  In this paper we consider the conditions under which a given ensemble of
two-qubit states can be optimally distinguished by local operations and
classical communication (LOCC). We begin by completing the \emph{perfect}
distinguishability problem of two-qubit ensembles - both for separable
operations and LOCC - by providing necessary and sufficient conditions for the
perfect discrimination of one pure and one mixed state. Then for the well-known
task of minimum error discrimination, it is shown that \textit{almost all}
two-qubit ensembles consisting of three pure states cannot be optimally
discriminated using LOCC. This is surprising considering that \textit{any} two
pure states can be distinguished optimally by LOCC. Special attention is given
to ensembles that lack entanglement, and we prove an easy sufficient condition
for when a set of three product states cannot be optimally distinguished by
LOCC, thus providing new examples of the phenomenon known as "non-locality
without entanglement". We next consider an example of $N$ parties who each
share the same state but who are ignorant of its identity. The state is drawn
from the rotationally invariant "trine ensemble", and we establish a tight
connection between the $N$-copy ensemble and Shor's "lifted" single-copy
ensemble. For any finite $N$, we prove that optimal identification of the
states cannot be achieved by LOCC; however as $N\to\infty$, LOCC can indeed
discriminate the states optimally. This is the first result of its kind.
Finally, we turn to the task of unambiguous discrimination and derive new lower
bounds on the LOCC inconclusive probability for symmetric states. When applied
to the double trine ensemble, this leads to a rather different
distinguishability character than when the minimum-error probability is
considered.
