Mutually unbiased measurements and Bell nonlocality

Mutually unbiased bases (MUBs) are highly useful finite-dimensional quantum measurements with myriad applications in quantum information theory and many connections to different areas of mathematics. We introduce a relaxation of the definition of MUBs, which does not depend on the Hilbert space dimension, only on the number of outcomes. We call these objects mutually unbiased measurements (MUMs), and we show that our definition corresponds to a natural operational definition of mutual unbiasedness. Moreover, d-outcome MUMs satisfy the same entropic uncertainty relations, and are incompatible to the same extent as d-dimensional MUBs. Indeed, d-outcome MUMs complemented by assuming a d-dimensional Hilbert space correspond exactly to d-dimensional MUBs. However, we show that while for d = 2 and 3, d-outcome MUMs always correspond to direct sums of d-dimensional MUBs, this is not the case for d = 4 and 5. Even more, for d = 4 and 5 we show that there exist d-outcome MUMs that cannot even be converted to d-dimensional MUBs using any completely positive unital map. Therefore our MUM notion is strictly more general than the notion of MUBs. Next, we devise a family of Bell inequalities, parametrised by an integer d >= 2, that device-independently certify a pair of d-outcome MUMs. Since there exist different pairs of MUBs that are not equivalent under a unitary transformation and complex conjugation, our certification does not constitute a self-test. Up to our knowledge, this is the first instance of an extremal point of the quantum set of correlations that is not a self-test. This raises the question: given an extremal point of the quantum set of correlations, to what extent does this fix the quantum realisation of a corresponding experiment? This talk is based on parts of our joint work with Armin Tavakoli, Denis Rosset, Jean-Daniel Bancal and Jędrzej Kaniewski, see the preprint at https://arxiv.org/abs/1912.03225