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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 ). 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〉.

In this paper, we present the explicit transformations of the optimal 1 → 3 universal quantum and the phase-covariant cloning in 3 dimensions (qutrits). The fidelities of two types of cloners reach the theoretical bounds.

We construct the explicit formulation of the probabilistically perfect quantum cloning machine that perfectly duplicates the input states chosen from the special set φ i i = 1 d consisting of the linearly independent and nonorthogonal quantum states with 〈φi ∣φj〉 = r ∈ (0, 1)(i ≠ j). The success probabilities of cloning the input states are equal and maximal. As two examples, we present the explicit transformations of the optimal 1 → 2 probabilistically perfect quantum cloning of the real states in 2 and 3 dimensions. The success probabilities of each of two cloning machines are equal and maximal.

We present the explicit transformation of the suboptimal asymmetric economical 1→2 phase-covariant quantum cloning in d-dimension without ancilla. We also present its telecloning protocol, which differs from the general ones in which the symmetry of the transformations are required.

We derive the explicit transformation of the optimal symmetric economical 1 → M phase-covariant quantum cloning machine (EPCCM), which works without ancilla, as well as the corresponding fidelities in 2-dimension. The fidelity ( 1 → M = 2 k + 1 EPCCM ) is covered with previous contributions [G.M. D'Ariano, et al., Phys. Rev. A 71 (2005) 042327]. Meanwhile, the explicit transformation of the optimal symmetric 1 → M phase-covariant quantum cloning machine (PCCM) (working with ancilla) as well as the corresponding fidelities in 2-dimension is also derived. The results of the PCCM agree with previous contributions [H. Fan, et al., Phys. Rev. A 65 (2001) 012304].

We propose an experimentally feasible scheme to implement the optimal general 1→2 phase-covariant quantum cloning machine based on cavity QED. In the scheme, the cavity is only virtually excited and thus the scheme is insensitive to the cavity field states and cavity decay.

We present an explicit transformation of optimal asymmetric $1\ensuremath{\rightarrow}2$ real state cloning in two dimensions. Interestingly, the distribution of the two fidelities of the cloning is the same as its counterpart in optimal asymmetric phase-covariant cloning. We also generalize the cloner to the $d$-dimensional case and derive the explicit transformation of the optimal symmetric $1\ensuremath{\rightarrow}2$ real state cloning in the $d$-dimensional system. The maximal fidelity of two clones obtained coincides with the theoretical value [P. Vavez and N. J. Cerf, Phys. Rev. A 68, 032313 (2003)].