Dephased Bell-state mixture (${\rho }_{\alpha }=\frac{1+\alpha }{2}|{\Phi }^{+}⟩⟨{\Phi }^{+}|+\frac{1-\alpha }{2}|{\Phi }^{-}⟩⟨{\Phi }^{-}|$)

For real $-1\le \alpha \le 1$, this two-qubit density operator has equal $00$ and $11$ populations and off-diagonal coherences $\alpha /2$. It is a pure state exactly when $\alpha =\pm 1$, and a separable quantum state exactly when $\alpha =0$. Its partial transpose has eigenvalues $1/2,1/2,\alpha /2,-\alpha /2$, directly witnessing entanglement for every nonzero admissible coherence.