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More than 50 years ago (1), John Bell proved that no theory of nature that obeys locality and realism (2) can reproduce all the predictions of quantum theory: in any local-realist theory, the correlations between outcomes of measurements on distant particles satisfy an inequality that can be violated if the particles are entangled. Numerous Bell inequality tests have been reported (3-13); however, all experiments reported so far required additional assumptions to obtain a contradiction with local realism, resulting in 'loopholes' (13-16). Here we report a Bell experiment that is free of any such additional assumption and thus directly tests the principles underlying Bell's inequality. We use an event-ready scheme (17-19) that enables the generation of robust entanglement between distant electron spins (estimated state fidelity of 0.92 ± 0.03). Efficient spin read-out avoids the fair-sampling assumption (detection loophole (14,15)), while the use of fast random-basis selection and spin read-out combined with a spatial separation of 1.3 kilometres ensure the required locality conditions (13). We performed 245 trials that tested the CHSH-Bell inequality (20) S ≤ 2 and found S = 2.42 ± 0.20 (where S quantifies the correlation between measurement outcomes). A null-hypothesis test yields a probability of at most P = 0.039 that a local-realist model for space-like separated sites could produce data with a violation at least as large as we observe, even when allowing for memory (16,21) in the devices. Our data hence imply statistically significant rejection of the local-realist null hypothesis. This conclusion may be further consolidated in future experiments; for instance, reaching a value of P= 0.001 would require approximately 700 trials for an observed S = 2.4. With improvements, our experiment could be used for testing less-conventional theories, and for implementing device-independent quantum-secure communication (22) and randomness certification (23,24)., We consider a Bell test in the form proposed by Clauser, Horne, Shimony and Holt (CHSH) (20) (Fig. 1a). The test involves two boxes labelled A and B. Each box [...]

Quantum metrology aims to use entanglement and other quantum resources to improve precision measurement (1). An interferometer using N independent particles to measure a parameter X can achieve at best the standard quantum limit of sensitivity, δ X [proportional] [N.sup.-1/2]. However, using N entangled particles and exotic states (2), such an interferometer (3) can in principle achieve the Heisenberg limit, δ X [proportional] [N.sup.-1]. Recent theoretical work (4-6) has argued that interactions among particles may be a valuable resource for quantum metrology, allowing scaling beyond the Heisenberg limit. Specifically, a k-particle interaction will produce sensitivity δ X [proportional] [N.sup.-k] with appropriate entangled states and δ X [proportional] [N.sup.-(k-1/2)] even without entanglement (7). Here we demonstrate 'super-Heisenberg' scaling of δ X [proportional] [N.sup.-3/2)] in a nonlinear, non-destructive (8,9) measurement of the magnetization (10,11) of an atomic ensemble (12). We use fast optical nonlinearities to generate a pairwise photon-photon interaction (13) (corresponding to k = 2) while preserving quantum-noise-limited performance (7,14). We observe super-Heisenberg scaling over two orders of magnitude in N, limited at large numbers by higher-order nonlinear effects, in good agreement with theory (13). For a measurement of limited duration, super-Heisenberg scaling allows the nonlinear measurement to overtake in sensitivity a comparable linear measurement with the same number of photons. In other situations, however, higher-order nonlinearities prevent this crossover from occurring, reflecting the subtle relationship between scaling and sensitivity in nonlinear systems. Our work shows that interparticle interactions can improve sensitivity in a quantum-limited measurement, and experimentally demonstrates a new resource for quantum metrology., The most precise instruments are interferometric in nature, and operate according to the laws of quantum mechanics. A collection of particles, for example photons or atoms, is prepared in a [...]