Note on purifications of a qubit
  This work provides a parametrization of the set of purifications of a 2x2 mixed state, and joint purifications (if any) of a given pair of mixed 2x2 states. The former is parametrized by SO(3, R), while the latter is parametrized by SO(2,R), except when the 2x2 state is pure. Using this parametrization some quantum information measures are obtained. In the process it is shown how to solve one variation of the classical Procustes problem. The appendix is devoted to the first step in extending this work to higher dimensions. 
Tiberiu Constantinescu and Viswanath Ramakrishna