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 $d\geq 3$. 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.
Let's analyze the situation: Having only access to the bits produced we can just perform statistics and play mathematics on them. And solely from the statistics of the box it is clearly impossible to say anything about its contents. End of story. Or, is it, really?
One of the most striking features of quantum physics is that, in some very particular situations, one can solve the above riddle. Through an experiment known as a Bell test, two or more boxes can become correlated in a very special way, which rules out all the above explanations, a phenomenon known as non-locality. When that Bell test gives the maximal score (maximal nonlocality), one can sometimes actually know what is inside the box!!! This is known as self-testing.
This statement may seem rather bold or puzzling. Of course, one could say that, if there is an atom of Caesium or an ion of Calcium inside the box that, in the process of generating the bits, behave exactly the same way, then it is impossible to tell them apart! But that is precisely the beauty of self-testing: it identifies all the states and measurements that behave the same way.
Pairs of particles prepared in the most quantum-correlated states are the easiest ones to self-test. These states are called maximally entangled and in their simpler version (qubits) the problem has been solved a long time ago: If the boxes inspect "uniformly" the state (performing the so-called mutually unbiased bases measurements), the correlations winning the Bell test with the best possible score are revealed. Then a mathematical proof shows that these states and measurements are the only ones that could have produced these correlations, thus revealing the relevant contents of the boxes.
However, for states of the same kind, albeit with more levels (qudits), designing a Bell test preserving these nice properties remains elusive. Here we manage to prove it for pairs of boxes that can produce an odd prime number of results each, and we provide a mathematical proof of the self-test when the number of possible results is three.
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