Local transformations of superpositions of entangled states

Suppose that we have two entangled states $\mathrm{\mid }{\varphi }_1⟩$, $\mathrm{\mid }{\psi }_1⟩$ that cannot be converted to any of other two states $\mathrm{\mid }{\varphi }_2⟩$, $\mathrm{\mid }{\psi }_2⟩$ by local operations and classical communication. We analyze the possibility of locally transforming a superposition of $\mathrm{\mid }{\varphi }_1⟩$ and $\mathrm{\mid }{\psi }_1⟩$ into a superposition of $\mathrm{\mid }{\varphi }_2⟩$ and $\mathrm{\mid }{\psi }_2⟩$. By using the Nielsen’s theorem we find the necessary and sufficient conditions for this conversion to be performed.