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arXiv:quant-ph/0702127 · Physical Review A

Monogamy inequality in terms of negativity for three-qubit states

Yong-Cheng Ou, Heng Fan

Published Jun 8, 2007• 280 citations
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Abstract

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.

DOI
10.1103/physreva.75.062308

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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-\ensuremathπ\ensuremath{\pi}\ensuremathπ in order to distinguish it from the three-tangle. The three-\ensuremathπ\ensuremath{\pi}\ensuremathπ is proved to be a natural entanglement measure. By contrast to the three-tangle, it is shown that the three-\ensuremathπ\ensuremath{\pi}\ensuremathπ 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.
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.
Bipartite quantum entanglement for qutrits and higher-dimensional objects is considered. We analyze the possibility of violation of monogamy inequality, introduced by Coffman, Kundu, and Wootters, for some systems composed of such objects. An explicit counterexample with a three-qutrit totally antisymmetric state is presented. Since three-tangle has been confirmed to be a natural measure of entanglement for qubit systems, our result shows that the three-tangle is no longer a legitimate measure of entanglement for states with three qutrits or higher-dimensional objects.
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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 obtain, analytically, the global negativity, partial KKK-way negativities (K=2,3)(K=2,3)(K=2,3), Wootter's tangle, and three-tangle for the generic three-qubit canonical state. It is found that the product of global negativity and partial three-way negativity is equal to the three-tangle, while the partial two-way negativity is related to the tangle of qubit pairs. We also calculate similar quantities for the state canonical to a single-parameter (0<q<1)(0<q<1)(0<q<1) pure state which is a linear combination of a Greenberger-Horne-Zeilinger state and a WWW state. In this case, for q=0.62685q=0.626\phantom{\rule{0.2em}{0ex}}85q=0.62685, the state has zero three-tangle and zero three-way negativity, having only WWW-like entanglement. The difference between the product of global and partial three-way negativities and the three-tangle for a given state is a quantitative measure of two-qubit coherences transformed by unitary transformations on the canonical state into three-qubit coherences. The global negativity and partial KKK-way negativities, obtained by selective partial transpositions on multiqubit state operators, satisfy inequalities which for three qubits are equivalent to the Coffman-Kundu-Wootter inequality.
We establish two new inequalities, the weighted strong monogamy (WSM) and the maximum residual strong monogamy (MRSM), which sharpen the generalized Coffman-Kundu-Wootters inequity for multiqubit states. The WSM inequality distinguishes itself from the strong monogamy (SM) conjecture [Phys. Rev. Lett. 113, 110501 (2014)] by using coefficients rather than exponents to modulate the weight allocated to various m-partite contributions. In contrast, the MRSM inequality is formulated using only the maximum m-partite entanglement. We find that the residual entanglement of the MRSM inequality can effectively distinguish the separable states. We also compare the tightness of various SM inequalities and provide examples using a four-qubit mixed state and a five-qubit pure state to illustrate the MRSM inequality. These examples characterize the trade-off relations among entanglement components involving varying numbers of qubits. Our results provide a rigorous framework to characterize and quantify the monogamy of multipartite entanglement.