Abstract
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 involvedmeasurements settings, each havingdoutcomes for an arbitrary prime numberd≥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.
Highlights
Nonlocality of quantum mechanics, in the sense first described by Einstein, Podolsky and Rosen [EPR35] and later formalised by Bell [Bel64], is arguably one of its most counterintuitive features
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
Bell violations are used to certify randomness produced in the experiment, which is often applied to device-independent cryptography, e.g. randomness generation/expansion [Col[06], PAM+10, CK11, VV12, MS16, BPPP14], quantum key distribution [BHK05, AGM06, ABG+07, RUV13, VV14, MS16, AFDF+18] or multi-party cryptography [SCA+11, KW16, RTK+18, RMW16, RMW18]
Summary
Nonlocality of quantum mechanics, in the sense first described by Einstein, Podolsky and Rosen [EPR35] and later formalised by Bell [Bel64], is arguably one of its most counterintuitive features. These results were combined to give certification schemes for higher-dimensional systems [YN13, CGS17, ŠCAA18] They are still based on the violation of many two-outcome Bell inequalities; in particular, the observables they use are constructed out of single-qubit measurements. In this work we introduce a generalisation of the CHSH Bell inequality which fills this gap To this aim we consider the Bell functional due to Buhrman and Massar (BM) [BM05] in which the settings and outcomes instead of being bits come from the set {0, 1, . We can compute the quantum value by first exhibiting a sum-of-squares (SOS) decomposition of the Bell operator and giving an explicit quantum realisation which saturates this bound This quantum realisation uses the maximally entangled state of local dimension d and rank-1 projective measurements which are pairwise mutually unbiased.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
