Probabilistically perfect quantum cloning and unambiguous state discrimination

We consider the N → M probabilistically perfect quantum cloning machine that perfectly produces M faithful copies from N identical input states, where the input states are selected, with prior probabilities η1and η2 = 1 − η1, from a given set of the two linearly independent states |ψ1〉⊗ N = (cosθ|0〉 + sinθ|1〉)⊗ N and |ψ2〉⊗ N = (sinθ|0〉 + cosθ|1〉)⊗ N (θ∈(0,π/2)θ∈0,π/2). We derive the optimal distribution of the success probabilities. When M approaches infinite, the probabilistically perfect quantum cloning can be regarded as a kind of the unambiguous state discrimination, and theoretically provides the upper bound of the unambiguous state discrimination. By using the optimal distribution of the success probabilities of the optimal asymmetric 1 → M probabilistically perfect quantum cloning, we can derive the maximum average success probability of the unambiguous discrimination of two nonorthogonal quantum states |ψ1〉and|ψ2〉. As an example, we give the explicit transformation of the optimal symmetric 1 → M probabilistically perfect quantum cloning to copy the two input states |ψ1〉 and |ψ2〉.