Distributing secret keys with informationtheoretic security is arguably one of the most important achievements of the field of quantum information processing and communications [1]. The rapid progress in this field has enabled quantum key distribution (QKD) in real-world conditions [2, 3] and commercial devices are now readily available. QKD systems based on continuous variables [4] present the major advantage that they only require standard telecommunication technology, and in particular, that they do not use photon counters. However, these systems were considered up till now unsuitable for longdistance communication [5][6][7]. Here, we overcome all previous limitations and demonstrate for the first time continuous-variable quantum key distribution over 80 km of optical fibre. The demonstration includes all aspects of a practical scenario, with real-time generation of secret keys, stable operation in a regular environment, and use of finite-size data blocks for secret information computation and key distillation. Our results correspond to an implementation guaranteeing the strongest level of security for QKD reported to date for such long distances and pave the way to practical applications of secure quantum communications.

Abstract. So far, most of the literature on (quantum) contextuality and the Kochen-Specker theorem seems either to concern particular examples of contextuality, or be considered as quantum logic. Here, we develop a general formalism for contextuality scenarios based on the combinatorics of hypergraphs which significantly refines a similar recent approach by Cabello, Severini and Winter (CSW). In contrast to CSW, we explicitly include the normalization of probabilities, which gives us a much finer control over the various sets of probabilistic models like classical, quantum and generalized probabilistic. In particular, our framework specializes to (quantum) nonlocality in the case of Bell scenarios, which arise very naturally from a certain product of contextuality scenarios due to Foulis and Randall. In the spirit of CSW, we find close relationships to several graph invariants. The recently proposed Local Orthogonality principle turns out to be a special case of a general principle for contextuality scenarios related to the Shannon capacity of graphs. Our results imply that it is strictly dominated by a low level of the Navascués-Pironio-Acín hierarchy of semidefinite programs, which we also apply to contextuality scenarios.We derive a wealth of results in our framework, many of these relating to quantum and supraquantum contextuality and nonlocality, and state numerous open problems. For example, we show that the set of quantum models on a contextuality scenario can in general not be characterized in terms of a graph invariant.In terms of graph theory, our main result is this: there exist two graphs G 1 and G 2

The goal of this paper is to extend the framework of finite-size analysis recently developed for quantum key distribution to continuous-variable protocols. We do not solve this problem completely here, and we mainly consider the finite-size effects on the parameter estimation procedure. Despite the fact that some questions are left open, we are able to give an estimation of the secret key rate for protocols which do not contain a postselection procedure. As expected, these results are significantly more pessimistic than those obtained in the asymptotic regime. However, we show that recent continuous-variable protocols are able to provide fully secure secret keys in the finite-size scenario, over distances larger than 50 km.