Quantum superreplication of states and gates

Although the no-cloning theorem forbids perfect replication of quantum information, it is sometimes possible to produce large numbers of replicas with vanishingly small error. This phenomenon, known as quantum superreplication, can occur for both quantum states and quantum gates. The aim of this paper is to review the central features of quantum superreplication and provide a unified view of existing results. The paper also includes new results. In particular, we show that when quantum superreplication can be achieved, it can be achieved through estimation up to an error of size O(M/N2), where N and M are the number of input and output copies, respectively. Quantum strategies still offer an advantage for superreplication in that they allow for exponentially faster reduction of the error. Using the relation with estimation, we provide i) an alternative proof of the optimality of Heisenberg scaling in quantum metrology, ii) a strategy for estimating arbitrary unitary gates with a mean square error scaling as log N/N2, and iii) a protocol that generates O(N2) nearly perfect copies of a generic pure state U |0〉 while using the corresponding gate U only N times. Finally, we point out that superreplication can be achieved using interactions among k systems, provided that k is large compared to M2/N2.