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arXiv:1204.5512 · Physical Review A

Genuine entanglement of four-qubit cluster diagonal states

Xiaoyu Chen, Ping Yu, Li-zhen Jiang, Ming Tian

Published Jan 22, 2013• 8 citations
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Abstract

We reduce the necessary and sufficient biseparable conditions of four-qubit cluster diagonal states to concise forms. Only 4 out of the 15 parameters are proved to be relevant in specifying the genuine entanglement of the states. Using the relative entropy of entanglement as entanglement measure, we analytically find the genuine entanglement of all the four-qubit cluster diagonal states. The formulas for the genuine entanglement are of five kinds, for seven different parameter regions of genuinely entangled states.

DOI
10.1103/physreva.87.012322

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We reduce the necessary and sufficient biseparable conditions of four-qubit cluster diagonal states to concise forms. Only 4 out of the 15 parameters are proved to be relevant in specifying the genuine entanglement of the states. Using the relative entropy of entanglement as entanglement measure, we analytically find the genuine entanglement of all the four-qubit cluster diagonal states. The formulas for the genuine entanglement are of five kinds, for seven different parameter regions of genuinely entangled states.
The characterization of genuine multiparticle entanglement is important for entanglement theory as well as experimental studies related to quantum-information theory. Here, we completely characterize genuine multiparticle entanglement for four-qubit states diagonal in the cluster-state basis. In addition, we give a complete characterization of multiparticle entanglement for all five-qubit graph states mixed with white noise, for states diagonal in the basis corresponding to the five-qubit Y\mathsf{Y}Y-shaped graph, and for a family of graph states with an arbitrary number of qubits.
We analytically prove the necessary and sufficient criterion for the full separability of three-qubit Greenberger-Horne-Zeilinger (GHZ) diagonal states. The corresponding entanglement is exactly calculable for some GHZ diagonal states and is tractable for the others using the relative entropy of entanglement. We show that the biseparable criterion and the genuine entanglement are determined only by the biggest GHZ diagonal element regardless of all the other smaller diagonal elements. We have completely solved the entanglement problems of three-qubit GHZ diagonal states.
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Abstract Quantifying genuine entanglement is a key task in quantum information theory. We study the quantification of genuine multipartite entanglement for four-qubit systems. Based on the concurrence of nine different classes of four-qubit states, with each class being closed under stochastic local operation and classical communication, we construct a concurrence tetrahedron. Proper genuine four-qubit entanglement measure is presented by using the volume of the concurrence tetrahedron. For non genuine entangled pure states, the four-qubit entanglement measure classifies the bi-separable entanglement. We show that the concurrence tetrahedron based measure of genuine four-qubit entanglement is not equivalent to the genuine four-partite entanglement concurrence. We illustrate the advantages of the concurrence tetrahedron by detailed examples.
Based on quantitative complementarity relations (QCRs), we analyze the multipartite correlations in four-qubit cluster-class states. It is proven analytically that the average multipartite correlation Ems{E}_{ms}Ems​ is entanglement monotone. Moreover, it is also shown that the mixed three-tangle is a correlation measure compatible with the QCRs in this kind of quantum state. More arrestingly, with the aid of the QCRs, a set of hierarchy entanglement measures is obtained rigorously in the present system.