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arXiv:1712.06801

Detecting qubit entanglement : an alternative to the PPT test

Joseph Samuel, Kumar Shivam, Supurna Sinha

Published Dec 19, 2017• 0 citations
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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 $(\mathbb{I}, \vecσ)$. 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 $λ_0, λ_1, λ_2, λ_3$ of $B$ are positive. With the choice of $λ_0$ as the dominant eigenvalue, the separable states satisfy $\sqrt{λ_1}+\sqrt{λ_2}+\sqrt{λ_3}\leq \sqrt{λ_0}$. 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.

DOI
10.48550/arxiv.1712.06801

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We show how to design families of operational criteria that distinguish entangled from separable quantum states. The simplest of these tests corresponds to the well-known Peres-Horodecki positive partial transpose (PPT) criterion, and the more complicated tests are strictly stronger. The new criteria are tractable due to powerful computational and theoretical methods for the class of convex optimization problems known as semidefinite programs. We successfully applied the results to many low-dimensional states from the literature where the PPT test fails. As a by-product of the criteria, we provide an explicit construction of the corresponding entanglement witnesses.
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It is well known that the positive partial transpose (PPT) criterion for determining separability is an operational separability criterion. However, in a high-dimensional (>6) situation, this criterion is not sufficient. How can we judge whether an entangled state in any high-dimensional quantum system is PPT or not? Here we propose a linear algebra method for checking the positivity of the partial transpose of a state and present a set of entangled mixed states with non-positive partial transpose.
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