In this paper, we derive the explicit transformations of the optimal 1 → 3, 4, 5 phase-covariant cloning in three dimensions, and then generalize them to the cases of 1 → M = 3n, 3n + 1, 3n + 2 (n ≥ 1 integer) cloning. The clone fidelities are coincident with the theoretical bounds found.
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)].
We consider cloning transformations of d-dimensional states of the form eiϕ0∣0>+eiϕ1∣1>+...+eiϕd−1∣d−1> that are covariant with respect to rotations of the phases ϕi's. The optimal cloning maps are easily obtained using a well-defined general characterization of state-dependent 1→2 cloning transformations in arbitrary dimensions. Our results apply to symmetric as well as asymmetric cloners, so that the balance between the fidelity of the two clones can be analyzed.
We generalize three symmetric cloners to the asymmetric cases and present three explicit cloning transformations as well as their corresponding fidelity distributions in d dimensions. The three asymmetric cloners, including the optimal asymmetric phase-covariant cloning (APCC) and the suboptimal asymmetric economical phase-covariant cloning (AEPCC) working without ancilla and the optimal asymmetric real state cloning (ARSC), together with the optimal asymmetric universal quantum cloning (AUQC) construct a generic cloning, where the quantum information of initial systems of different pure input states in d dimensions with their information not completely known can be optimally distributed to different final systems. By comparison of the fidelity distributions of the four asymmetric cloners, some interesting results can be obtained.