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Journal of Modern Optics

Quantum entanglement criteria

Aini Syahida Sumairi, Nor Hazmin Sabri, C. H. Raymond Ooi

Published Apr 1, 2013• 10 citations
Publisher

Abstract

We review the main criteria used to detect entanglement in quantum systems. The main properties of each criteria are summarized depending on whether the criteria provides a sufficient or necessary condition, whether it involves density matrix or operators, or if the criteria is phase sensitive. We show that several criteria have much in common and they could be related mathematically. We also discuss the features of entanglement which are useful in quantum information technology.

DOI
10.1080/09500340.2013.796016

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We review the main criteria used to detect entanglement in quantum systems. The main properties of each criteria are summarized depending on whether the criteria provides a sufficient or necessary condition, whether it involves density matrix or operators, or if the criteria is phase sensitive. We show that several criteria have much in common and they could be related mathematically. We also discuss the features of entanglement which are useful in quantum information technology.
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We introduce a class of inequalities based on low-order correlations of operators to detect entanglement in bipartite systems. The operators may either be Hermitian or non-Hermitian and are applicable to any physical system or class of states. Testing the criteria on example systems reveals that they outperform other common correlation-based criteria, such as those by Duan-Giedke-Cirac-Zoller and Hillery-Zubairy. One unusual feature of the criteria is that the correlations include the density matrix itself, which is related to the purity of the system. We discuss how such a term could be measured in relation to the criteria.
We develop a method to construct entanglement criteria for arbitrary multipartite systems of discrete or continuous variables and hybrid combinations of both. While any set of local operators generates a sufficient condition for entanglement of arbitrary quantum states, a suitable set leads to a necessary and sufficient criterion for pure states. The criteria are readily implementable with existing technology and reveal entanglement that remains undetected by the respective state-of-the-art methods for discrete and continuous variables.
Entanglement plays a key role in quantum information. The separability problem arises: how to prove that a given quantum state is entangled? Covariance-matrix-based entanglement criteria are the most important and widely used approach to this problem in continuous variable systems. Here, the separability problem is systematically studied in the more experimentally realistic setting that the questioned state is only partly known. A simple general formalism is proposed for this variant of the separability problem. It builds relations between certain classes of criteria and bounds the power of these criteria. In particular, it shows the equivalence of many seemingly different covariance-matrix-based criteria; it offers guidance on the study of and reveals the inherent limitations of covariance-matrix-based criteria. Also, a Shannon entropic criterion is proposed for m ⊕ n-mode states. It unifies and improves many known covariance-matrix-based criteria and Shannon entropic criteria; it is necessary and sufficient for 1 ⊕ n-mode Gaussian states and m ⊕ n-mode bisymmetric Gaussian states.
Abstract Quantum entanglement plays a key role in quantum computation and quantum information processing. It is of great significance to find efficient and experimentally friend separability criteria to detect entanglement. In this paper, we firstly propose two easily used entanglement criteria based on matrix moments. The first entanglement criterion only uses the first two realignment moments of a density matrix. The second entanglement criterion is based on the moments related to the partially transposed matrix. By detailed examples we illustrate the effectiveness of these criteria in detecting entanglement. Moreover, we provide an experimentally measurable lower bound of concurrence based on these moments. Finally, we present both bipartite and genuine tripartite entanglement measures based on the moments of the reduced states. By detailed examples, we show that our entanglement measures characterize the quantum entanglement in a more fine ways than the existing measures.