Probabilistic cloning and identification of linearly independent quantum states

We construct a probabilistic quantum cloning machine by a general unitary-reduction operation. With a postselection of the measurement results, the machine yields faithful copies of the input states. It is shown that the states secretly chosen from a certain set S = {\Psi(1)], \Psi(2)],...,\Psi(n)]} can be probabilistically cloned if and only if \Psi(1)], \Psi(2)],..., and [Psi(n)] are linearly independent. We derive the best possible cloning efficiencies. Probabilistic cloning has a close connection with the problem of identification of a set of states, which is a type of n + 1 outcome measurement on n linearly independent states. The optimal efficiencies for this type of measurement are obtained.