Probabilistic quantum cloning of a subset of linearly dependent states

It is well known that a quantum state, secretly chosen from a certain set, can be probabilistically cloned with positive cloning efficiencies if and only if all the states in the set are linearly independent. In this paper, we focus on probabilistic quantum cloning of a subset of linearly dependent states. We show that a linearly-independent subset of linearly-dependent quantum states {| Ψ 1 ⟩,| Ψ 2 ⟩,…,| Ψ n ⟩} can be probabilistically cloned if and only if any state in the subset cannot be expressed as a linear superposition of the other states in the set {| Ψ 1 ⟩,| Ψ 2 ⟩,…,| Ψ n ⟩}. The optimal cloning efficiencies are also investigated. Graphical abstract