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 S0ex0ex=0ex0ex|Ψ₁〉,|Ψ₂〉,,|Ψₙ〉 can be probabilistically cloned if and only if |Ψ₁〉,|Ψ₂〉,,and|Ψₙ〉 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.