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Bell inequalities are an important tool in device-independent quantuminformation processing because their violation can serve as a certificate ofrelevant quantum properties. Probably the best known example of a Bellinequality is due to Clauser, Horne, Shimony and Holt (CHSH), which is definedin the simplest scenario involving two dichotomic measurements and whose allkey properties are well understood. There have been many attempts to generalisethe CHSH Bell inequality to higher-dimensional quantum systems, however, formost of them the maximal quantum violation---the key quantity for mostdevice-independent applications---remains unknown. On the other hand, theconstructions for which the maximal quantum violation can be computed, do notpreserve the natural property of the CHSH inequality, namely, that the maximalquantum violation is achieved by the maximally entangled state and measurementscorresponding to mutually unbiased bases. In this work we propose a novelfamily of Bell inequalities which exhibit precisely these properties, and whosemaximal quantum violation can be computed analytically. In the simplestscenario it recovers the CHSH Bell inequality. These inequalities involve $d$measurements settings, each having $d$ outcomes for an arbitrary prime number$d\geq 3$. We then show that in the three-outcome case our Bell inequality canbe used to self-test the maximally entangled state of two-qutrits and threemutually unbiased bases at each site. Yet, we demonstrate that in the case ofmore outcomes, their maximal violation does not allow for self-testing in thestandard sense, which motivates the definition of a new weak form ofself-testing. The ability to certify high-dimensional MUBs makes theseinequalities attractive from the device-independent cryptography point of view.

Maximal nonlocality from maximal entanglement and mutually unbiased bases, and self-testing of two-qutrit quantum systems