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arXiv:1607.00135 · Physical Review A

Testing the monogamy relations via rank-2 mixtures

Eylee Jung, DaeKil Park

Published Oct 20, 2016• 2 citations
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Abstract

We introduce two tangle-based four-party entanglement measures ${t}_{1}$ and ${t}_{2}$, and two negativity-based measures ${n}_{1}$ and ${n}_{2}$, which are derived from the monogamy relations. These measures are computed for three four-qubit maximally entangled and $W$ states explicitly. We also compute these measures for the rank-2 mixture ${\ensuremath{\rho}}_{4}=p|{\text{GHZ}}_{4}\ensuremath{\rangle}\ensuremath{\langle}{\text{GHZ}}_{4}|+(1\ensuremath{-}p)|{W}_{4}\ensuremath{\rangle}\ensuremath{\langle}{W}_{4}|$ by finding the corresponding optimal decompositions. It turns out that ${t}_{1}({\ensuremath{\rho}}_{4})$ is trivial and the corresponding optimal decomposition is equal to the spectral decomposition. Probably, this triviality is a sign of the fact that the corresponding monogamy inequality is not sufficiently tight. We fail to compute ${t}_{2}({\ensuremath{\rho}}_{4})$ due to the difficulty in the calculation of the residual entanglement. The negativity-based measures ${n}_{1}({\ensuremath{\rho}}_{4})$ and ${n}_{2}({\ensuremath{\rho}}_{4})$ are explicitly computed and the corresponding optimal decompositions are also derived explicitly.

DOI
10.1103/physreva.94.042330

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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-$\ensuremath{\pi}$ 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 report a set of monogamy constraints on one-tangle, two-tangles, three-tangles and four-way correlations of a general four-qubit pure state. It is found that given a two-qubit marginal ρ\rhoρ of an NNN qubit pure state $\left\vert \Psi_{N}\right\rangle $, the non-Hermitian matrix ρρ~\rho \widetilde{\rho}ρρ​ where ρ~\widetilde{\rho}ρ​ $=\left( \sigma_{y}\otimes \sigma_{y}\right) \rho^{\ast}\left( \sigma_{y}\otimes\sigma_{y}\right) $, contains information not only about the entanglement properties of the two-qubits in that state but also about the correlations of the pair of qubits with $\left( N-2\right) $ qubits in $\left\vert \Psi_{N}\right\rangle $. To extract information about tangles of a four-qubit state $\left\vert \Psi_{4}\right\rangle $, the coefficients in the characteristic polynomial of matrix ρρ~\rho\widetilde{\rho}ρρ​ are analytically expressed in terms of 2×22\times22×2 matrices of state coefficients. A general monogamy relation satisfied by one-tangle, two-tangles, three-tangles, and residual correlations in a multiqubit state $\left\vert \Psi_{N}\right\rangle $ is also given.
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.
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 examine the various properties of the three four-qubit monogamy relations, all of which introduce the power factors in the three-way entanglement to reduce the tripartite contributions. On the analytic ground as much as possible we try to find the minimal power factors, which make the monogamy relations hold if the power factors are larger than the minimal powers. Motivated to the three-qubit monogamy inequality we also examine whether those four-qubit monogamy relations provide the SLOCC-invariant four-way entanglement measures or not. Our analysis indicate that this is impossible provided that the monogamy inequalities are derived merely by introducing weighting power factors.