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 /phi(1)>( multiply sign in circle N)+/phi 2 >( multiply sign in circle N), where the number of subsystems N is very large, but the states of the individual subsystems have large overlap, /<phi 1/phi 2 >/ 2=1-epsilon 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>( multiply sign in circle n)+/1>( multiply sign in circle 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 epsilon 2.