum networks. The introduction of a 3-grade m-star quantum network, a natural extension of star-shaped networks, features a hierarchy of nodes including A, m stars B 1, …, B m centered around A, and m stars C 1 j , … , C m j centered around each B j , and m + m 2 independent sources. The strong locality of this network has been explored, assuming that each party possesses only two observables. In this paper, we put forward a significant extension to the strong locality of 3-grade m-star quantum networks. Specifically, each C j k performs n dichotomic measurements, while A and each B j conduct 2 n−1 dichotomic measurements. We establish a generalized strong locality inequality that holds for any value of n, and subsequently conduct a thorough analysis to determine the optimal quantum violation of established inequality. Through specific examples, we confirm the feasibility of achieving the optimal quantum violation of the generalized strong locality inequality for any n. We notice that for n > 3, a single copy of a two-qubit entangled state may not be sufficient to reveal the non-strong locality, whereas utilizing multiple copies can trigger this property.
Abstract The interest in quantum network correlations has surged, as Bell inequalities originating from a single source fall short of capturing the intricate many-body correlations within generic quantum networks. The introduction of a 3-grade m-star quantum network, a natural extension of star-shaped networks, features a hierarchy of nodes including A, m stars B 1, …, B m centered around A, and m stars C 1 j , … , C m j centered around each B j , and m + m 2 independent sources. The strong locality of this network has been explored, assuming that each party possesses only two observables. In this paper, we put forward a significant extension to the strong locality of 3-grade m-star quantum networks. Specifically, each C j k performs n dichotomic measurements, while A and each B j conduct 2 n−1 dichotomic measurements. We establish a generalized strong locality inequality that holds for any value of n, and subsequently conduct a thorough analysis to determine the optimal quantum violation of established inequality. Through specific examples, we confirm the feasibility of achieving the optimal quantum violation of the generalized strong locality inequality for any n. We notice that for n > 3, a single copy of a two-qubit entangled state may not be sufficient to reveal the non-strong locality, whereas utilizing multiple copies can trigger this property.
Abstract The correlations in quantum networks have attracted strong interest due to the fact that linear Bell inequalities derived from one source are useless for characterizing multipartite correlations of general quantum networks. In this paper, a type of multi-star-shaped quantum networks are introduced and discussed. Such a network consists of three-grade nodes: the first grade is named party (node) A , the second one consists of m nodes marked B 1 , B 2 , …, B m , which are stars of A and the third one consists of m 2 nodes C k j ( j , k = 1 , 2 , … , m ) , where C k j ( k = 1 , 2 , … , m ) are stars of B j . We call such a network a three-grade m -star quantum network and denoted by SQN (3, m ), being as a natural extension of bilocal networks and star-shaped networks. We introduce and discussed the locality and strong locality of a SQN (3, m ) and derive the related nonlinear Bell inequalities, called (3, m )-locality inequalities and (3, m )-strong locality inequalities. To compare with the bipartite locality of quantum states, we define the separability of SQN (3, m ) that imply the locality and then locality of SQN (3, m ). When all of the shared states of the network are pure ones, we prove that SQN (3, m ) is nonlocal if and only if it is entangled.
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 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.
Source-independent quantum networks are considered as a natural generalization to the Bell scenario where we investigate the nonlocal properties of quantum states distributed and measured in a network. Considering the simplest network of entanglement swapping, recently Gisin et al. [Phys. Rev. A 96, 020304(R) (2017)] and Andreoli [New. J. Phys. 19, 113020 (2017)] independently provided a systematic characterization of the set of quantum states leading to violation of the so-called bilocality inequality. In this work, we consider the complexities in quantum networks with an arbitrary number of parties distributed in chain-shaped and star-shaped networks. We derive the maximal violation of the ``n-local'' inequality that can be achieved by arbitrary two-qubit states for such chain- and star-shaped networks. This would further provide a deeper understanding of quantum correlations in complex structures.
Abstract This work investigates network nonlocality sharing for a broad class of bipartite Bell inequalities in generalized star networks with an ( n , m , k ) configuration, comprising n independent branches, m sequential Alices per branch, and k measurement settings per party. On each branch, the intermediate Alices implement weak measurements, whereas the final Alice and the central Bob perform sharp projective measurements. Network nonlocality sharing is witnessed when the quantum values of the network correlations associated with relevant parties simultaneously violate a star-network Bell inequality generated from the given class of bipartite Bell inequalities. We streamline the calculation of the quantum values of the network correlations and derive an analytical expression for the bipartite quantum correlator, valid for arbitrary two-qubit source states, measurement settings and weak-measurement strengths. Network nonlocality sharing for Vértesi inequalities has been studied within the framework, and simultaneous violations are found in the ( 2 , 2 , 6 ) and ( 2 , 2 , 465 ) cases, with the latter exhibiting greater robustness. Our approach suggests a practical route to studying network nonlocality sharing by utilizing diverse bipartite Bell inequalities beyond the commonly used CHSH-type constructions.
The nonlocal correlations of noisy quantum systems are important for both understanding nature and developing quantum technology. We consider the correlations of star and chain quantum networks in which noisy entanglement sources are measured by nonsignaling parties. We derive the necessary and sufficient conditions for when a broad family of star and chain n-locality inequalities achieve their maximal quantum violation assuming that each party's dichotomic observables are separable across qudit systems. When pairs of local dichotomic observables are considered on qubit systems, we derive maximal n-local violations that are larger and more robust to noise than the maximal n-locality violations reported previously. To obtain these larger values, we consider observables that we have not found in previous studies. Thus, we gain insights into self-testing entanglement sources and measurements in star and chain networks.