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 \=\left\{\left| \Psi_1\right> ,\left| \Psi_2\right> ,... ,\left| \Psi_n\right> \right\} $can be probabilistically cloned if and only if$% \left| \Psi_1\right>$,$\left| \Psi_2\right>$,$... ,$and$\left| \Psi_n\right>$are linearly-independent. We derive the best possible cloning efficiencies. Probabilistic cloning has close connection with the problem of identification of a set of states, which is a type of$n+1$outcome measurement on$n$ linearly independent states. The optimal efficiencies for this type of measurement are obtained.