 Solution

The average of $z_j{z}_k^{\ast }{z}_{j^{\prime }}^{\ast }{z}_{k^{\prime }}$ vanishes unless the exponent of every independent fourth root is balanced modulo four. The surviving index patterns are $j=k,j^{\prime }=k^{\prime }$ and $j=j^{\prime },k=k^{\prime }$. Their intersection $j=k=j^{\prime }=k^{\prime }$ has been counted twice. Hence, writingthe phase average isSolving for the projector onto the maximally entangled state gives