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Zhongguo kexue. Wulixue Lixue Tianwenxue

Quantum separable criterion based on characteristic function

CHEN XiaoYu, JIANG LiZhen

Published Oct 18, 2017• 3 citations
Publisher

Abstract

Entanglement is the most important feature of quantum information science and it is closely related to the foundation of quantum mechanics. To determine a given bipartite quantum state to be entangled or not is by no means easy. We propose a separable criterion for bipartite quantum states based on their characteristic function or characteristic matrix. We use local filtering transform and local rotation transform to simplify the characteristic matrix, while keeping the positivity and entanglement property of the quantum state invariant. The development of Heisenberg-Weyl matrix basis makes the transforms possible. The local transforms greatly reduce the number of none-zero elements of characteristic matrix. The new criterion of separability then is constructed on the none-zero elements. We compare the entanglement detecting properties of our criterion with realignment criterion and modified correlation matrix criterion on $3\times~3$ bound entangled states. It is shown that our criterion is better than realignment criterion for most of the states.

DOI
10.1360/sspma2017-00203

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Entanglement is the most important feature of quantum information science and it is closely related to the foundation of quantum mechanics. To determine a given bipartite quantum state to be entangled or not is by no means easy. We propose a separable criterion for bipartite quantum states based on their characteristic function or characteristic matrix. We use local filtering transform and local rotation transform to simplify the characteristic matrix, while keeping the positivity and entanglement property of the quantum state invariant. The development of Heisenberg-Weyl matrix basis makes the transforms possible. The local transforms greatly reduce the number of none-zero elements of characteristic matrix. The new criterion of separability then is constructed on the none-zero elements. We compare the entanglement detecting properties of our criterion with realignment criterion and modified correlation matrix criterion on 3× 33\times~33× 3 bound entangled states. It is shown that our criterion is better than realignment criterion for most of the states.
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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.
Entanglement as a vital resource for information processing can be described by special properties of the quantum state. Using the well-known Weyl basis we propose a new Bloch decomposition of the quantum state and study its separability problem. This decomposition enables us to find an alternative characterization of the separability based on the correlation matrix. We show that the criterion is effective in detecting entanglement for the isotropic states, Bell-diagonal states and some PPT entangled states. We also use the Weyl operators to construct an detecting operator for quantum teleportation.
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.