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We consider the effective spin Hamiltonian describing a mixture of twospecies of pseudo-spin-1/2 Bose gases with interspecies spin exchange. First weanalyze the stability of the fixed points of the corresponding classicaldynamics, of which the signature is found in quantum dynamics with adisentangled initial state. Focusing on the case without an external potential,we find all the ground states by taking into account quantum fluctuationsaround the classical ground state in each parameter regime. The nature ofentanglement and its relation with classical bifurcation is investigated. Whenthe total spins of the two species are unequal, the maximal entanglement at theparameter point of classical bifurcation is possessed by the excited statecorresponding to the classical fixed point which bifurcates, rather than by theground state.

Binary mixture of pseudospin-1/2 Bose gases with interspecies spin exchange: from classical fixed points and ground states to quantum ground states