Bell's theorem was a cornerstone for our understanding of quantum theory and the establishment of Bell non-locality played a crucial role in the development of quantum information. Recently, its extension to complex networks has been attracting growing attention, but a deep characterization of quantum behavior is still missing for this novel context. In this work we analyze quantum correlations arising in the bilocality scenario, that is a tripartite quantum network where the correlations between the parties are mediated by two independent sources of states. First, we prove that non-bilocal correlations witnessed through a Bell-state measurement in the central node of the network form a subset of those obtainable by means of a local projective measurement. This leads us to derive the maximal violation of the bilocality inequality that can be achieved by arbitrary two-qubit quantum states and arbitrary local projective measurements. We then analyze in details the relation between the violation of the bilocality inequality and the CHSH inequality. Finally, we show how our method can be extended to the n -locality scenario consisting of n two-qubit quantum states distributed among n + 1 nodes of a star-shaped network.
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 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.
While quantum networks have been extensively studied as a natural extension of the standard Bell scenario with a richer correlation structure, general constructions of nonlinear Bell inequalities with self-testing properties are still largely lacking. In this work, we present a general method for constructing such inequalities in the simplest network scenario, in which two independent sources distribute bipartite quantum states to three spatially separated observers. These inequalities allow for arbitrary numbers of binary measurements and are maximally violated by maximally entangled states of the corresponding local dimensions together with sets of pairwise anticommuting Clifford observables. Importantly, their maximal quantum values can be determined analytically, which makes them particularly promising for device-independent applications. In particular, we prove that these Bell inequalities can be used to device-independently certify the underlying quantum network, including both the quantum states produced by the sources and the observables measured by all parties. To the best of our knowledge, this is the first self-testing result for quantum networks that relies solely on the maximal violation of a nonlocality witness.
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 nature of quantum correlations in networks featuring independent sources of entanglement remains poorly understood. Here, focusing on the simplest network of entanglement swapping, we start a systematic characterization of the set of quantum states leading to violation of the so-called "bilocality" inequality. First, we show that all possible pairs of entangled pure states can violate the inequality. Next, we derive a general criterion for violation for arbitrary pairs of mixed two-qubit states. Notably, this reveals a strong connection between the CHSH Bell inequality and the bilocality inequality, namely that any entangled state violating CHSH also violates the bilocality inequality. We conclude with a list of open questions.