  We review the problem of state reconstruction in classical and in quantum
physics, which is rarely considered at the textbook level. We review a method
for retrieving a classical state in phase space, similar to that used in
medical imaging known as Computer Aided Tomography. We explain how this method
can be taken over to quantum mechanics, where it leads to a description of the
quantum state in terms of the Wigner function which, although may take on
negative values, plays the role of the probability density in phase space in
classical physics. We explain another approach to quantum state reconstruction
based on the notion of Mutually Unbiased Bases, and indicate the relation
between these two approaches. Both are for a continuous, infinite-dimensional
Hilbert space. We then study the finite-dimensional case and show how the
second method, based on Mutually Unbiased Bases, can be used for state
reconstruction.
