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

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