Linearly-independent quantum states can be cloned

Abstract

A fundamental question in quantum mechanics is, whether it is possible to replicate an arbitrary unknown quantum state. Then famous quantum no-cloning theorem [Nature 299, 802 (1982)] says no to the question. But it leaves open the following question: If the state is not arbitrary, but secretly chosen from a certain set $\$={ | Ψ_1> ,| Ψ_2> ,... ,| Ψ_n> } $, whether is the cloning possible? This question is of great practical significance because of its applications in quantum information theory. If the states $| Ψ_1>, | Ψ_2>,...$ and $| Ψ_n> $ are linearly-dependent, similar to the proof of the no-cloning theorem, the linearity of quantum mechanics forbids such replication. In this paper, we show that, if the states $| Ψ_1>, | Ψ_2>, ...$ and $| Ψ_n> $ are linearly-independent, they do can be cloned by a unitary-reduction process.