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arXiv:0711.4477 · New Journal of Physics

Three-tangle for mixtures of generalized GHZ and generalized W states

C Eltschka, A Osterloh, J Siewert, A Uhlmann

Published Apr 10, 2008• 83 citations
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Abstract

We give a complete solution for the three-tangle of mixed three-qubit states composed of a generalized Greenberger–Horne–Zeilinger (GHZ) state, a |000⟩+ b |111⟩, and a generalized W state, c |001⟩+ d |010⟩+ f |100⟩. Using the methods introduced by Lohmayer et al (2006 Phys. Rev. Lett. 97 260502), we provide explicit expressions for the mixed-state three-tangle and the corresponding optimal decompositions for this more general case. Moreover, as a special case, we obtain a general solution for a family of states consisting of a generalized GHZ state and an orthogonal product state.

DOI
10.1088/1367-2630/10/4/043014

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We give a complete solution for the three-tangle of mixed three-qubit states composed of a generalized Greenberger–Horne–Zeilinger (GHZ) state, a |000⟩+ b |111⟩, and a generalized W state, c |001⟩+ d |010⟩+ f |100⟩. Using the methods introduced by Lohmayer et al (2006 Phys. Rev. Lett. 97 260502), we provide explicit expressions for the mixed-state three-tangle and the corresponding optimal decompositions for this more general case. Moreover, as a special case, we obtain a general solution for a family of states consisting of a generalized GHZ state and an orthogonal product state.
We give a complete solution for the three-tangle of mixed three-qubit states composed of a generalized Greenberger-Horne-Zeilinger (GHZ) state, a vertical bar 000 > + b vertical bar 111 >, and a generalized W state, c vertical bar 001 > + d vertical bar 010 > + f vertical bar 100 >. Using the methods introduced by Lohmayer et al (2006 Phys. Rev. Lett. 97 260502), we provide explicit expressions for the mixed-state three-tangle and the corresponding optimal decompositions for this more general case. Moreover, as a special case, we obtain a general solution for a family of states consisting of a generalized GHZ state and an orthogonal product state.
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.
Three-tangle for a rank-3 mixture composed of Greenberger-Horne-Zeilinger, WWW, and flipped-WWW states is analytically calculated. The optimal decompositions in the full range of parameter space are constructed by making use of the convex-roof extension. We also provide an analytical technique, which determines whether or not an arbitrary rank-3 state has vanishing three-tangle. This technique is developed by making use of the Bloch sphere S8{S}^{8}S8 of the qutrit system. The Coffman-Kundu-Wootters inequality is discussed by computing one-tangle and concurrences. It is shown that the one-tangle is always larger than the sum of squared concurrences and three-tangle. The physical implication of three-tangle is briefly discussed.
We provide a complete analysis of mixed three-qubit states composed of a Greenberger-Horne-Zeilinger state and a W state orthogonal to the former. We present optimal decompositions and convex roofs for the three-tangle. Further, we provide an analytical method to decide whether or not an arbitrary rank-2 state of three qubits has vanishing three-tangle. These results highlight intriguing differences compared to the properties of two-qubit mixed states, and may serve as a quantitative reference for future studies of entanglement in multipartite mixed states. By studying the Coffman-Kundu-Wootters inequality we find that, while the amounts of inequivalent entanglement types strictly add up for pure states, this "monogamy" can be lifted for mixed states by virtue of vanishing tangle measures.
Some mixed states composed of only Greenberger-Horne-Zeilinger (GHZ) states can be expressed in terms of only WWW states. This fact implies that such states have vanishing three-tangle. One of such rank-3 states, ${\ensuremath{\Pi}}_{GHZ}$, is explicitly presented in this Rapid Communication. These results are used to compute analytically the three-tangle of a rank-4 mixed state $\ensuremath{\sigma}$ composed of four GHZ states. This analysis with considering Bloch sphere S16{S}^{16}S16 of d=4d=4d=4 qudit system allows us to derive the hyperpolyhedron. It is shown that the states in this hyperpolyhedron have vanishing three-tangle. Computing the one-tangles for ${\ensuremath{\Pi}}_{GHZ}$ and $\ensuremath{\sigma}$, we prove the monogamy inequality explicitly. Making use of the fact that the three-tangle of ${\ensuremath{\Pi}}_{GHZ}$ is zero, we try to explain why the WWW class in the whole mixed states is not of measure zero contrary to the case of pure states.
Recently, a new type of symmetry for three-qubit quantum states was introduced, the so-called Greenberger-Horne-Zeilinger (GHZ) symmetry. It includes the operations which leave the three-qubit standard GHZ state unchanged. This symmetry is powerful as it yields families of mixed states that are, on the one hand, complex enough from the physics point of view and, on the other hand, simple enough mathematically so that their properties can be characterized analytically. We show that by using the properties of GHZ-symmetric states it is straightforward to derive optimal witnesses for three-qubit entanglement.