  Inspired by the method that can deterministically generated the massive
entanglement through phase transitions, we study the ground state properties of
a spin-1 condensate mixture, under the premise that the heteronuclear
spin-exchange collision is taken into account. We developed a effective model
to analyze the binary-coupled two-level system and studied the ground state
phase transitions. Three representative quantum states with the same number
distribution are studied and distinguished through the number fluctuations. We
demonstrate that there will be the GreenbergerHorne-Zeilinger (GHZ) state in
the mixture if the the extra magnetic field is specifically selected or
adiabatically adjusted. One advantage of preparing entangled states in mixtures
is that we only need to adjust the external magnetic field, instead of
considering the microwaves-magnetic cooperation. Finally we estimate the
feasibility of experimentally generating the heteronuclear many-body
entanglement in the alkali-metal atomic mixture.
