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arXiv:2606.21362

Bell inequalities tailored to optimal global randomness certification

Ignacio Perito, Raffaele D'Avino, Michał Jung, Piotr Mironowicz, Antonio Acín, Remigiusz Augusiak

quant-phSubmitted Jun 19, 2026• Announced Jun 22, 2026• 0 citations• Revised Jun 24, 2026 (v2)
v1 · Jun 19, 2026v2 · Jun 24, 2026v3 · Jul 21, 2026
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Abstract

We present two novel families of bipartite Bell inequalities designed to achieve optimal global randomness certification for an arbitrary number of outputs $d$. We first use symmetry arguments to argue that their maximal quantum violations certify $2\log d$ random bits. For the first family, we construct a quantum realization using $d\times d$ maximally entangled states which provides a quantum violation that we conjecture to be optimal for any $d$. It is then numerically shown that the obtained quantum violation certifies optimal global randomness, up to numerical precision, for $d=3,4$. For the second family, we provide the optimal quantum violation and its quantum realization for any $d$, again using $d\times d$ maximally entangled states and projective measurements over at least two unbiased bases on one of the parties. We self-test this realization for $d=3$, which implies the optimal certification of two fully random trits.

Comments
See also the related works by R. D'Avino et al and M. Farkas et al. today on arXiv.org

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We present two novel families of bipartite Bell inequalities designed to achieve optimal global randomness certification for an arbitrary number of outputs ddd. We first use symmetry arguments to argue that their maximal quantum violations certify 2log⁡d2\log d2logd random bits. For the first family, we construct a quantum realization using d×dd\times dd×d maximally entangled states which provides a quantum violation that we conjecture to be optimal for any ddd. It is then numerically shown that the obtained quantum violation certifies optimal global randomness, up to numerical precision, for d=3,4d=3,4d=3,4. For the second family, we provide the optimal quantum violation and its quantum realization for any ddd, again using d×dd\times dd×d maximally entangled states and projective measurements over at least two unbiased bases on one of the parties. We self-test this realization for d=3d=3d=3, which implies the optimal certification of two fully random trits.
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Certification of quantum systems and their properties has become a field of intensive study. Here, taking advantage of the one-sided device-independent (1SDI) scenario (also known as the quantum steering scenario), we propose a self-testing scheme for all bipartite entangled states using a single family of steering inequalities with the minimal number of two measurements per party. Building on this scheme we then show how to certify all rank-one extremal measurements, including nonprojective d2{d}^{2}d2-outcome measurements, which in turn can be used for certification of the maximal amount of randomness, that is, 2log2d2{\mathrm{log}}_{2}d2log2​d bits. Finally, in the particular case of d=3d=3d=3, we propose an extended Bell scenario that transforms a 1SDI scheme to ``almost device independent.''
Bell inequalities are an important tool in device-independent quantum information processing because their violation can serve as a certificate of relevant quantum properties. Probably the best known example of a Bell inequality is due to Clauser, Horne, Shimony and Holt (CHSH), which is defined in the simplest scenario involving two dichotomic measurements and whose all key properties are well understood. There have been many attempts to generalise the CHSH Bell inequality to higher-dimensional quantum systems, however, for most of them the maximal quantum violation---the key quantity for most device-independent applications---remains unknown. On the other hand, the constructions for which the maximal quantum violation can be computed, do not preserve the natural property of the CHSH inequality, namely, that the maximal quantum violation is achieved by the maximally entangled state and measurements corresponding to mutually unbiased bases. In this work we propose a novel family of Bell inequalities which exhibit precisely these properties, and whose maximal quantum violation can be computed analytically. In the simplest scenario it recovers the CHSH Bell inequality. These inequalities involve d measurements settings, each having d outcomes for an arbitrary prime number d ≥ 3 . We then show that in the three-outcome case our Bell inequality can be used to self-test the maximally entangled state of two-qutrits and three mutually unbiased bases at each site. Yet, we demonstrate that in the case of more outcomes, their maximal violation does not allow for self-testing in the standard sense, which motivates the definition of a new weak form of self-testing. The ability to certify high-dimensional MUBs makes these inequalities attractive from the device-independent cryptography point of view.
Bell nonlocality provides a device-independent (DI) way to certify quantum randomness, based on which true random numbers can be extracted from the observed correlations without detail characterizations on devices for quantum state preparation and measurement. However, the efficiency of current strategies for DI randomness certification is still heavily constrained when it comes to non-maximal Bell values, especially for multiple parties. Here, we present a family of multipartite Bell inequalities that allows to certify optimal quantum randomness and self-test GHZ (Greenberger-Horne-Zeilinger) states, which are inspired from the stabilizer group of the GHZ state. Due to the simple representation of stabilizer group for GHZ states, this family of Bell inequalities is of simple structure and can be easily expanded to more parties. Compared with the Mermin-type inequalities, this family of Bell inequality is more efficient in certifying quantum randomness when non-maximal Bell values achieved. Meanwhile, the general analytical upper bound for the Holevo quantity is presented, and achieves better performance compared with the MABK (Mermin-Ardehali-Belinskii-Klyshko) inequality, Parity-CHSH (Clauser-Horne-Shimony-Holt) inequality and Holz inequality at N=3N=3N=3, which is of particular interests for experimental researches on DI quantum cryptography in quantum networks.