More than 50 years ago, John Bell proved that no theory of nature that obeys locality and realism 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; however, all experiments reported so far required additional assumptions to obtain a contradiction with local realism, resulting in 'loopholes'. 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 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), 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. We performed 245 trials that tested the CHSH-Bell inequality 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 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 and randomness certification.

Quantum entanglement between spatially separated objects is one of the most intriguing phenomena in physics. The outcomes of independent measurements on entangled objects show correlations that cannot be explained by classical physics. Besides being of fundamental interest, entanglement is a unique resource for quantum information processing and communication. Entangled qubits can be used to establish private information or implement quantum logical gates [1,2]. Such capabilities are particularly useful when the entangled qubits are spatially separated [3][4][5], opening the opportunity to create highly connected quantum networks [6] or extend quantum cryptography to long distances [7,8]. Here we present a key experiment towards the realization of long-distance quantum networks with solid-state quantum registers. We have entangled two electron spin qubits in diamond that are separated by a three-meter distance. We establish this entanglement using a robust protocol based on local creation of spin-photon entanglement and a subsequent joint measurement of the photons. Detection of the photons heralds the projection of the spin qubits onto an entangled state. We verify the resulting non-local quantum correlations by performing single-shot readout [9] on the qubits in different bases. The long-distance entanglement reported here can be combined with recently achieved initialization, readout and entanglement operations [9-13] on local long-lived nuclear spin registers, enabling deterministic long-distance teleportation, quantum repeaters and extended quantum networks.A quantum network can be constructed by using entanglement to connect local processing nodes, each containing a register of well-controlled and long-lived qubits [6]. Solids are an attractive platform for such registers, as the use of nanofabrication and material design may enable well-controlled and scalable qubit systems [14]. The potential impact of quantum networks on science and technology has recently spurred research efforts towards generating entangled states of distant solid-state qubits [15][16][17][18][19][20][21].A prime candidate for a solid-state quantum register is the nitrogen-vacancy (NV) defect centre in diamond. The NV centre combines a long-lived electronic spin (S=1) with a robust optical interface, enabling measurement and high-fidelity control of the spin qubit [15,[22][23][24]. Furthermore, the NV electron spin can be used to access and manipulate nearby nuclear spins [9-13], thereby forming a multi-qubit register. To use such registers in a quantum network requires a mechanism to coherently connect remote NV centres.Here we demonstrate the generation of entanglement between NV centre spin qubits in distant setups. We achieve this breakthrough by combining recently established spin initialization and single-shot readout techniques [9] with efficient resonant optical detection and feedback-based control over the optical transitions, all in a single experiment and executed with high fidelity. These results put solid-state qubits on ...

Harnessing the full power of nascent quantum processors requires the efficient management of a limited number of quantum bits with finite coherent lifetimes. Hybrid algorithms, such as the variational quantum eigensolver (VQE), leverage classical resources to reduce the required number of quantum gates. Experimental demonstrations of VQE have resulted in calculation of Hamiltonian ground states, and a new theoretical approach based on a quantum subspace expansion (QSE) has outlined a procedure for determining excited states that are central to dynamical processes. We use a superconducting-qubit-based processor to apply the QSE approach to the H 2 molecule, extracting both ground and excited states without the need for auxiliary qubits or additional minimization. Further, we show that this extended protocol can mitigate the effects of incoherent errors, potentially enabling larger-scale quantum simulations without the need for complex error-correction techniques.