Past exam of the laser science and quantum information processing course of the University of Oxford / 2021 / 6 / d

Suppose the conditional atomic pure state were a product stateComparing its four probability amplitudes with $(|eg⟩+|ge⟩)/\sqrt{2}$ givesThe last two equations make all four factors nonzero, contradicting either of the first two. Equivalently, the coefficient matrix has determinant $-1/2$, while a product state has a rank-one coefficient matrix and zero determinant. Thus the atoms are entangled; indeed each reduced density matrix is $I/2$, making the conditional state maximally entangled.