Quantum mechanics predicts the existence of intrinsically random processes. Contrary to classical randomness, this lack of predictability can not be attributed to ignorance or lack of control. Here we propose a method to quantify the amount of randomness that can be extracted in two scenarios: (i) the quantum steering scenario, where two parties measure a bipartite system in an unknown state but one of them does not trust his measurement apparatus, and (ii) the prepare-and-measure scenario, where additionally the quantum state is known. We use our methods to compute the maximal amount of local randomness that can be certified by measuring systems subject to noise and losses and show that randomness can be certified from a single measurement if and only if the detectors used in the test have detection efficiency higher than 50%. One of the most distinct features of quantum mechanics is its intrinsically random character. While in classical mechanics lack of predictability can always be associated to ignorance or lack of control of the probed systems, the rules of quantum physics say that one can not predict the outcome of a measurement even if all the variables of a system are known. This inherent unpredictability has been exploited in different applications such as quantum random number generation [1] and quantum key distribution [2] . Recent results have shown that the randomness observed in quantum mechanics can be certified even without relying on any modelling of the quantum devices used for the generation of the random data. In fact, by analysing the data obtained in experiments involving local measurements on bipartite entangled systems one can prove that no one could have predicted this data in advance whenever a Bell inequality violation is observed [3, 4] . This is called device-independent randomness certification [5, 6] . The device-independent approach has the practical advantage that it does not rely on the exact description of the experimental set-up. This is crucial when implementing cryptographic protocols as an adversary can use a mismatch between the theoretical description and the actual implementation of the set-up to fake its performance [7] [8] [9] . However, deviceindependent protocols require and low levels of noise [4] , which make them very demanding experimentally. An intermediate scenario is that of quantum steering [11, 12] . It refers to the case where two parties, say Alice and Bob, apply local measurements on an unknown bipartite system. While one of them, Bob, has complete knowledge of his measurement apparatuses, Alice does not, and treats her measuring device as a black box with classical inputs and outputs. Quantum steering has been receiving lot of attention recently due to the fact that it allows for entanglement detection which is more robust to noise and experimental imperfections than Bell nonlocality [12, 13] . Moreover, quantum steering was shown to be useful for one-sided device independent quantum key distribution [14] The main result of our paper is a general method that quantifies the optimal amount of randomness can be certified from a single measurement in a steering experiment. We use this method to show that randomness can be certified provided that the detectors used have efficiency higher than 50%. Our method can be seen as the generalisation of the ideas of [26, 27] from the deviceindependent scenario to the semi-device-independent scenario. We compare the results obtained there compared to those obtained here, in terms of the amount of randomness that can be obtained by measuring systems subjected to white noise, and find substantial benefits can be obtained in the present setting. There are several motivations to quantify the amount of randomness in the steering scenario. From a fundamental point of view, it is important to understand how much randomness can be maintained if we give up partial information about the specific description of the systems [15, 28, 29] . From a practical point of view, the amount of randomness obtained in the steering scenario gives an upper bound to what Alice and Bob would obtain in a fully device-independent setting, regardless of the number of measurements Bob would apply. Furthermore, it is a scenario that appears naturally in some asymmetric applications. For instance the present results give a way of quantifying the amount of randomness in remote untrusted stations. This is relevant, for instance, when the provider of a quantum-random-number generator wants to remotely check if the devices they provided are still functioning properly. arXiv:1504.08302v1 [quant-ph] 30 Apr 2015
This work is built on a series of works in one-sided device-independent randomness certification, with REF establishing tools based on semi-definite programming.
Optimal randomness certification in the quantum steering and prepare-and-measure scenarios
New J. Phys. 17, 113010 (2015)
Physics