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arXiv:0810.5403 · Physical Review A

Three-tangle for rank-three mixed states: Mixture of Greenberger-Horne-Zeilinger, W , and flipped- W states

Eylee Jung, Mi-Ra Hwang, DaeKil Park, Jin-Woo Son

Published Feb 11, 2009• 48 citations
arXivHTMLPDF

Abstract

Three-tangle for a rank-3 mixture composed of Greenberger-Horne-Zeilinger, $W$, and flipped-$W$ 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 ${S}^{8}$ 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.

DOI
10.1103/physreva.79.024306

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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.
An algorithm is proposed that serves to handle full-rank density matrices when coming from a lower-rank method to compute the convex roof. This is in order to calculate an upper bound for any polynomial SL-invariant multipartite entanglement measure EEE. This study exemplifies how this algorithm works based on a method for calculating convex roofs of rank-2 density matrices. It iteratively considers the decompositions of the density matrix into two states each, exploiting the knowledge for the rank-2 case. The algorithm is therefore quasiexact as far as the rank-2 case is concerned, and it also hints where it should include more states in the decomposition of the density matrix. Focusing on the measure of three-way entanglement of qubits (called three-tangle), I show the results the algorithm gives for two states, one of which is the Greenberger-Horne-Zeilinger--Werner (GHZ-WWW) state, for which the exact convex roof is known. It overestimates the three-tangle in the state, thereby giving insight into the optimal decomposition the GHZ-WWW state has. As a proof of principle, I have run the algorithm for the three-tangle on the transverse quantum Ising model. I give qualitative and quantitative arguments why the convex roof should be close to the upper bound found here.
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 quantification of multipartite entanglement remains a long-standing open challenge in quantum information theory with major implications for quantum foundations and quantum technologies. In the context of three-qubit pure states, the concurrence triangle underlies geometric interpretations for a variety of multipartite entanglement measures. In particular, the concurrence fill, which is proportional to the square root of the area of the concurrence triangle, has been shown to be a valid measure of genuine tripartite entanglement. As a result, there is now significant interest in extending the concurrence triangle and concurrence fill to three-qubit mixed states. In this work, we show that any rank-2 three-qubit mixed state can be visualized in terms of concurrence prisms associated with their pure state decompositions. Thus, the concurrence fill of the mixed state obtained through convex-roof extension is proportional to the total prism fill area, defined as the square root of the base area times the height, associated with the prisms minimized over all pure state decompositions. We then consider three-qubit mixed states that are incoherent mixtures of Greenberger-Horne-Zeilinger (GHZ) and W states and analytically perform their convex-roof extension to obtain a closed-form expression for the concurrence fill of these rank-2 mixtures. Finally, we apply the geometric picture of concurrence prisms to these mixtures and their pure state decomposition corresponding to minimum concurrence fill as a tool for visualization.