Detecting qubit entanglement : an alternative to the PPT test

TL;DR

This paper introduces a new entanglement detection method for two-qubit systems using a Partial Lorentz Transformation test, which is both necessary and sufficient, offering an alternative to the PPT test with explicit example demonstrations.

Contribution

The paper presents a novel entanglement detection criterion based on eigenvalues of a transformed matrix, providing a necessary and sufficient condition as an alternative to the PPT test.

Findings

The inequality involving eigenvalues effectively detects entanglement.

The method is both necessary and sufficient for two-qubit states.

Explicit examples demonstrate the test's applicability.

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 $4\times4$ 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 $\sqrt{\lambda_1}+\sqrt{\lambda_2}+\sqrt{\lambda_3}\leq \sqrt{\lambda_0}$. 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.