Local transformations of superpositions of entangled states

Abstract

Suppose that we have two entangled states $\ket {ϕ_1}$, $\ket{ψ_1}$ that cannot be converted to any of other two states $\ket{ϕ_2}$, $\ket{ψ_2}$ by local operations and classical communication. We analyze the possibility of locally transforming a superposition of $\ket{ϕ_1}$ and $\ket{ψ_1}$ into a superposition of $\ket{ϕ_2}$ and $\ket{ψ_2}$. By using the Nielsen's theorem we find the necessary and sufficient conditions for this conversion to be performed.