  We develop a measurement operator formalism to handle quantum nondemolition
(QND) measurement induced entanglement generation between two atomic gases. We
first derive how the QND entangling scheme reduces to a positive
operator-valued measure, and consider its limiting case when it can be used to
construct a projection operator that collapses the state to a total spin
projection state. We then analyze how a stroboscopic sequence of such
projections made in the x and z basis evolves the initial wave function. Such a
sequence of QND projections can enhance the entanglement between the atomic
ensembles and makes the state converge towards a highly entangled state. We
show several mathematical identities which greatly simplify the state evolution
in the projection sequence and allow one to derive the exact state in a highly
efficient manner. Our formalism does not use the Holstein-Primakoff
approximation as is conventionally done, and treats the spins of the atomic
gases in an exact way.
