Effective Size of Certain Macroscopic Quantum Superpositions

Several experiments and experimental proposals for the production of macroscopic superpositions naturally lead to states of the general form $|{\varphi }_1{⟩}^{\otimes N}+|{\varphi }_2{⟩}^{\otimes N}$, where the number of subsystems $N$ is very large, but the states of the individual subsystems have large overlap, $|⟨{\varphi }_1|{\varphi }_2⟩{|}^2=1-{ϵ}^2$. We propose two different methods for assigning an effective particle number to such states, using ideal Greenberger-Horne-Zeilinger states of the form $|0{⟩}^{\otimes n}+|1{⟩}^{\otimes n}$ as a standard of comparison. The two methods are based on decoherence and on a distillation protocol, respectively. Both lead to an effective size $n$ of the order of $N{ϵ}^2$.