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 $S=\{|{\Psi }_1〉,|{\Psi }_2〉,\dots ,|{\Psi }_n〉\}$ can be probabilistically cloned if and only if $|{\Psi }_1〉,|{\Psi }_2〉,\dots ,\mathrm{and}|{\Psi }_n〉$ 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 $n+1\mathrm{}$ outcome measurement on $n$ linearly independent states. The optimal efficiencies for this type of measurement are obtained.