Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0606071 · Physical Review Letters

Entangled Three-Qubit States without Concurrence and Three-Tangle

Robert Lohmayer, Andreas Osterloh, Jens Siewert, Armin Uhlmann

Published Dec 29, 2006• 187 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.97.260502

Related papers

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

Eylee Jung, Mi-Ra Hwang, DaeKil Park +1 more

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.

arXiv:0810.5403•Physical Review A•Feb 11, 2009•48 citations

Entangled multi-qubit states without higher-tangle

Heng Fan, Yong-Cheng Ou, Vwani Roychowdhury

We analyze mixed multi-qubit states composed of a W class state and a product state with all qubit in |0>. We find the optimal pure state decomposition and convex roofs for higher-tangle with bipartite partition between one qubit and the rest qubits for those mixed states. The optimality of the decomposition is ensured by the Coffman-Kundu-Wootters (CKW) inequality which describes the monogamy of quantum entanglement. The generalized monogamy inequality is found to be true for W class states with arbitrary partitions between one qubit and multi-qubit.

arXiv:0707.1578•Jul 11, 2007•3 citations

Three-tangle does not properly quantify tripartite entanglement for Greenberger-Horne-Zeilinger-type states

Eylee Jung, DaeKil Park, Jin-Woo Son

Some mixed states composed of only Greenberger-Horne-Zeilinger (GHZ) states can be expressed in terms of only $W$ 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 ${S}^{16}$ of $d=4$ 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 $W$ class in the whole mixed states is not of measure zero contrary to the case of pure states.

arXiv:0901.2620•Physical Review A•Jul 8, 2009•29 citations

Three-tangle of assistance

Xi-Jun Ren, Zhen Wang, Xingxiang Zhou +2 more

In this report, we calculate the three tangle of assistance and give the corresponding best possible pure state decomposition of rank-two mixed states for the three-qubit Greenberger-Horne-Zeilinger (GHZ) state and $W$ state. As the three tangle can be transformed into bipartite concurrence with assistance, we achieve a lower bound of the assisted concurrence with local operations and classical communication (LOCC) constraints via these pure state decompositions. It is smaller than the total assisted concurrence, which shows that local assistance alone is not sufficient for maximizing bipartite entanglement. The increase of three tangle is attributed to the genuine multipartite entanglement in the purified state. A genuine four-qubit entanglement measure is calculated for the purified four-qubit state and compared to the three tangle of assistance.

Physical Review A•Apr 13, 2010•3 citations

Monogamy inequality and residual entanglement of three qubits under decoherence

Thiago Rodrigues de Oliveira

Exploring an analytical expression for the convex roof of the pure state squared concurrence for rank 2 mixed states the entanglement of a system of three particles under decoherence is studied, using the monogamy inequality for mixed states and the residual entanglement obtained from it. The monogamy inequality is investigated both for the concurrence and the negativity in the case of local independent phase damping channel acting on generalized Greenberger-Horne-Zeilinger states of three particles and the local independent amplitude damping channel acting on generalized $\ensuremath{\mid}\mathrm{W}⟩=\ensuremath{\mid}001⟩+\ensuremath{\mid}010⟩+\ensuremath{\mid}100⟩$ state of three particles. It is shown that the bipartite entanglement between one qubit and the rest has a qualitative similar behavior to the entanglement between individual qubits, and that the residual entanglement in terms of the negativity cannot be a good entanglement measure for mixed states, since it can increase under local decoherence.

arXiv:0903.0019•Physical Review A•Aug 24, 2009•9 citations

Monogamy inequality in terms of negativity for three-qubit states

Yong-Cheng Ou, Heng Fan

We propose an entanglement measure to quantify three-qubit entanglement in terms of negativity. A monogamy inequality analogous to the Coffman-Kundu-Wootters inequality is established. This consequently leads to a definition of residual entanglement, which is referred to as the three-$\ensuremath{\pi}$ in order to distinguish it from the three-tangle. The three-$\ensuremath{\pi}$ is proved to be a natural entanglement measure. By contrast to the three-tangle, it is shown that the three-$\ensuremath{\pi}$ always gives greater than zero values for pure states belonging to the $W$ and Greenberger-Horne-Zeilinger classes, implying that three-way entanglement always exists for them; the three-tangle generally underestimates the three-way entanglement of a given system. This investigation will offer an alternative tool to understand genuine multipartite entanglement.

arXiv:quant-ph/0702127•Physical Review A•Jun 8, 2007•280 citations
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.
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 analyze mixed multi-qubit states composed of a W class state and a product state with all qubit in |0>. We find the optimal pure state decomposition and convex roofs for higher-tangle with bipartite partition between one qubit and the rest qubits for those mixed states. The optimality of the decomposition is ensured by the Coffman-Kundu-Wootters (CKW) inequality which describes the monogamy of quantum entanglement. The generalized monogamy inequality is found to be true for W class states with arbitrary partitions between one qubit and multi-qubit.
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.
In this report, we calculate the three tangle of assistance and give the corresponding best possible pure state decomposition of rank-two mixed states for the three-qubit Greenberger-Horne-Zeilinger (GHZ) state and WWW state. As the three tangle can be transformed into bipartite concurrence with assistance, we achieve a lower bound of the assisted concurrence with local operations and classical communication (LOCC) constraints via these pure state decompositions. It is smaller than the total assisted concurrence, which shows that local assistance alone is not sufficient for maximizing bipartite entanglement. The increase of three tangle is attributed to the genuine multipartite entanglement in the purified state. A genuine four-qubit entanglement measure is calculated for the purified four-qubit state and compared to the three tangle of assistance.
Exploring an analytical expression for the convex roof of the pure state squared concurrence for rank 2 mixed states the entanglement of a system of three particles under decoherence is studied, using the monogamy inequality for mixed states and the residual entanglement obtained from it. The monogamy inequality is investigated both for the concurrence and the negativity in the case of local independent phase damping channel acting on generalized Greenberger-Horne-Zeilinger states of three particles and the local independent amplitude damping channel acting on generalized $\ensuremath{\mid}\mathrm{W}⟩=\ensuremath{\mid}001⟩+\ensuremath{\mid}010⟩+\ensuremath{\mid}100⟩$ state of three particles. It is shown that the bipartite entanglement between one qubit and the rest has a qualitative similar behavior to the entanglement between individual qubits, and that the residual entanglement in terms of the negativity cannot be a good entanglement measure for mixed states, since it can increase under local decoherence.
We propose an entanglement measure to quantify three-qubit entanglement in terms of negativity. A monogamy inequality analogous to the Coffman-Kundu-Wootters inequality is established. This consequently leads to a definition of residual entanglement, which is referred to as the three-$\ensuremath{\pi}$ in order to distinguish it from the three-tangle. The three-$\ensuremath{\pi}$ is proved to be a natural entanglement measure. By contrast to the three-tangle, it is shown that the three-$\ensuremath{\pi}$ always gives greater than zero values for pure states belonging to the WWW and Greenberger-Horne-Zeilinger classes, implying that three-way entanglement always exists for them; the three-tangle generally underestimates the three-way entanglement of a given system. This investigation will offer an alternative tool to understand genuine multipartite entanglement.