where we introduced q n / √ 2r s as the real part of c n and p n / √ 2r s as the imaginary part of c n . This phase e iΦ is a constant. The transformation in Eq. 46 provides the connection between a given pair of q n and p n and the angle variables of the generalized spin coherent states {θ n , ϕ n } through Eq. 18. Because we want to convert 2N -2 real independent variables of {θ n , ϕ n } into 2N variables {q n , p n } that are subjects to a total population constraint (see Eq. 56 or Eq. 57), we need to introduce one more independent variable e iΦ to define {q n , p n }. Further, this phase is necessary to introduce q 1 and p 1 variables, because 1|Ω = cos θ1 2 in Eq. 18 is purely real. Using the coefficients defined in Eq. 46, the expectation value of the spin operator (Eq. 23) is expressed as