Mutually unbiased bases and symmetric informationally complete measurements in Bell experiments: Bell inequalities, device-independent certification and applications

Abstract

Mutually unbiased bases (MUBs) and symmetric informationally complete projectors (SICs) are central to many conceptual and practical aspects of quantum theory. In this work, we investigate their role in quantum nonlocality. For every integer \(d\geq 2\), we introduce Bell inequalities for which pairs of \(d\)-dimensional MUBs and SICs, respectively, produce the largest violations allowed in quantum mechanics. To investigate whether these inequalities can be used for the purpose of device-independent certification of measurements, we show that the concepts of MUBs and SICs admit a natural operational interpretation which does not depend on the dimension of the underlying Hilbert space. We prove that the maximal quantum violations certify precisely these operational notions. In the case of MUBs we also show that the maximal violation certifies the presence of a maximally entangled state of local dimension \(d\) and that the maximal violation is achieved by a unique probability distribution. This constitutes the first example of an extremal point of the quantum set which admits physically inequivalent quantum realisations, i.e.~is not a self-test. Finally, we investigate the performance of our Bell inequalities in two tasks of practical relevance. We show that the Bell inequalities for MUBs guarantee the optimal key rate in a device-independent quantum key distribution protocol with \(d\) outcomes. Moreover, using the Bell inequalities for SICs, we show how qubit and qutrit systems can generate more device-independent randomness than higher-dimensional implementations based on standard projective measurements. We also investigate the robustness of the key and randomness generation schemes to noise. The results establish the relevance of MUBs and SICs for both fundamental and applied considerations in quantum nonlocality.