Testing randomness of series generated in an optical Bell’s experiment

Abstract

The generation of series of random numbers is an important and difficult problem. Appropriate measurements on entangled states have been proposed as the definitive solution to produce series of certified randomness, and quantum optical systems play a major role. However, several reports indicate that random number generators based on quantum measurements have a high rate of series rejected by standard tests of randomness. This is believed to be caused by experimental imperfections and is usually solved by using classical algorithms to extract randomness. This is acceptable to generate random numbers in a single place. In quantum key distribution (QKD) instead, if the extractor is known by an eavesdropper (a situation that cannot be ruled out), the key’s security may be menaced. We use a not-loophole-free, “toy” all-fiber-optic-based setup, mimicking a QKD one operating in the field, to generate binary series and evaluate their level of randomness according to Ville’s principle. The series are tested with a battery of indicators of statistical and algorithmic randomness and nonlinear analysis. The good performance of a simple method to get random series from rejected ones, previously reported by Solis et al. is confirmed and supported with additional arguments. Incidentally, a theoretically predicted relationship between complexity and entropy is verified. Regarding QKD, the level of randomness of series, obtained by applying Toeplitz’s extractor to rejected series, is found to be indistinguishable from the level of non-rejected raw ones.

Binary quantum random number generator based on value indefinite observables

“…In our case, both U x in (1) as well as U ′ U x in (5) have one eigenvalue −1 , and a double eigenvalue 1. More explicitly, the matrix satisfies the equality V T U x V = U ′ U x : this proves that the matrix U x defined in (1) is unitarily equivalent to the matrix combination U ′ U x in (6).…”

“…Can we construct QRNGs "provably better" than PRNGs? There are two types of QRNGs "theoretically certified": by the Bell inequalities [5][6][7] and by the Located Kochen-Specker Theorem 8,9 , a form of the Kochen-Specker Theorem, see [10][11][12] for detailed reviews.To date, only the second type of QRNGs has been mathematically proven to be better than any PRNG 8,13,14 . These QRNGs are three-dimensional: Since two-dimensional analogs of the Kochen-Specker Theorem as well as the Located Kochen-Specker Theorem are false, the generated sequences must be at least ternary 15 .…”

“…Can we construct QRNGs "provably better" than PRNGs? There are two types of QRNGs "theoretically certified": by the Bell inequalities [5][6][7] and by the Located Kochen-Specker Theorem 8,9 , a form of the Kochen-Specker Theorem, see [10][11][12] for detailed reviews.…”

Binary Quantum Random Number Generator Based on Value Indefinite Observables

“…Can we construct QRNGs "provably better" than PRNGs? There are two types of QRNGs "theoretically certified" by the Bell inequalities [10][11][12] and by the Located Kochen-Specker Theorem [1,13], a form of the Kochen-Specker Theorem, see [14,15] for detailed reviews. To date, only the second type of QRNGs have been mathematically proved better than any PRNG, namely [1][2][3].…”