Abstract
Mutually unbiased bases (MUBs) are a crucial ingredient for many protocols in quantum information processing. Measurements performed in these bases are unbiased to the maximally possible extent, which is used to prove randomness or secrecy of measurement results. In this work we show that certain properties of sets of MUBs crucially depend on their specific choice, including, somewhat surprisingly, measurement outcome labelling. If measurements are chosen in a coherent way, the secrecy of the result can be completely lost for specific sets of MUB measurements, while partially retained for others. This can potentially impact a broad spectrum of applications, where MUBs are utilized.