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arXiv:1405.3989 · Physical Review Letters

Strong Monogamy Conjecture for Multiqubit Entanglement: The Four-Qubit Case

Bartosz Regula, Sara Di Martino, Soojoon Lee, Gerardo Adesso

Published Sep 10, 2014• 93 citations
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Abstract

We investigate the distribution of bipartite and multipartite entanglement in multiqubit states. In particular, we define a set of monogamy inequalities sharpening the conventional Coffman-Kundu-Wootters constraints, and we provide analytical proofs of their validity for relevant classes of states. We present extensive numerical evidence validating the conjectured strong monogamy inequalities for arbitrary pure states of four qubits.

DOI
10.1103/physrevlett.113.110501

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We investigate the distribution of bipartite and multipartite entanglement in multiqubit states. In particular, we define a set of monogamy inequalities sharpening the conventional Coffman-Kundu-Wootters constraints, and we provide analytical proofs of their validity for relevant classes of states. We present extensive numerical evidence validating the conjectured strong monogamy inequalities for arbitrary pure states of four qubits.
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We establish two new inequalities, the weighted strong monogamy (WSM) and the maximum residual strong monogamy (MRSM), which sharpen the generalized Coffman-Kundu-Wootters inequity for multiqubit states. The WSM inequality distinguishes itself from the strong monogamy (SM) conjecture [Phys. Rev. Lett. 113, 110501 (2014)] by using coefficients rather than exponents to modulate the weight allocated to various m-partite contributions. In contrast, the MRSM inequality is formulated using only the maximum m-partite entanglement. We find that the residual entanglement of the MRSM inequality can effectively distinguish the separable states. We also compare the tightness of various SM inequalities and provide examples using a four-qubit mixed state and a five-qubit pure state to illustrate the MRSM inequality. These examples characterize the trade-off relations among entanglement components involving varying numbers of qubits. Our results provide a rigorous framework to characterize and quantify the monogamy of multipartite entanglement.
Monogamy is a defining feature of entanglement, having far-reaching applications. Recently, Regula et al. [Phys. Rev. Lett. 113, 110501 (2014)] proposed a stronger version of monogamy relation for concurrence. We have extended the strong monogamy inequality for another entanglement measure, viz., negativity. In particular, we have concentrated on the four-qubit system and provided a detailed study on the status of strong monogamy on pure states. Further, we have analytically provided some classes of states for which negativity and squared negativity satisfy strong monogamy. Numerical evidences have also been shown in proper places. Our analysis also provides cases where strong monogamy is violated.
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.