Regular Article

Probabilistic quantum cloning of a subset of linearly dependent states

¹ Anhui Xinhua University, Hefei 230088, P.R. China
² School of Physics & Material Science, Anhui University, Hefei 230601, P.R. China

^a e-mail: wenzhang@ahu.edu.cn

Received: 8 October 2017
Received in final form: 26 November 2017
Published online: 6 February 2018

Abstract

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 {|Ψ₁⟩,|Ψ₂⟩,…,|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 {|Ψ₁⟩,|Ψ₂⟩,…,|Ψ_n⟩}. The optimal cloning efficiencies are also investigated.