  We map neutrinos to qubit and qutrit states of quantum information theory by
constructing the Poincar\'e sphere using SU(2) Pauli matrices and SU(3)
Gell-Mann matrices, respectively. The construction of the Poincar\'e sphere in
the two-qubit system enables us to construct the Bloch matrix, which yields
valuable symmetries in the Bloch vector space of two neutrino systems. By
identifying neutrinos with qutrits, we calculate the measures of qutrit
entanglement for neutrinos. We use SU(3) Gell-Mann matrices tensor products to
construct the Poincar\'e sphere of two qutrits neutrino systems. The comparison
between the entanglement measures of bipartite qubits and bipartite qutrits in
the two neutrino system are shown. The result warrants a study of two qutrits
entanglement in the three neutrino system.
