Mutually unbiased bases and symmetric informationally complete measurements in Bell experiments

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 that admits physically inequivalent quantum realizations. Our results elaborately demonstrate the foundational and practical relevance of the two most important discrete Hilbert space structures to the field of quantum nonlocality.

Automated summary

The text discusses the importance of mutually unbiased bases (MUBs) and symmetric informationally complete projectors (SICs) in quantum theory, particularly in the context of quantum nonlocality. It highlights the development of Bell inequalities, device-independent certification, and protocols for quantum key distribution and quantum random number generation using MUBs and SICs. Additionally, it presents the first example of an extremal point in the quantum set of correlations with physically inequivalent quantum realizations.