  When an informationally incomplete set of observables is considered there are
several solutions to the quantum state reconstruction problem using von Neumann
measurements. The set of solutions are known as Pauli partners, which are not
easy to find even numerically. We present, in a self-contained paper, a new way
to find this solutions using the physical imposition operator. We show that
every Pauli partner is an attractive fixed point of this operator, which means
that we can find complete sets of Pauli partners very efficiently. As a
particular case, we found numerically 24 mutually unbiased bases in dimension
N=23 in less than 30 seconds in a standard PC. We hope that the algorithm
presented can be adapted to construct MU Constellations, SIC-POVMs, Equiangular
Tight Frames and Quantum t-Designs, which could open new possibilities to find
numerical solutions to these open problems related with quantum information
theory.
