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arXiv:0806.0887 · Physical Review A

Partial K -way negativities and three-tangle for three-qubit states

S. Shelly Sharma, Naresh Kumar Sharma

Published Jul 28, 2008• 13 citations
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Abstract

We obtain, analytically, the global negativity, partial $K$-way negativities $(K=2,3)$, 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 $(0<q<1)$ pure state which is a linear combination of a Greenberger-Horne-Zeilinger state and a $W$ state. In this case, for $q=0.626\phantom{\rule{0.2em}{0ex}}85$, the state has zero three-tangle and zero three-way negativity, having only $W$-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 $K$-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.

DOI
10.1103/physreva.78.012113

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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 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 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.
We propose to quantify three-qubit entanglement using global negativity along with KKK-way negativities, where K=2K=2K=2 and 3. The principle underlying the definition of KKK-way negativity for pure and mixed states of NNN subsystems is a positive partial transpose sufficient condition. However, KKK-way partial transpose with respect to a subsystem is defined so as to shift the focus to KKK-way coherences instead of KKK subsystems of the composite system. A quantum state of a three-qubit system is characterized by the coherences measured by global, two-way, and three-way negativities. For a canonical state of three-qubit system, entanglement measures for genuine tripartite entanglement, W-like entanglement, and bipartite entanglement can be related to two-way and three-way negativities.
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.
A characterization of multipartite quantum states having NNN subsystems, based on negativities of matrices obtained by selective partial transposition of the state operator, is proposed. The KKK-way partial transpose with respect to a subsystem is constructed by imposing constraints involving the states of KKK subsystems of the multipartite composite system. The KKK-way negativity, defined as the negativity of KKK-way partial transpose, quantifies the KKK-way coherences of the composite system. For an NNN-partite system the fraction of KKK-way negativity $(2\ensuremath{\le}K\ensuremath{\le}N)$, contributing to global negativity, is obtained. The entanglement measures for a given state $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}$ are identified as the partial KKK-way negativities of the corresponding canonical state.
A classification of N-partite states, based on K-way negativities (K=2 to N), is proposed. The K-way partial transpose with respect to a subsystem is defined so as to shift the focus to K-way coherences instead of K subsystems of the composite system. For an N-partite system, the fraction of K-way negativity contributing to global negativity, is obtained. After minimizing K-way negativities through local unitary qubit rotations, a combined analysis of 2-way, 3-way and global negativities is shown to provide distinct measures of genuine tripartite, W-state like and bipartite entanglement, for three qubit composite system. To illustrate the point, entanglement of three qubit GHZ class states, W-class states, three boson state and noisy states is analysed. While genuine N-partite entanglement of a composite system is generated by N-way coherences, N-partite entanglement in general can be present due to (K