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One of the differences between classical and quantum world is that in theformer we can always perform a measurement that gives certain outcomes for allpure states, while such a situation is not possible in the latter. The degreeof randomness of the distribution of the measurement outcomes can be quantifiedby the Shannon entropy. While it is well known that this entropy, as a functionof quantum states, needs to be minimized by some pure states, we would like toaddress the question how 'badly' can we end by choosing initially any purestate, i.e., which pure states produce the maximal amount of uncertainty undergiven measurement. We find these maximizers for all highly symmetric POVMs indimension 2, and for all SIC-POVMs in any dimension.

Pure states of maximum uncertainty with respect to a given POVM