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arXiv:quant-ph/0605196 · Physical Review A

Classification of GHZ-type, W -type, and GHZ- W -type multiqubit entanglement

Lin Chen, Yi-Xin Chen

Published Dec 18, 2006• 36 citations
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Abstract

We propose the concept of SLOCC-equivalent basis in the multiqubit space. In particular, two special SEBs, the Greenberger-Horne-Zeilinger--(GHZ-) type and the $W$-type basis are introduced. They can make up a more general family of multiqubit states, the GHZ-$W$-type states, which is a useful kind of entanglement for quantum teleportation and error correction. We completely characterize the property of this type of state, and mainly classify the GHZ-type and the $W$-type states in a regular way, which is related to the enumerative combinatorics. Many concrete examples are given to exhibit our method of classification. We also propose the condition on which two GHZ-$W$-type states are interconvertible with probability 1.

DOI
10.1103/physreva.74.062310

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We propose the concept of SLOCC-equivalent basis in the multiqubit space. In particular, two special SEBs, the Greenberger-Horne-Zeilinger--(GHZ-) type and the WWW-type basis are introduced. They can make up a more general family of multiqubit states, the GHZ-WWW-type states, which is a useful kind of entanglement for quantum teleportation and error correction. We completely characterize the property of this type of state, and mainly classify the GHZ-type and the WWW-type states in a regular way, which is related to the enumerative combinatorics. Many concrete examples are given to exhibit our method of classification. We also propose the condition on which two GHZ-WWW-type states are interconvertible with probability 1.
In this paper, we study the entanglement classification of the two-dimensional subspace spanned by nnn-qubit Greenberger-Horne-Zeilinger (GHZ) and WWW states under stochastic local operations and classical communication (SLOCC). We show that for $n\ensuremath{\ge}4$ such a system can have infinite SLOCC inequivalent classes. Specifically, two states, ${|\mathrm{GHZ}\ensuremath{\rangle}}_{n}+x{|W\ensuremath{\rangle}}_{n}$ and ${|\mathrm{GHZ}\ensuremath{\rangle}}_{n}+y{|W\ensuremath{\rangle}}_{n}$, from the subspace spanned by nnn-qubit ${|W\ensuremath{\rangle}}_{n}$ and ${|\mathrm{GHZ}\ensuremath{\rangle}}_{n}$ are SLOCC equivalent if and only if (xy)n=1{(\frac{x}{y})}^{n}=1(yx​)n=1.
The hierarchy of nonlocality and entanglement in multipartite systems is one of the fundamental problems in quantum physics. We study this topic in three-qubit systems considering the entanglement classification of stochastic local operations and classical communication (SLOCC). The equivalence under SLOCC divides three-qubit states into separable, biseparable, WWW, and Greenberger-Horne-Zeilinger (GHZ) classes. The WWW and GHZ are two subclasses of genuine tripartite entanglement. We adopt the family of GHZ-symmetric states as a research subject, which share the symmetries of the GHZ state and have a complete characterization of SLOCC classes. In the biseparable region (with bipartite entanglement), there exist GHZ-symmetric states that are found to be fully local. In addition, there are bilocal states in both the WWW and GHZ classes. That is, neither of the subclasses of genuine tripartite entanglement can ensure genuinely tripartite nonlocality.
Similar to the three-qubit Greenberger-Horne-Zeilinger (GHZ) symmetry we explore the four-qubit GHZ symmetry group and its subgroup, the restricted GHZ symmetry group. While the set of symmetric states under the whole group transformation is represented by three real parameters, the set of symmetric states under the subgroup transformation is represented by two real parameters. After comparing the symmetric states for whole group and subgroup, the entanglement is examined for the latter set. It is shown that the set has only two stochastic local operations and classical communication classes, Labc2{L}_{ab{c}_{2}}Labc2​​ and Gabcd{G}_{abcd}Gabcd​. Extension to the multiqubit system is briefly discussed.
The first characterization of mixed-state entanglement was achieved for two-qubit states in Werner's seminal work [Phys. Rev. A 40, 4277 (1989)]. A physically important extension concerns mixtures of a pure entangled state [such as the Greenberger-Horne-Zeilinger (GHZ) state] and the unpolarized state. These mixed states serve as benchmark for the robustness of multipartite entanglement. They share the symmetries of the GHZ state. We call such states GHZ symmetric. Here we give a complete description of the entanglement in the family of three-qubit GHZ-symmetric states and, in particular, of the three-qubit generalized Werner states. Our method relies on the appropriate parametrization of the states and on the invariance of entanglement properties under general local operations. An application is the definition of a symmetrization witness for the entanglement class of arbitrary three-qubit states.
We study the interconversion of multipartite symmetric NNN-qubit states under stochastic local operations and classical communication (SLOCC). We demonstrate that if two symmetric states can be connected with a nonsymmetric invertible local operation (ILO), then they belong necessarily to the separable, W, or Greenberger-Horne-Zeilinger (GHZ) entanglement class, establishing a practical method of discriminating subsets of entanglement classes. Furthermore, we prove that there always exists a symmetric ILO connecting any pair of symmetric NNN-qubit states equivalent under SLOCC, simplifying the requirements for experimental implementations of local interconversion of those states.
We investigate genuine multiqubit nonlocality in families of entangled 3- and 4-qubit pure states by analyzing a Bell-type inequality that is violated only if all qubits are nonlocally correlated. We present detailed numerical studies of the relationship between entanglement and violation of the Svetlichny Bell-type inequality in an experimentally accessible set of 3-qubit pure states, and identify the special nonlocality property of the maximal slice states in the space of all 3-qubit pure states. We also analyze nonlocal correlations in 3-qubit generalized Greenberger–Horne–Zeilinger (GHZ) states and extend our analysis to the case of 4-qubit generalized GHZ states. We show that like the 3-qubit case, some 4-qubit generalized GHZ states do not violate a Bell inequality that tests for genuine 4-qubit nonlocality. Furthermore, the location of the boundary between the states that do violate the inequality and those that do not is the same for the 3- and 4-qubit generalized GHZ states.