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arXiv:1308.5488

Analytically computable tangle for three-qubit mixed states

Hiroyasu Tajima

Published Aug 26, 2013• 0 citations
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Abstract

We present a new tripartite entanglement measure for three-qubit mixed states. The new measure $t_{\mathrm{r}}(ρ)$, which we refer to as the r-tangle, is given as a kind of the tangle, but has a feature which the tangle does not have; if we can derive an analytical form of $ t_{\mathrm{r}}(ρ)$ for a three-qubit mixed state $ρ$, we can also derive $t_{\mathrm{r}}(ρ')$ analytically for any states $ρ'$ which are SLOCC-equivalent to the state $ρ$. The concurrence of two-qubit states also satisfies the feature, but the tangle does not. These facts imply that the r-tangle $t_{\mathrm{r}}$ is the appropriate three-partite counterpart of the concurrence. We also derive an analytical form of the r-tangle $t_{\mathrm{r}}$ for mixtures of a generalized GHZ state and a generalized W state, and hence for all states which are SLOCC-equivalent to them.

DOI
10.48550/arxiv.1308.5488

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We present a new tripartite entanglement measure for three-qubit mixed states. The new measure tr(ρ)t_{\mathrm{r}}(ρ)tr​(ρ), which we refer to as the r-tangle, is given as a kind of the tangle, but has a feature which the tangle does not have; if we can derive an analytical form of tr(ρ) t_{\mathrm{r}}(ρ)tr​(ρ) for a three-qubit mixed state ρρρ, we can also derive tr(ρ′)t_{\mathrm{r}}(ρ')tr​(ρ′) analytically for any states ρ′ρ'ρ′ which are SLOCC-equivalent to the state ρρρ. The concurrence of two-qubit states also satisfies the feature, but the tangle does not. These facts imply that the r-tangle trt_{\mathrm{r}}tr​ is the appropriate three-partite counterpart of the concurrence. We also derive an analytical form of the r-tangle trt_{\mathrm{r}}tr​ for mixtures of a generalized GHZ state and a generalized W state, and hence for all states which are SLOCC-equivalent to them.
This study presents an analytical investigation of entanglement quantification in three-qubit pure states through the Acín canonical representation, which serves as a generalization of the two-qubit Schmidt decomposition. Driven by the intricacies of multipartite entanglement and the shortcomings of current measures, we employ a global concurrence measure derived from the generalized concurrence of various bipartitions within the system. The characteristics of this measure are explored both analytically and numerically across a range of SLOCC entanglement classes, such as product, biseparable, GHZ-type, and W-type states. When compared to the three-tangle and tripartite negativity, our findings indicate that this measure yields complementary insights into multipartite correlations. The outcomes underscore the significance of this measure in elucidating the structure and distribution of entanglement within three-qubit quantum systems.
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.
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.
We construct an important missing piece in the entanglement theory of pure three-qubit states, which is a polynomial measure of W-entanglement. This works in parallel to the three-tangle, which is a polynomial measure of GHZ-entanglement, and to the bipartite concurrence, which is a polynomial measure of bipartite entanglement. We also show that these entanglement measures are ordered, the bipartite measure is larger than the W measure, which is larger than the GHZ measure. It is meaningful then to consider these three types of three-qubit entanglement, which are also ordered, bipartite is weaker than W, which is weaker than GHZ, in parallel to the order of the three kinds of equivalence classes of entangled three-qubit states.
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.