  In recent years, informationally complete measurements have attracted
considerable attention, especially in the context of classical shadows. In the
particular case of informationally over-complete measurements, for which the
number of possible outcomes exceeds the dimension of the space of linear
operators in Hilbert space, the dual POVM operators used to interpret the
measurement outcomes are not uniquely defined. In this work, we propose a
method to optimize the dual operators after the measurements have been carried
out in order to produce sharper, unbiased estimations of observables of
interest. We discuss how this procedure can produce zero-variance estimations
in cases where the classical shadows formalism, which relies on so-called
canonical duals, incurs exponentially large measurement overheads. We also
analyze the algorithm in the context of quantum simulation with randomized
Pauli measurements, and show that it can significantly reduce statistical
errors with respect to canonical duals on multiple observable estimations.
