  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.
