(a) The sketch of a two-player 2 × 2 quantum game. Here J and J† are two-qubits entangling gates. U_A and U_B are strategy operators carried out by players Alice and Bob, respectively. M_A and M_B are projective measurements in bases { |0〉, |1〉 }. Through unitary operations V_A and V_B, players can adjust their measurement basis according to their decision policies. (b) When |ψ〉_AB = |ψ〉_A ⊗ |ψ〉_B and J = J† = I, a two-player 2 × 2 quantum game is reduced to a classical game.