  In this thesis we study the problem of unambiguously discriminating two mixed
quantum states. We first present reduction theorems for optimal unambiguous
discrimination of two generic density matrices. We show that this problem can
be reduced to that of two density matrices that have the same rank $r$ in a
2$r$-dimensional Hilbert space. These reduction theorems also allow us to
reduce USD problems to simpler ones for which the solution might be known. As
an application, we consider the unambiguous comparison of $n$ linearly
independent pure states with a simple symmetry. Moreover, lower bounds on the
optimal failure probability have been derived. For two mixed states they are
given in terms of the fidelity. Here we give tighter bounds as well as
necessary and sufficient conditions for two mixed states to reach these bounds.
We also construct the corresponding optimal measurement. With this result, we
provide analytical solutions for unambiguously discriminating a class of
generic mixed states. This goes beyond known results which are all reducible to
some pure state case. We however show that examples exist where the bounds
cannot be reached. Next, we derive properties on the rank and the spectrum of
an optimal USD measurement. This finally leads to a second class of exact
solutions. Indeed we present the optimal failure probability as well as the
optimal measurement for unambiguously discriminating any pair of geometrically
uniform mixed states in four dimensions. This class of problems includes for
example the discrimination of both the basis and the bit value mixed states in
the BB84 QKD protocol with coherent states.
