Mutually Unbiased Bases and Symmetric Informationally Complete Measurements in Bell Experiments
Mutually unbiased bases and symmetric informationally complete measurements in Bell experiments Armin Tavakoli,1 Máté Farkas,2, 3 Denis Rosset,4 Jean-Daniel Bancal,1 and J˛edrzejKaniewski5 1Department of Applied Physics, University of Geneva, 1211 Geneva, Switzerland 2Institute of Theoretical Physics and Astrophysics, National Quantum Information Centre, Faculty of Mathematics, Physics and Informatics, University of Gdansk, 80-952 Gdansk, Poland 3International Centre for Theory of Quantum Technologies, University of Gdansk, 80-308 Gdansk, Poland 4Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, N2L 2Y5, Canada 5Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland (Dated: October 21, 2020) Mutually unbiased bases (MUBs) and symmetric informationally complete projectors (SICs) are crucial to many conceptual and practical aspects of quantum theory. Here, we develop their role in quantum nonlocality by: i) introducing families of Bell inequalities that are maximally violated by d-dimensional MUBs and SICs respectively, ii) proving device-independent certification of natural operational notions of MUBs and SICs, and iii) using MUBs and SICs to develop optimal-rate and nearly optimal-rate protocols for device independent quantum key distribution and device-independent quantum random number generation respectively. Moreover, we also present the first example of an extremal point of the quantum set of correlations which admits physically inequivalent quantum realisations. Our results elaborately demonstrate the foundational and practical relevance of the two most important discrete Hilbert space structures to the field of quantum nonlocality. One-sentence summary — Quantum nonlocality is de- a single measurement. Since MUBs and SICs are both con- veloped based on the two most celebrated discrete struc- ceptually natural, elegant and (as it turns out) practically im- tures in quantum theory.
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