We employ an approach wherein vacuum entanglement is directly probed in a controlled manner. The approach consists of having a pair of initially nonentangled detectors locally interact with the field for a finite duration, such that the two detectors remain causally disconnected, and then analyzing the resulting detector mixed state. It is demonstrated that the correlations between arbitrarily far-apart regions of the vacuum of a relativistic free scalar field cannot be reproduced by a local hidden-variable model, and that as a function of the distance L between the regions, the entanglement decreases at a slower rate than ∼ exp(−(L/cT ) 3 ).

The majority of recent works investigating the link between non-locality and randomness, e.g. in the context of device-independent cryptography, do so with respect to some specific Bell inequality, usually the CHSH inequality. However, the joint probabilities characterizing the measurement outcomes of a Bell test are richer than just the degree of violation of a single Bell inequality. In this work we show how to take this extra information into account in a systematic manner in order to optimally evaluate the randomness that can be certified from non-local correlations. We further show that taking into account the complete set of outcome probabilities is equivalent to optimizing over all possible Bell inequalities, thereby allowing us to determine the optimal Bell inequality for certifying the maximal amount of randomness from a given set of non-local correlations.

Device-independent randomness generation and quantum key distribution protocols rely on a fundamental relation between the non-locality of quantum theory and its random character. This relation is usually expressed in terms of a trade-off between the probability of guessing correctly the outcomes of measurements performed on quantum systems and the amount of violation of a given Bell inequality. However, a more accurate assessment of the randomness produced in Bell experiments can be obtained if the value of several Bell expressions is simultaneously taken into account, or if the full set of probabilities characterizing the behavior of the device is considered. We introduce protocols for device-independent randomness generation secure against classical side information, that rely on the estimation of an arbitrary number of Bell expressions or even directly on the experimental frequencies of measurement outcomes. Asymptotically, this results in an optimal generation of randomness from experimental data (as measured by the min-entropy), without having to assume beforehand that the devices violate a specific Bell inequality.

In the distrustful quantum cryptography model the parties have conflicting interests and do not trust one another. Nevertheless, they trust the quantum devices in their labs. The aim of the device-independent approach to cryptography is to do away with the latter assumption, and, consequently, significantly increase security. It is an open question whether the scope of this approach also extends to protocols in the distrustful cryptography model, thereby rendering them "fully" distrustful. In this Letter, we show that for bit commitment-one of the most basic primitives within the model-the answer is positive. We present a device-independent (imperfect) bit-commitment protocol, where Alice's and Bob's cheating probabilities are ≃0.854 and 3/4, which we then use to construct a device-independent coin flipping protocol with bias ≲0.336.

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