Bell's theorem plays a crucial role in quantum information processing and thus several experimental investigations of Bell inequalities violations have been carried out over the years. Despite their fundamental relevance, however, previous experiments did not consider an ingredient of relevance for quantum networks: the fact that correlations between distant parties are mediated by several, typically independent sources. Here, using a photonic setup, we investigate a quantum network consisting of three spatially separated nodes whose correlations are mediated by two distinct sources. This scenario allows for the emergence of the so-called non-bilocal correlations, incompatible with any local model involving two independent hidden variables. We experimentally witness the emergence of this kind of quantum correlations by violating a Bell-like inequality under the fair-sampling assumption. Our results provide a proof-of-principle experiment of generalizations of Bell's theorem for networks, which could represent a potential resource for quantum communication protocols.

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 a growing attention, but a deep characterization of quantum behaviour 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 separable 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 projective separable 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 n-locality scenario consisting of n two-qubit quantum states distributed among n + 1 nodes of a star-shaped network.

Non-classical correlations arising in complex quantum networks are attracting growing interest, both from a fundamental perspective and for potential applications in information processing. In particular, in an entanglement swapping scenario a new kind of correlations arise, the so-called nonbilocal correlations that are incompatible with local realism augmented with the assumption that the sources of states used in the experiment are independent. In practice, however, bilocality tests impose strict constraints on the experimental setup and in particular to presence of shared reference frames between the parties. Here, we experimentally address this point showing that false positive nonbilocal quantum correlations can be observed even though the sources of states are independent. To overcome this problem, we propose and demonstrate a new scheme for the violation of bilocality that does not require shared reference frames and thus constitute an important building block for future investigations of quantum correlations in complex networks.

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Quantum teleportation establishes a correspondence between an entangled state shared by two separate parties that can communicate classically and the presence of a quantum channel connecting the two parties. The standard benchmark for quantum teleportation, based on the average fidelity between the input and output states, indicates that some entangled states do not lead to channels which can be certified to be quantum. It was recently shown that if one considers a finer-tuned witness, then all entangled states can be certified to produce a non-classical teleportation channel. Here we experimentally demonstrate a complete characterization of a new family of such witnesses, of the type proposed in Phys. Rev. Lett. 119, 110501 (2017) under different conditions of noise. Furthermore, we show non-classical teleportation using quantum states that can not achieve average teleportation fidelity above the classical limit. Our results have fundamental implications in quantum information protocols and may also lead to new applications and quality certification of quantum technologies.Introduction. -The role of entanglement in quantum information processing is of utmost importance, but it is also subject of debate. Entanglement is today the core of many key discoveries ranging from quantum teleportation [3], to quantum dense coding [4], quantum computation [5-7] and quantum cryptography [8,9]. Quantum communication protocols such as device-independent quantum key distribution [10] are heavily based on entanglement to reach nonlocalitybased communication security [11].The prototype for quantum information transfer using entanglement as a communication channel is the quantum teleportation protocol [3], where a sender and a receiver share a maximally entangled state which they can use to perfectly transfer an unknown quantum state. This protocol represents a milestone in theoretical quantum information science [12][13][14] and lies at the basis of many technological application such as quantum communication via quantum repeaters [16,17] or gate teleportation [15]. It has been implemented over hundreds of kilometers in free-space [18,19] and more recently in a ground-to-satellite experiment [20]. Employed platforms include mainly photonic qubits [21][22][23][24][25][26][27], but also nuclear magnetic resonance [28], trapped atoms [31,32], atomic ensembles [29,30] and solid-state systems [33][34][35].To fully understand the role of entanglement in quantum teleportation, it is necessary to gauge what is the actual entanglement content (if any) that must be involved in order to upgrade a classical channel to a quantum channel. Indeed, it is known that not all entangled states allow a teleportation fidelity to be reached which is higher than the one achievable using only classical communication [36], with the notable example of bound entangled states [37,38]. This mismatch between the nonclassicality of the shared state and nonclassicality of teleportation can be resolved by taking into account the full available information inste...