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arXiv:1410.7925 · Chinese Physics B

Monogamous nature of symmetricN-qubit states of the W class: Concurrence and negativity tangle

P. J. Geetha, K. O. Yashodamma, Sudha

Published Nov 1, 2015• 5 citations
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Abstract

Using Majorana representation of symmetric N -qubit pure states, we have examined the monogamous nature of the family of states with two-distinct spinors, the W class of states. We have evaluated the N -concurrence tangle and showed that all the states in this family have vanishing concurrence tangle. The negativity tangle for the W class of states is shown to be non-zero, illustrating the fact that the concurrence tangle is always lesser than or equal to the negativity tangle in a pure N -qubit state.

DOI
10.1088/1674-1056/24/11/110302

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Using Majorana representation of symmetric N -qubit pure states, we have examined the monogamous nature of the family of states with two-distinct spinors, the W class of states. We have evaluated the N -concurrence tangle and showed that all the states in this family have vanishing concurrence tangle. The negativity tangle for the W class of states is shown to be non-zero, illustrating the fact that the concurrence tangle is always lesser than or equal to the negativity tangle in a pure N -qubit state.
We evaluate the monogamy inequality for symmetric, non-symmetric pure states of importance in terms of squared concurrence, squared entanglement of formation, squared negativity of partial transpose and compare the corresponding tangles. We show that though concurrence and concurrence tangle are zero for two special classes of mixed entangled states, both negativity tangle and entanglement of formation (EOF) tangle turn out to be non-zero. A comparison of different tangles is carried out in each case and it is shown that while the concurrence tangle captures the genuine multiqubit entanglement in N -qubit pure states with N distinct spinors (containing GHZ and superposition of W-, obverse W states) either negativity tangle or EOF tangle is to be used as a better measure of entanglement in the W-class of states with two distinct spinors and in the special classes of mixed multiqubit states.
The limitation on the shareability of quantum entanglement over several parties, the so-called monogamy of entanglement, is an issue that has received considerable attention from the quantum information community over the last decade. A natural question of interest in this connection is whether monogamy of correlations is true for correlations other than entanglement. This issue is examined here by choosing quantum deficit, proposed by A. K. Rajagopal and R. W. Rendell [Phys. Rev. A 66, 022104 (2002)], an operational measure of correlations. In addition to establishing the polygamous nature of the class of three-qubit symmetric pure states characterized by two distinct Majorana spinors (to which the WWW states belong), those with three distinct Majorana spinors [to which Greenberger-Horne-Zeilinger (GHZ) states belong] are shown to either obey or violate monogamy relations. While the generalized WWW states can be monogamous or polygamous, the generalized GHZ states exhibit monogamy with respect to quantum deficit. The issue of using monogamy conditions based on quantum deficit to witness the states belonging to stochastic local operations and classical communication (SLOCC) inequivalent classes is discussed in light of these results.
A novel use of Majorana geometric representation brings out distinct entanglement families of permutation symmetric states of qubits. The paradigmatic W and GHZ (Greenberger-Horne-Zeilinger) states of three qubits respectively contain two and three independent Majorana spinors. Another unique state with three distinct Majorana spinors -- constructed through a permutation symmetric superposition of two up qubits and one down qubit (W state) and its obverse state Wbar) exhibits genuine three-party entanglement, which is robust under loss of a qubit. While the GHZ state has irreducible correlations and cannot be determined from its parts, we show here that the correlation information of the W-superposition state is imprinted uniquely in its two party reduced states. This striking example sheds light on the contrasting irreducibility features of interconvertible states.
We derive a family of entanglement monotones, one member of which turns out to be the negativity. Two others are shown to be lower bounds on the I-concurrence, and on the I-tangle, respectively [P. Rungta and C. M. Caves, to appear in Phys. Rev. A]. We compare these bounds with the I-concurrence and I-tangle on the isotropic states, and on rank-two density operators resulting from a Tavis-Cummings interaction. Our results provide a global structure relating several different entanglement measures. Additionally, they possess analytic forms which are easily evaluated in the most general cases.
In this paper, we investigate two different entanglement measures, the negativity and concurrence, in the case of pure and mixed states of two-qubit system basing on the spin coherent states. For two-qubit pure states, the negativity is the same as the concurrence. For mixed states, using a simplified expression of concurrence in Wootters' measure of entanglement, we write the bounds of Verstraete et al.1as a function of some new parameters and we compare the both measures for a class of mixed states.
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.