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A density operator is a pure state exactly when its matrix rank is one. Within the admissible interval from part (a), one of $(1+\alpha )/2$ and $(1-\alpha )/2$ must vanish. Therefore the pure-state parameters and vectors arewhere the Bell states are $|{\Phi }^{\pm }⟩=(|00⟩\pm |11⟩)/\sqrt{2}$. Equivalently, the purity of a density operator is $\mathrm{Tr}{\rho }_{AB}^2=(1+{\alpha }^2)/2$, reaching one exactly at these two endpoints.

The eigenvalues found in part (ii) give the purity of a density operatorThe displayed truncation lies between zero and one when $|\lambda |\le 2\sqrt{2}{\omega }^2$. Perturbation theory requires the stronger condition $|\lambda |\ll {\omega }^2$; the exact coupled oscillator is stable only for $|\lambda |<{\omega }^2$. The reduction in purity records entanglement between $A$ and $B$.