Quantum superreplication of states and gates

While the no-cloning theorem forbids the 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 take place both for quantum states and quantum gates. The aim of this paper is to review the central features of quantum superreplication, providing a unified view on the 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 vanishing with a power law. Quantum strategies still offer an advantage for superreplication, in that they allow for an exponentially faster reduction of the error. Using the relation with estimation, we provide i) an alternative proof of the optimality of the Heisenberg scaling of quantum metrology, ii) a strategy to estimate arbitrary unitary gates with Heisenberg scaling, up to a logarithmic overhead, and iii) a protocol that generates M nearly perfect copies of a generic pure state with a number of queries to the corresponding unitary gate scaling as the square root of M. Finally, we point out that superreplication can be achieved using interactions among k systems, provided that k is large compared to square of the ratio between the numbers of input and output copies.