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

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.

14 pages, latex2e; v2: some equations in the appendix are modified

References in corpus (3)

• Quantum entanglement
• Universal observable detecting all two-qubit entanglement and determinant based separability tests
• Manipulating sudden death of entanglement of two-qubit X-states in thermal reservoirs

Cited by in corpus (1)

• Moment-Based Evidence for Simple Rational-Valued Hilbert-Schmidt Generic 2 x 2 Separability Probabilities