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.

Inferring causal relations from experimental observations is of primal importance in science. Instrumental tests provide an essential tool for that aim, as they allow to estimate causal dependencies even in the presence of unobserved common causes. In view of Bell's theorem, which implies that quantum mechanics is incompatible with our most basic notions of causality, it is of utmost importance to understand whether and how paradigmatic causal tools obtained in a classical setting can be carried over to the quantum realm. Here we show that quantum effects imply radically different predictions in the instrumental scenario. Among other results, we show that an instrumental test can be violated by entangled quantum states. Furthermore, we demonstrate such violation using a photonic setup with active feed-forward of information, thus providing an experimental proof of this new form of non-classical behavior. Our findings have fundamental implications in causal inference and may also lead to new applications of quantum technologies.Instrumental variables were originally invented to estimate parameters in econometric models of supply and demand [1] and since then have found a wide range of applications in various other fields [2, 3]. Remarkably, an instrument allows one to estimate the strength of causal influences between two variables solely from observed data [4, 5], without any assumptions on the functional dependence among them. This is the approach known in quantum information science as "deviceindependent" [6]. For that, an instrumental test is crucial, since it provides empirically testable inequalities allowing one to check whether one has a valid instrument [7].Instrumental inequalities as well as the estimation of causal dependencies are derived from classical notions of cause and effect that, since Bell's theorem [8], we know cannot be taken for granted in quantum phenomena. Given this mismatch between classical and quantum predictions, it is natural to ask how fundamental tools in causal inference behave in a quantum scenario. This has motivated the emerging field of quantum causal modeling [9][10][11][12][13][14][15][16][17], which has provided sophisticated generalizations of the classical theory of causality [5] to the quantum realm, thereby discovering, for example, exciting quantum advantages for causal inference [18][19][20]. Within this new framework, it was shown [13] that a paradigmatic class of instrumental inequalities [7] are satisfied by quantum mechanics. However, it is not known whether other instrumental inequalities may admit quantum violations. Moreover, even if a given observed statistics is compatible with a classical instrumental causal model, it may well still be the case that quantum effects do offer some sort of enhancement.In this article, we show that the quantum predictions for the instrumental scenario are radically different from those of classical causality theory. Firstly, we show that a standard measure of causation -the average causal effect (ACE) [4, 5, 21] -can be largely...

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.

Light beams having a vectorial field structure, or polarization, that varies over the transverse profile and a central optical singularity are called vector vortex (VV) beams and may exhibit specific properties such as focusing into “light needles” or rotation invariance. VV beams have already found applications in areas ranging from microscopy to metrology, optical trapping, nano-optics, and quantum communication. Individual photons in such beams exhibit a form of single-particle quantum entanglement between different degrees of freedom. On the other hand, the quantum states of two photons can be also entangled with each other. Here, we combine these two concepts and demonstrate the generation of quantum entanglement between two photons that are both in VV states: a form of entanglement between two complex vectorial fields. This result may lead to quantum-enhanced applications of VV beams as well as to quantum information protocols fully exploiting the vectorial features of light