Solution to the Mean Kingʹs Problem in Prime Power Dimensions Using Discrete Tomography

We solve a state reconstruction problem which arises in quantum information and which generalizes a problem introduced by Vaidman, Aharonov, and Albert in 1987. The task is to correctly predict the outcome of a measurement which an experimenter performed secretly in a lab. Using tomographic reconstruction based on summation over lines in an affine geometry, we show that this is possible whenever the measurements form a maximal set of mutually unbiased bases. Using a different approach we show that if the dimension of the system is large, the measurement result as well as the secretly chosen measurement basis can be inferred with high probability. This can be achieved even when the meanspirited King is unwilling to reveal the measurement basis at any point in time.