  Mixed states of a quantum system, represented by density operators, can be
decomposed as a statistical mixture of pure states in a number of ways where
each decomposition can be viewed as a different preparation recipe. However the
fact that the density matrix contains full information about the ensemble makes
it impossible to estimate the preparation basis for the quantum system. Here we
present a measurement scheme to (seemingly) improve the performance of unsharp
measurements. We argue that in some situations this scheme is capable of
providing statistics from a single copy of the quantum system, thus making it
possible to perform state tomography from a single copy. One of the byproduct
of the scheme is a way to distinguish between different preparation methods
used to prepare the state of the quantum system. However, our numerical
simulations disagree with our intuitive predictions. We show that a
counter-intuitive property of a biased classical random walk is responsible for
the proposed mechanism not working.
