  We study the measurement for the unambiguous discrimination of two mixed
quantum states that are described by density operators $\rho_1$ and $\rho_2$ of
rank d, the supports of which jointly span a 2d-dimensional Hilbert space.
Based on two conditions for the optimum measurement operators, and on a
canonical representation for the density operators of the states, two equations
are derived that allow the explicit construction of the optimum measurement,
provided that the expression for the fidelity of the states has a specific
simple form. For this case the problem is mathematically equivalent to
distinguishing pairs of pure states, even when the density operators are not
diagonal in the canonical representation. The equations are applied to the
optimum unambiguous discrimination of two mixed states that are similar states,
given by $\rho_2= U\rho_1 U^{\dag}$, and that belong to the class where the
unitary operator U can be decomposed into multiple rotations in the d mutually
orthogonal two-dimensional subspaces determined by the canonical
representation.
