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arXiv:0709.3766 · Physical Review A

Entanglement detection beyond the computable cross-norm or realignment criterion

Chengjie Zhang, Yong-Sheng Zhang, Shun Zhang, Guang‐Can Guo

Published Jun 2, 2008• 93 citations
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Abstract

Separability problem, to decide whether a given state is entangled or not, is a fundamental problem in quantum information theory. We propose a powerful and computationally simple separability criterion, which allows us to detect the entanglement of many bound entangled states. The criterion is strictly stronger than the criterion based on Bloch representations, the computable cross-norm or realignment criterion and its optimal nonlinear entanglement witnesses. Furthermore, this criterion can be generalized to an analog of permutation separability criteria in the even-partite systems.

DOI
10.1103/physreva.77.060301

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Separability problem, to decide whether a given state is entangled or not, is a fundamental problem in quantum information theory. We propose a powerful and computationally simple separability criterion, which allows us to detect the entanglement of many bound entangled states. The criterion is strictly stronger than the criterion based on Bloch representations, the computable cross-norm or realignment criterion and its optimal nonlinear entanglement witnesses. Furthermore, this criterion can be generalized to an analog of permutation separability criteria in the even-partite systems.
Quantum properties, such as entanglement and coherence, are indispensable resources in various quantum information processing tasks. However, given a quantum state, it is not a simple question to identify whether it is entangled or not. We study the entanglement of quantum systems in arbitrary dimensions using the Bloch representation of density matrix. This approach enables us to consider quantum entanglement geometrically. We first analyze the effect of local Bloch vector on determining whether a quantum state is entangled or separable, and obtain an entanglement criterion related to local Bloch vector, which is stronger than the criterion that only considers the correlation matrix. This criterion is equivalent to the optimized computable cross norm criterion in [14]. Our proof is considered from the perspective of Bloch representation, and there is no need to know the matrix relignment in advance. This criterion also be extended to the problem of the separability of tripartite quantum states. We then combine the local filtering transformation with this entanglement criterion. After applying a suitable local filtering transformation, the quantum state has at most one nonzero Bloch vector, and examples are given to illustrate that the local filtering transformation can help detect some bound states where the original entanglement criterion fails. Moreover, for a finite-dimensional quantum system in which one party is a qubit, we give an analytical form of the combined entanglement criterion.
We present a framework for deciding whether a quantum state is separable or entangled using covariance matrices of locally measurable observables. This leads to the covariance matrix criterion as a general separability criterion. We demonstrate that this criterion allows us to detect many states where the familiar criterion of the positivity of the partial transpose fails. It turns out that a large number of criteria that have been proposed to complement the positive partial transpose criterion---the computable cross norm or realignment criterion, the criterion based on local uncertainty relations, criteria derived from extensions of the realignment map, and others---are in fact a corollary of the covariance matrix criterion.