Self-testing mutually unbiased bases in the prepare-and-measure scenario

Mutually unbiased bases (MUBs) constitute the canonical example of incompatible quantum measurements. One standard application of MUBs is the task known as quantum random access code (QRAC), in which classical information is encoded in a quantum system, and later part of it is recovered by performing a quantum measurement. We analyse a specific class of QRACs, known as the $2^{d} \to 1$ QRAC, in which two classical dits are encoded in a $d$-dimensional quantum system. It is known that among rank-1 projective measurements MUBs give the best performance. We show (for every $d$) that this cannot be improved by employing non-projective measurements. Moreover, we show that the optimal performance can only be achieved by measurements which are rank-1 projective and mutually unbiased. In other words, the $2^{d} \to 1$ QRAC is a self-test for a pair of MUBs in the prepare-and-measure scenario. To make the self-testing statement robust we propose measures which characterise how well a pair of (not necessarily projective) measurements satisfies the MUB conditions and show how to estimate these measures from the observed performance. Similarly, we derive explicit bounds on operational quantities like the incompatibility robustness or the amount of uncertainty generated by the uncharacterised measurements. For low dimensions the robustness of our bounds is comparable to that of currently available technology, which makes them relevant for existing experiments. Lastly, our results provide essential support for a recently proposed method for solving the long-standing existence problem of MUBs.