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

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 involve $d$ measurements settings, each having $d$ outcomes for an arbitrary prime number $d\ge 3$. 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.

arXiv comment19 pages, no figures, accepted in Quantum

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• Device-independent certification of maximal randomness from pure entangled two-qutrit states using non-projective measurements
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• Robust certification of arbitrary outcome quantum measurements from temporal correlations
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• Scalable Bell inequalities for graph states of arbitrary prime local dimension and self-testing
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• Self-testing quantum systems of arbitrary local dimension with minimal number of measurements
• A universal scheme to self-test any quantum state or measurement
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• Counterexamples in self-testing
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• Translating Bell Non-Locality to Prepare-and-Measure Scenarios under Dimensional Constraints
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• Frustration graph formalism for qudit observables
• Qudit Clauser-Horne-Shimony-Holt Inequality and Nonlocality from Wigner Negativity
• Certification of multi-qubit quantum systems with temporal inequalities
• Any gate of a quantum computer can be certified device-independently
• Quantifying incompatibility of quantum measurements through non-commutativity
• Trading symmetry for Hilbert-space dimension in Bell-inequality violation
• Controlling nonlocality of bipartite qubit states via quantum channels
• Designing generalized elegant Bell inequalities in higher dimensions from a Tsirelson bound
• Communication scenario enables robust self-testing of n-party Greenberger-Horne-Zeilinger basis measurements