  We consider the problem of correctly classifying a given quantum two-level
system (qubit) which is known to be in one of two equally probable quantum
states. We assume that this task should be performed by a quantum machine which
does not have at its disposal a complete classical description of the two
template states, but can only have partial prior information about their level
of purity and mutual overlap. Moreover, similarly to the classical supervised
learning paradigm, we assume that the machine can be trained by $n$ qubits
prepared in the first template state and by $n$ more qubits prepared in the
second template state. In this situation we are interested in the optimal
process which correctly classifies the input qubit with the largest probability
allowed by quantum mechanics. The problem is studied in its full generality for
a number of different prior information scenarios and for an arbitrary size $n$
of the training data. Finite size corrections around the asymptotic limit
$n\rightarrow \infty$ are also derived. When the states are assumed to be pure,
with known overlap, the problem is also solved in the case of d-level systems.

Quantum Physics