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In this paper, we show a direct method of deriving the Peres-Horodeckicriterion for the two-qubit states from the Hill-Wootters formula for theentanglement of formation. Although the Peres-Horodecki criterion and theHill-Wootters formula are established results in the field of quantuminformation theory, they are proved independently and connections between themare not discussed precisely. In this paper, we clarify these connections.First, we replace the original Peres-Horodecki criterion with its anotherequivalent statement found by Augusiak et al. Second, we obtain an analyticalform of the concurrence of an arbitrary two-qubit state $\rho$, using Ferrari'smethod to solve a quartic equation for eigenvalues $\rho\tilde{\rho}$. Finally,with the above preparations, we accomplish the direct derivation of thePeres-Horodecki criterion from the Hill-Wootters formula.

Direct derivation of the Peres-Horodecki criterion for the two-qubit states from the Hill-Wootters formula for the entanglement of formation