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 S0ex{0ex}=0ex{0ex}|{₁〉, |₂〉, , |₍〉} can be probabilistically cloned if and only if |₁〉, |₂〉, , and|₍〉 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.