The explicit transformations of the 1→ 3 optimal universal quantum cloning and the optimal phase-covariant quantum cloning in d-dimensions are presented, and the dimensionalities of their ancillary systems are both d-dimensions. As d→∞ , their clone fidelities move toward 1/3, showing a classical limit for the fidelity of quantum cloning. Based on the reduction of the unitary transformation of quantum cloning, the transformation of the 1→M=d+1 optimal economical phase-covariant quantum cloning in d-dimensions is derived, and the clone fidelity is covered by the theoretical value.
The entanglement property of two identical atoms, initially entangled in Bell states, coupled to a single-mode cavity is considered. Based on the reduced non-perturbative quantum master equation method, the entanglement evolution of the two atoms with decay is investigated beyond the conventional rotating-wave approximation. We show that the counter-rotating wave terms, usually neglected, have a great influence on the disentanglement behaviour of the system. The phenomena of entanglement sudden death and entanglement sudden birth will occur. In addition, we show that the entanglement can be strengthened by introducing the dipole-dipole interaction of the two atoms.
