No-cloning of nonorthogonal states does not require inner product preserving

Abstract

The no-cloning theorem says there is no quantum copy machine which can copy any one-qubit state. Inner product preserving was always used to prove the no-cloning of nonorthogonal states. In this paper we show that the no-cloning of nonorthogonal states does not require inner product preserving and discuss the minimal properties which a linear operator possesses to copy two different states at the same device. In this paper, we obtain the following necessary and sufficient condition. For any two different states ∣ψ〉 = a∣0〉+b∣1〉∣ψ〉=a∣0〉+b∣1〉 and ∣ϕ〉 = c∣0〉+d∣1〉∣ϕ〉=c∣0〉+d∣1〉, assume that a linear operator LL can copy them, that is, L(∣ψ,0〉) = ∣ψ,ψ〉L(∣ψ,0〉)=∣ψ,ψ〉 and L(∣ϕ,0〉) = ∣ϕ,ϕ〉L(∣ϕ,0〉)=∣ϕ,ϕ〉. Then the two states are orthogonal if and only if L(∣0,0〉)L(∣0,0〉) and L(∣1,0〉)L(∣1,0〉) are unit length states. Thus we only need linearity and that L(∣0,0〉)L(∣0,0〉) and L(∣1,0〉)L(∣1,0〉) are unit length states to prove the no-cloning of nonorthogonal states. It implies that inner product preserving is not necessary for the no-cloning of nonorthogonal states.