Amplifying the Randomness of Weak Sources Correlated With Devices

Abstract

The problem of device-independent randomness amplification against no-signaling adversaries has so far been studied under the assumption that the weak source of randomness is uncorrelated with the (quantum) devices used in the amplification procedure. In this work, we relax this assumption, and reconsider the original protocol of Colbeck and Renner using a Santha-Vazirani (SV) source. To do so, we introduce an SV-like condition for devices, namely that any string of SV source bits remains weakly random conditioned upon any other bit string from the same SV source and the outputs obtained when this further string is input into the devices. Assuming this condition, we show that a quantum device using a singlet state to violate the chained Bell inequalities leads to full randomness in the asymptotic scenario of a large number of settings, for a restricted set of SV sources (with 0 ≤ ε < (2 (1/12) − 1)/(2(2 (1/12) + 1)) ≈ 0.0144). We also study a device-independent protocol that allows for correlations between the sequence of boxes used in the protocol and the SV source bits used to choose the particular box from whose output the randomness is obtained. Assuming the SV-like condition for devices, we show that the honest parties can achieve amplification of the weak source, for the parameter range 0 ≤ ε < 0.0132, against a class of attacks given as a mixture of product box sequences, made of extremal nosignaling boxes, with additional symmetry conditions. Composable security proof against this class of attacks is provided.

Certified randomness in quantum physics

“…Other works [23,25,28,29,30,31,32,33] have focused on the second point and study how arbitrarily good randomness can be generated in Bell setups using sources of imperfect randomness. These protocols are often known as randomness amplification protocols [28].…”

“…Under only the assumption of no-signalling, the violation of Bell inequalities certifies the presence of randomness, but requires some initial randomness. Full randomness amplification protocols [23,25,29,30,31,32,33] are not able to completely break this circularity, but relax it as much as possible.…”

Randomness in quantum mechanics: philosophy, physics and technology

“…The final result was again asymptotic, in the sense that to attain full randomness amplification the user now requires an infinite number of devices. Randomness amplification protocols have been studied by several other works, see for instance (Bouda et al, 2014;Brandão et al, 2016;Chung et al, 2014;Coudron and Yuen, 2013;Grudka et al, 2014;Mironowicz et al, 2015;Ramanathan et al, 2016;Wojewódka et al, 2016). As above, the scope of this section is not to provide a complete description of all the works studying the problem of randomness amplification, but rather to provide a general framework that encompasses most of them.…”

Randomness Amplification under Minimal Fundamental Assumptions on the Devices

“…Moreover, we also assume that the structure of the box p(x, z|u , w) is fixed independently of the SV source p(u, t, e), i.e., the box is an unknown and arbitrary input-output channel independent of the SV source. This precludes malicious correlations such as in the scenario where for each bit string u taken from the source, a different (possibly local) box tuned to u is supplied, in which case the Bell test may be faked by local boxes [17]. Finally, it is worth noting that no randomness may be extracted under the assumptions stated above in a classical setting, whereas the Bell violation by quantum boxes allows to amplify randomness in a device-independent setting.…”

“…Important open questions still remain. One interesting question is whether the requirement of strict independence between the SV source and the devices can be relaxed to only require limited independence [17]. Another is to amplify the randomness of more general minentropy sources that do not possess the structure of the Santha-Vazirani source.…”