QLunch: Máté Farkas

Abstract: Mutually unbiased bases (MUBs) correspond to widely useful measurements in quantum information, used in state determination, quantum cryptography, quantum communication and other tasks. In this work we introduce a generalisation of MUBs that we call mutually unbiased measurements (MUMs). MUMs are defined through a complementarity property: if a measurement yields a definite outcome on a quantum state, then a measurement unbiased to it yields a uniformly random outcome on the same state. MUMs exactly correspond to MUBs whenever the Hilbert space dimension matches the outcome number. In general, MUMs admit the same incompatibility robustness and the same entropic uncertainty relations as MUBs. We characterise MUMs via block matrices that we call Hadamard matrices of unitaries. We show that a pair of MUMs is a direct sum of MUBs if and only if all blocks of the corresponding Hadamard matrix commute. Using this characterisation, we show that there exist MUMs that are not direct sums of MUBs, and we give explicit constructions through a correspondence with quaternionic Hadamard matrices. We further show that there exist MUMs that cannot be mapped to MUBs via any completely positive unital map, and that the number of MUMs with a fixed number of outcomes is unbounded, in stark contrast with the number of MUBs in dimension d, which is restricted to at most d+1.

We introduce a family of Bell inequalities whose maximal violation certify precisely the MUM property of a pair of measurements. Due to the fact that there exist unitarily inequivalent MUBs, this result also implies that the quantum correlation maximally violating these Bell inequalities is in general an extremal point but not a self-test, the first example of such a correlation. We further show that the maximal violation certifies log(d) bits of device-independent secret key that can be achieved by sharing a locally d-dimensional maximally entangled state. Then, we generalise the inequalities to an arbitrary number of MUMs (instead of two) and by numerically optimising these inequalities in a fixed dimension we tackle the long-standing problem of the number of MUBs in composite dimensions, known as Zauner's conjecture.

This talk is based on:

• A. Tavakoli, M. Farkas, D. Rosset, J-D. Bancal, J. Kaniewski, Science Advances 7, eabc3847 (2021)
• M. Prat Colomer, L. Mortimer, I. Frérot, M. Farkas, A. Acín, Quantum 6, 778 (2022)
• M. Farkas, J. Kaniewski, A. Nayak, IEEE Transactions on Information Theory 69(6), 3814–3824 (2023)