Bell inequalities tailored to the Greenberger-Horne-Zeilinger states of arbitrary local dimension

Abstract

In device-independent quantum information processing Bell inequalities are not only used as detectors of nonlocality, but also as certificates of relevant quantum properties. In order for these certificates to work, one very often needs Bell inequalities that are maximally violated by specific quantum states. Recently, in [A. Salavrakos et al., Phys. Rev. Lett. 119, 040402 (2017)] a general class of Bell inequalities, with arbitrary numbers of measurements and outcomes, has been designed, which are maximally violated by the maximally entangled states of two quantum systems of arbitrary dimension. In this work, we generalize these results to the multipartite scenario and obtain a general class of Bell inequalities maximally violated by the Greenberger-Horne-Zeilinger states of any number of parties and any local dimension. We then derive analytically their maximal quantum and nonsignaling values. We also obtain analytically the bound for detecting genuine nonlocality and compute the fully local bound for a few exemplary cases. Moreover, we consider the question of adapting this class of inequalities to partially entangled GHZ-like states for some special cases of low dimension and small number of parties. Through numerical methods, we find classes of inequalities maximally violated by these partially entangled states.

arXiv comment22 pages (18.2 + appendices)

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• Self-testing of multipartite GHZ states of arbitrary local dimension with arbitrary number of measurements per party
• Custom Bell inequalities from formal sums of squares
• Detecting the dimensionality of genuine multi-particle entanglement
• Scalable Bell inequalities for graph states of arbitrary prime local dimension and self-testing
• Tight upper bound of the maximal quantum violation of Gisin's elegant Bell inequality and its application in randomness certification
• Bounding the fidelity of quantum many-body states from partial information
• Deriving three-outcome permutationally invariant Bell inequalities
• Linear maps as sufficient criteria for entanglement depth and compatibility in many-body systems
• "All-versus-nothing" proof of genuine tripartite steering and entanglement certification in the two-sided device-independent scenario
• Single Bell inequality to detect genuine nonlocality in three-qubit genuinely entangled states