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arXiv:1801.07846

Four-Qubit Monogamy and Four-Way Entanglement

DaeKil Park

Published Jan 24, 2018• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.1801.07846

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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.
We investigate possible generalizations of the Coffman-Kundu-Wootters monogamy inequality to four qubits, accounting for multipartite entanglement in addition to the bipartite terms. We show that the most natural extension of the inequality does not hold in general, and we describe the violations of this inequality in detail. We investigate alternative ways to extend the monogamy inequality to express a constraint on entanglement sharing valid for all four-qubit states, and perform an extensive numerical analysis of randomly generated four-qubit states to explore the properties of such extensions.
We investigate monogamy relations related to quantum entanglement for n−n-n−qubit quantum systems. General monogamy inequalities are presented to the β\betaβth (β∈(0,2))(\beta\in(0,2))(β∈(0,2)) power of concurrence, negativity and the convex-roof extended negativity, as well as the β\betaβth (β∈(0,2))(\beta\in(0,\sqrt{2}))(β∈(0,2​)) power of entanglement of formation. These monogamy relations are complementary to the existing ones with different regions of parameter β\betaβ. In additions, new monogamy relations are also derived which include the existing ones as special cases.
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We investigate the monogamy relations related to the concurrence and the entanglement of formation. General monogamy inequalities given by the $\ensuremath{\alpha}\text{th}$ power of concurrence and entanglement of formation are presented for NNN-qubit states. The monogamy relation for entanglement of assistance is also established. Based on these general monogamy relations, the residual entanglement of concurrence and entanglement of formation are studied. Some relations among the residual entanglement, entanglement of assistance, and three tangle are also presented.
We discuss the possibility to interpret the residual entanglement for more than three qubits in terms of distributed multipartite entanglement or, in other words, possible extensions of the Coffman-Kundu-Wootters monogamy equality to higher qubit numbers. Existing knowledge on entanglement in multipartite systems puts narrow constraints on the form of such extensions. We study various examples for families of pure four-qubit states for which the characterization of three-qubit and four-qubit entanglement in terms of polynomial invariants is known. These examples indicate that, although families with such extensions do exist, a generalized monogamy equality cannot be found along those lines.