Probabilistically perfect quantum cloning and unambiguous state discrimination

Abstract

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 η 1and η 2 = 1 − η 1, from a given set of the two linearly independent states | ψ 1〉 ⊗ N = (cos θ|0〉 + sin θ|1〉) ⊗ N and | ψ 2〉 ⊗ N = (sin θ|0〉 + cos θ|1〉) ⊗ N ( 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 | ψ 1〉and| ψ 2〉. As an example, we give the explicit transformation of the optimal symmetric 1 → M probabilistically perfect quantum cloning to copy the two input states | ψ 1〉 and | ψ 2〉.