  We consider the Unambiguous State Discrimination (USD) of two mixed quantum
states. We study the rank and the spectrum of the elements of an optimal USD
measurement. This naturally leads to a partial fourth reduction theorem. This
theorem shows that either the failure probability equals its overall lower
bound given in term of the fidelity or a two-dimensional subspace can be split
off from the original Hilbert space. We then use this partial reduction theorem
to derive the optimal solution for any two equally probable Geometrically
Uniform (GU) states $\rho_0$ and $\rho_1=U\rho_0 U^\dagger$, $U^2={\openone}$,
in a four-dimensional Hilbert space. This represents a second class of
analytical solutions for USD problems that cannot be reduced to some pure state
cases. We apply our result to answer two questions that are relevant in
implementations of the Bennett and Brassard 1984 quantum key distribution
protocol using weak coherent states.
