Xi'an's Og

The New York Times (of 09 June) pointed me to a Nature paper of 27 May 2026 that I had most curiously missed! It is called Experimental randomness amplification and it presents a technique aimed at integrally correcting the bias in quantum random bit generators. Independently from the device used. This is quite interesting, even though I am definitely missing a lot.

“…randomness amplification protocols make use of a Bell test. Bell tests consist of measurements performed on entangled systems. Their purpose is to prove that, under natural locality assumptions, there cannot exist any variables that determine the outcomes of these measurements” Kulikov et al.

Not so perfect then, if relying on a test, a statistical test, cannot provide or prove certainty about the improved predictability of the random generator… The core of the method is described as follows:

“Any source of random bits B₁ … B_n can be characterized by a pair of parameters (μ, ε). Like any realistic device, the source may fail with some probability ε, in which case nothing is guaranteed about the randomness of B₁ … B_n (…) [Adopting] the Santha–Vazirani (SV) model⁵, considering an adversary attempting to predict the output bit B_i of the source, we say the source is a μ-SV source if  P_guess(B_i∣E)≤½+μ  holds for all i. Here E denotes any (potentially quantum) side information that is available before the bit B_i is produced (…) and P_guess(B_i∣E) denotes the probability of guessing B_i, given access to E (…) The case μ = 0 corresponds to a perfectly random (unbiased) source, while larger values of μ (…) bias μ quantifies the predictive power of a potential adversary, [with] the assumption that a source is (μ, ε)-random [can] be falsified by a statistical test that yields an observed bias μ_obs larger than μ. Therefore, if μ_obs satisfies μ_obs < μ, we say that the test result is compatible with the (μ, ε)-randomness assumption (…) In this work, we demonstrate that, for any input source with μ ≤ 0.75%, our experimental set-up (…) yields completely unbiased output randomness (…) The residual increase of the failure probability, ε, can in principle be made arbitrarily small at the cost of consuming more input randomness (…) To achieve this, we use two spatially separated sources of randomness [and] treat their concatenated outputs (…) as a single μ-SV source” Kulikov et al.

An assumption of importance in the proof (which I did not check!) is that both (independent) sources of randomness share the same bias μ, furthermore assumed to be constant over time, which should be a concern with physical devices. I also do not understand why a mere concatenation of the outputs saves the day, but quickly browsing the complete paper (aka supplementary information) I found that the final output K is via a two-process extractor that involves several n x n binary matrices even though it achieves a O(n log n) runtime. The NYT article reports that it took nine hours to generate 45 million bits, yes bits… Last remark about checking for pure, uniform, randomness in the experiment by relying on Marsaglie’s Diehard set of tests:

“we generate a random bitstring K consisting of m = 45,025,658 bits, starting from 5,368,709,120 low-quality random bits (…) chosen such that a failure probability of the protocol as low as ε = 10⁻¹² is guaranteed  (…) provided that the bias μ (…) is below 0.75%. Although it is fundamentally impossible to verify the unpredictability of a bitstring by analysing the string itself, we (…) run the NIST statistical test suite and the Diehard batteries of statistical tests [and] the result passes all the tests for which the string is sufficiently long.” Kulikov et al.