Detecting qubit entanglement : an alternative to the PPT test

Abstract

We propose a Partial Lorentz Transformation (PLT) test for detecting entanglement in a two qubit system. One can expand the density matrix of a two qubit system in terms of a tensor product of $(\mathbb{I},\vec{\sigma })$. The matrix $A$ of the coefficients that appears in such an expansion can be "squared" to form a $4\times 4$ matrix $B$. It can be shown that the eigenvalues ${\lambda }_0,{\lambda }_1,{\lambda }_2,{\lambda }_3$ of $B$ are positive. With the choice of ${\lambda }_0$ as the dominant eigenvalue, the separable states satisfy $\sqrt{{\lambda }_1}+\sqrt{{\lambda }_2}+\sqrt{{\lambda }_3}\le \sqrt{{\lambda }_0}$. Violation of this inequality is a test of entanglement. Thus, this condition is both necessary and sufficient and serves as an alternative to the celebrated Positive Partial Transpose (PPT) test for entanglement detection. We illustrate this test by considering some explicit examples.