  We theoretically investigate schemes to discriminate between two
nonorthogonal quantum states given multiple copies. We consider a number of
state discrimination schemes as applied to nonorthogonal, mixed states of a
qubit. In particular, we examine the difference that local and global
optimization of local measurements makes to the probability of obtaining an
erroneous result, in the regime of finite numbers of copies $N$, and in the
asymptotic limit as $N \rightarrow \infty$. Five schemes are considered:
optimal collective measurements over all copies, locally optimal local
measurements in a fixed single-qubit measurement basis, globally optimal fixed
local measurements, locally optimal adaptive local measurements, and globally
optimal adaptive local measurements. Here, adaptive measurements are those for
which the measurement basis can depend on prior measurement results. For each
of these measurement schemes we determine the probability of error (for finite
$N$) and scaling of this error in the asymptotic limit. In the asymptotic
limit, adaptive schemes have no advantage over the optimal fixed local scheme,
and except for states with less than 2% mixture, the most naive scheme (locally
optimal fixed local measurements) is as good as any noncollective scheme. For
finite $N$, however, the most sophisticated local scheme (globally optimal
adaptive local measurements) is better than any other noncollective scheme, for
any degree of mixture.
