Device-independent randomness extraction from an arbitrarily weak min-entropy source

Abstract

Expansion and amplification of weak randomness plays a crucial role in many security protocols. Using quantum devices, such procedure is possible even without trusting the devices used, by utilizing correlations between outcomes of parts of the devices. We show here how to extract random bits with an arbitrarily low bias from a single arbitrarily weak min-entropy source in a device independent setting. To do this we use Mermin devices that exhibit super-classical correlations. Number of devices used scales polynomially in the length of the random sequence n. Our protocol is robust, it can tolerate devices that malfunction with a probability dropping polynomially in n at the cost of a minor increase of the number of devices used.High quality randomness is a very useful resource in many computation and cryptographic tasks. In fact it has been shown that many protocols (including quantum ones) vitally require perfect randomness for their security [1][2][3].Unfortunately, at the same time perfect randomness is very rare. In the classical world the true randomness, i.e. independent uniformly distributed random bits, cannot be produced at all. The only available resource is pseudo-randomness, sequences that appear random to all observers (often referred to as adversaries) not having full information about the whole environment. Thus classical randomness generators produce pseudorandom numbers stemming from external sources and fluctuations, hoping that the adversary will not be able to reconstruct all the background information. Sources producing imperfect randomness even taking into account the limited capabilities of the adversary are called weak random sources. To enhance the quality and security of these sources, randomness extractors are used. These are devices that combine more sources of randomness to obtain fewer bits of higher quality [4].On the other hand, theoretically the production of true randomness is possible, if one assumes Quantum theory to be valid: Preparation of a pure state and measurement in its complementary basis will yield a perfectly random result. This is due to the inherent randomness present in Quantum theory itself -this principle is being used in the design commercially available devices [5]. The assumption, however, is high quality and stability of quantum devices in an adversarial setting, which is far from trivial to achieve [6].In addition, quantum devices in reality act more like black boxes that are inaccessible for users except for providing them classical inputs and obtaining classical outputs from them. It is very hard, if not impossible, to directly test what these devices actually do, whether they perform operations and measurements as promised and whether their outputs really come from quantum measurements. Therefor it is crucial to test these devices even during their activity -satisfying these tests shall guarantee that the devices are correctly designed and manufactured and they work as desired. This is possible by utilizing super-classical correlations of certain quantum...

Quantum randomness extraction for various levels of characterization of the devices

“…It is remarkable that even this assumption can be partially relaxed, giving rise to the possibility of randomness amplification [12][13][14][15][16][17][18][19], but we don't consider this task in this paper.…”

Measurement-dependent locality beyond independent and identically distributed runs

“…The first results [14][15][16] were obtained for a slightly stronger model of random sources, the so-called Santha-Vazirani sources, but subsequent results have claimed the possibility of amplifying even a min-entropy source [17,18]. However, all these protocols require multi-partite entanglement and are not robust to deviations from an ideal quantum state; it is currently an open problem to devise a randomness amplification protocol that can be implemented with existing devices.…”

“…This task is provably impossible with classical information processing, but it becomes possible if the weak source is used to choose the inputs (including the state) in a Bell test, whose outcomes are taken as the new random numbers [14][15][16][17][18]. One may wonder why the optimised approach of Pütz and coworkers has not yet been applied to improve the bounds on randomness amplification.…”

A versatile construction of Bell inequalities for the multipartite scenario

“…They offer, however, a much richer source of correlations than the bipartite set-up, and have already been proven useful for several tasks. Either for a better use of the potential provided by multipartite systems-which might be particularly interesting for tasks on quantum networks-or simply to explore scenarios that go beyond the standard bipartite set-up, the study of multipartite scenarios is nowadays a central problem [11][12][13][14][15][16][17][18][19].…”