Unpredictability, or randomness, of the outcomes of measurements made on an entangled state can be certified provided that the statistics violate a Bell inequality. In the standard Bell scenario where each party performs a single measurement on its share of the system, only a finite amount of randomness, of at most 4 log 2 d bits, can be certified from a pair of entangled particles of dimension d. Our work shows that this fundamental limitation can be overcome using sequences of (nonprojective) measurements on the same system. More precisely, we prove that one can certify any amount of random bits from a pair of qubits in a pure state as the resource, even if it is arbitrarily weakly entangled. In addition, this certification is achieved by near-maximal violation of a particular Bell inequality for each measurement in the sequence.Introduction.-Bell's theorem [1] has shown that the predictions of quantum mechanics demonstrate non-locality. That is, they cannot be described by a theory in which there are objective properties of a system prior to measurement that satisfy the no-signalling principle (sometimes referred to as "local realism"). Thus, if one requires the no-signalling principle to be satisfied at the operational level then the outcomes of measurements demonstrating non-locality must be unpredictable [1][2][3]. This unpredictability, or randomness, is not the result of ignorance about the system preparation but is intrinsic to the theory.Although the connection between quantum non-locality (via Bell's theorem) and the existence of intrinsic randomness is well known [1][2][3][4] it was analyzed in a quantitative way only recently [5,6]. It was shown how to use nonlocality (probability distributions that violate a Bell inequality) to certify the unpredictability of the outcomes of certain physical processes. This was termed device-independent randomness certification, because the certification only relies on the statistical properties of the outcomes and not on how they were produced. The development of information protocols exploiting this certified form of randomness, such as deviceindependent randomness expansion [5][6][7] and amplification protocols [8,9], followed.Entanglement is a necessary resource for quantum nonlocality, which in turn is required for randomness certification. It is thus crucial to understand qualitatively and quantitatively how these three fundamental quantities relate to one another. In our work, we focus on asking how much certifiable randomness can be obtained from a single entangled state as a resource. Progress has been made in this direction for entangled states shared between two parties, Alice (A) and Bob (B), in the standard scenario where each party makes a single measurement on his share of the system and then discards it. An argument adapted from Ref. [10] shows that either of the two parties, A or B can certify at most 2log 2 d bits of randomness [11], where d is the dimension of the local Hilbert space the state lives in, which in turn implies a bound of 4log 2 d b...

We show that for all n ≥ 3, an example of an n-partite quantum correlation that is not genuinely multipartite nonlocal but rather exhibiting anonymous nonlocality, that is, nonlocal but biseparable with respect to all bipartitions, can be obtained by locally measuring the n-partite GreenbergerHorne-Zeilinger (GHZ) state. This anonymity is a manifestation of the impossibility to attribute unambiguously the underlying multipartite nonlocality to any definite subset(s) of the parties, even though the correlation can indeed be produced by nonlocal collaboration involving only such subsets. An explicit biseparable decomposition of these correlations is provided for any partitioning of the n parties into two groups. Two possible applications of these anonymous GHZ correlations in the device-independent setting are discussed: multipartite secret sharing between any two groups of parties and bipartite quantum key distribution that is robust against nearly arbitrary leakage of information.Quantum correlations that violate a Bell inequality [1], a constraint first derived in the studies of local-hiddenvariable-theories, were initially perceived only as a counterintuitive feature with no classical analog. With the discovery of quantum information science, these intriguing correlations have taken the new role as a resource. For instance, in nonlocal games [2], the presence of Bellinequality-violating (hereafter referred as nonlocal) correlation signifies the usage of strategies that cannot be achieved using only shared randomness. They are also an indispensable resource in quantum information and communication tasks such as the reduction of communication complexity [3], the distribution of secret keys using untrusted devices [4,5], as well as the certification and expansion of randomness [6] etc. (see [7] for a review).Thus far, prior studies of quantum nonlocality have focussed predominantly on the bipartite setup. However, as with quantum entanglement [8,9], correlations between measurement outcomes can exhibit a much richer structure in the multipartite setup. Consider a multipartite Bell-type experiment with the i-th party's choice of measurement setting (input) denoted by x i = 0, 1 and the corresponding outcome (output) by a i = ±1. Already in the tripartite setting [10], quantum mechanics allow for correlations -a collection of joint conditional probability distributions P = {P ( a| x)} = {P (a 1 a 2 a 3 |x 1 x 2 x 3 )} -that cannot be reproduced even when a subset of the parties are allowed to share some nonlocal resource R [11,12].

While all bipartite pure entangled states are known to generate correlations violating a Bell inequality, and are therefore nonlocal, the quantitative relation between pure state entanglement and nonlocality is poorly understood. In fact, some Bell inequalities are maximally violated by non-maximally entangled states and this phenomenon is also observed for other operational measures of nonlocality.In this work, we study a recently proposed measure of nonlocality defined as the probability that a pure state displays nonlocal correlations when subjected to random measurements. We first prove that this measure satisfies some natural properties for an operational measure of nonlocality. Then, we show that for pure states of two qubits the measure is monotonic with entanglement for all correlation two-outcome Bell inequalities: for all these inequalities, the more the state is entangled, the larger the probability to violate them when random measurements are performed. Finally, we extend our results to the multipartite setting.

The generation of (Bell-)nonlocal correlations, i.e., correlations leading to the violation of a Bell-like inequality, requires the usage of a nonlocal resource, such as an entangled state. When given a correlation (a collection of conditional probability distributions) from an experiment or from a theory, it is desirable to determine the extent to which the participating parties would need to collaborate nonlocally for its (re)production. Here, we propose to achieve this via the minimal group size (MGS) of the resource, i.e., the smallest number of parties that need to share a given type of nonlocal resource for the above-mentioned purpose. In addition, we provide a general recipe-based on the lifting of Bell-like inequalities-to construct MGS witnesses for nonsignaling resources starting from any given ones. En route to illustrating the applicability of this recipe, we also show that when restricted to the space of full-correlation functions, nonsignaling resources are as powerful as unconstrained signaling resources. Explicit examples of correlations where their MGS can be determined using this recipe and other numerical techniques are provided.

The characterization of quantum correlations in terms of information-theoretic resource has been a fruitful approach to understand the power of quantum correlations as a resource. While bipartite entanglement and Bell inequality violation in this setting have been extensively studied, relatively little is known about their multipartite counterpart. In this paper, we apply and adapt the recently proposed definitions of multipartite nonlocality [Phys. Rev. A 88, 014102] to the three-and four-partite scenario to gain new insight on the resource aspect of multipartite nonlocal quantum correlations. Specifically, we show that reproducing certain tripartite quantum correlations requires mixtures of classical resources -be it the ability to change the groupings or the time orderings of measurements. Thus, when seen from the perspective of biseparable one-way classical signaling resources, certain tripartite quantum correlations do not admit a definite causal order. In the fourpartite scenario, we obtain a superset description of the set of biseparable correlations which can be produced by allowing two groups of bipartite non-signaling resources. Quantum violation of the resulting Bell-like inequalities are investigated. As a byproduct, we obtain some new examples of device-independent witnesses for genuine four-partite entanglement, and also device-independent witnesses that allows one to infer the structure of the underlying multipartite entanglement.

The outcomes of local measurements made on entangled systems can be certified to be random provided that the generated statistics violate a Bell inequality. This way of producing randomness relies only on a minimal set of assumptions because it is independent of the internal functioning of the devices generating the random outcomes. In this context it is crucial to understand both qualitatively and quantitatively how the three fundamental quantities -entanglement, non-locality and randomness -relate to each other. To explore these relationships, we consider the case where repeated (non projective) measurements are made on the physical systems, each measurement being made on the post-measurement state of the previous measurement. In this work, we focus on the following questions: For systems in a given entangled state, how many nonlocal correlations in a sequence can we obtain by measuring them repeatedly? And from this generated sequence of non-local correlations, how many random numbers is it possible to certify? In the standard scenario with a single measurement in the sequence, it is possible to generate non-local correlations between two distant observers only and the amount of random numbers is very limited. Here we show that we can overcome these limitations and obtain any amount of certified random numbers from an entangled pair of qubit in a pure state by making sequences of measurements on it. Moreover, the state can be arbitrarily weakly entangled. In addition, this certification is achieved by near-maximal violation of a particular Bell inequality for each measurement in the sequence. We also present numerical results giving insight on the resistance to imperfections and on the importance of the strength of the measurements in our scheme.Digital Object Identifier 10.4230/LIPIcs...

Local measurements acting on entangled quantum states give rise to a rich correlation structure in the multipartite scenario. We introduce a versatile technique to build families of Bell inequalities witnessing different notions of multipartite nonlocality for any number of parties. The idea behind our method is simple: a known Bell inequality satisfying certain constraints, for example the Clauser-Horne-Shimony-Holt inequality, serves as the seed to build new families of inequalities for more parties. The constructed inequalities have a clear operational meaning, capturing an essential feature of multipartite correlations: their violation implies that numerous subgroups of parties violate the inequality chosen as seed. The more multipartite nonlocal the correlations, the more subgroups can violate the seed. We illustrate our construction using different seeds and designing Bell inequalities to detect k-way nonlocal multipartite correlations, in particular, genuine multipartite nonlocal correlations-the strongest notion of multipartite nonlocality. For one of our inequalities we prove analytically that a large class of pure states that are genuine multipartite entangled (GME) exhibit genuine multipartite nonlocality for any number of parties, even for some states that are almost product. We also provide numerical evidence that this family is violated by all GME pure states of three and four qubits. Our results make us conjecture that this family of Bell inequalities can be used to prove the equivalence between genuine multipartite pure-state entanglement and nonlocality for any number of parties.

We present the first complete implementation of a randomness and privacy amplification protocol based on Bell tests. This allows the building of device-independent random number generators which output provably unbiased and private numbers, even if using an uncharacterised quantum device potentially built by an adversary. Our generation rates are linear in the runtime of the quantum device and the classical randomness post-processing has quasi-linear complexity -making it efficient on a standard personal laptop. The statistical analysis is tailored for real-world quantum devices, making it usable as a quantum technology today.We then showcase our protocol on the quantum computers from the IBM-Q experience. Although not purposely built for the task, we show that quantum computer can run faithful Bell tests by adding minimal assumptions. At a high level, these amount to trusting that the quantum device was not purposely built to trick the user, but otherwise remains mostly uncharacterised. In this semi-device-independent manner, our protocol generates provably private and unbiased random numbers on today's quantum computers. Contents I. OverviewA. Introduction B. Results C. Relation to previous work II. Idea of the protocol A. Setup B. Interaction with the quantum device -data collection C. Verification / certification D. Randomness post-processing III. Main tools and ingredients A. What is randomness? B. Imperfect random number generators C. Quantum devices, Bell tests, and guessing probabilities D. Bell tests with imperfect random inputs E. Statistical analysis F. Post-processing randomness G. List of assumptions IV. Protocol and concrete numerical examples A. Steps of the protocol B. Efficiency of the protocol C. Fine tuning the randomness post-processing V. Implementation on IBM's quantum computers A. Overview B. Quantum computers for Bell experiments C. Bell inequality violations D.