  Pattern recognition is a central topic in Learning Theory with numerous
applications such as voice and text recognition, image analysis, computer
diagnosis. The statistical set-up in classification is the following: we are
given an i.i.d. training set $(X_{1},Y_{1}),... (X_{n},Y_{n})$ where $X_{i}$
represents a feature and $Y_{i}\in \{0,1\}$ is a label attached to that
feature. The underlying joint distribution of $(X,Y)$ is unknown, but we can
learn about it from the training set and we aim at devising low error
classifiers $f:X\to Y$ used to predict the label of new incoming features.
  Here we solve a quantum analogue of this problem, namely the classification
of two arbitrary unknown qubit states. Given a number of `training' copies from
each of the states, we would like to `learn' about them by performing a
measurement on the training set. The outcome is then used to design mesurements
for the classification of future systems with unknown labels. We find the
asymptotically optimal classification strategy and show that typically, it
performs strictly better than a plug-in strategy based on state estimation.
  The figure of merit is the excess risk which is the difference between the
probability of error and the probability of error of the optimal measurement
when the states are known, that is the Helstrom measurement. We show that the
excess risk has rate $n^{-1}$ and compute the exact constant of the rate.
