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Chinese Physics B

Comparative analysis of entanglement measures based on monogamy inequality

P. J. Geetha, Sudha, K. S. Mallesh

Published May 1, 2017• 4 citations
Publisher

Abstract

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.

DOI
10.1088/1674-1056/26/5/050301

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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.
We prove exactly that the squared entanglement of formation, which quantifies the bipartite entanglement, obeys a general monogamy inequality in an arbitrary multiqubit mixed state. Based on this kind of exotic monogamy relation, we are able to construct two sets of useful entanglement indicators: the first one can detect all genuine multiqubit entangled states even in the case of the two-qubit concurrence and n-tangles being zero, while the second one can be calculated via quantum discord and applied to multipartite entanglement dynamics. Moreover, we give a computable and nontrivial lower bound for multiqubit entanglement of formation.
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.
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.
Unified-(q, s) entanglement [Formula: see text] is a generalized bipartite entanglement measure, which encompasses Tsallis-q entanglement, Rényi-q entanglement, and entanglement of formation as its special cases. We first provide the extended (q; s) region of the generalized analytic formula of [Formula: see text]. Then, the monogamy relation based on the squared [Formula: see text] for arbitrary multiqubit mixed states is proved. The monogamy relation proved in this paper enables us to construct an entanglement indicator that can be utilized to identify all genuine multiqubit entangled states even the cases where three tangle of concurrence loses its efficiency. It is shown that this monogamy relation also holds true for the generalized W-class state. The αth power [Formula: see text] based general monogamy and polygamy inequalities are established for tripartite qubit states.