Upper Bounds for the Number of Quantum Clones under Decoherence

Universal quantum cloning machines (UQCMs), sometimes called quantum cloners, generate many outputs with identical density matrices, with as close a resemblance to the input state as is allowed by the basic principles of quantum mechanics. Any experimental realization of a quantum cloner has to cope with the effects of decoherence which terminate the coherent evolution demanded by a UQCM. We examine how many clones can be generated within a decoherence time. We compare the time that a quantum cloner implemented with trapped ions requires to produce M copies from N identical pure state inputs and the decoherence time during which the probability of spontaneous emission becomes non-negligible. We find a method to construct an N → M cloning circuit, and estimate the number of elementary logic gates required. It turns out that our circuit is highly vulnerable to spontaneous emission as the number of gates in the circuit is exponential with respect to the number of qubits involved.