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

TL;DR

This paper establishes a direct analytical connection between the Peres-Horodecki criterion and the Hill-Wootters formula for two-qubit states, enhancing understanding of quantum entanglement criteria.

Contribution

It provides a direct derivation of the Peres-Horodecki criterion from the Hill-Wootters formula, linking two fundamental entanglement measures in quantum information theory.

Findings

Derived an analytical form of concurrence for arbitrary two-qubit states.

Connected the Peres-Horodecki criterion with the Hill-Wootters formula.

Clarified the relationship between two key entanglement criteria.

Abstract

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.