On the effective size of certain "Schrödinger cat'' like states

Abstract

Several experiments and experimental proposals for the production of macroscopic superpositions naturally lead to states of the general form $|ϕ_1>^{\otimes N}+|ϕ_2>^{\otimes N}$, where the number of subsystems $N$ is very large, but the states of the individual subsystems have large overlap, $|łϕ_1|ϕ_2 \r|^2=1-ε^2$. We propose two different methods for assigning an effective particle number to such states, using ideal Greenberger--Horne--Zeilinger (GHZ)-- states of the form $|0\r^{\otimes n}+|1\r^{\otimes n}$ 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 $N ε^2$.