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arXiv:1508.07673 · Physical Review A

Negativity and strong monogamy of multiparty quantum entanglement beyond qubits

Jin Hyuk Choi, Jeong San Kim

Published Oct 8, 2015• 63 citations
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Abstract

We propose the square of convex-roof extended negativity (SCREN) as a powerful candidate to characterize strong monogamy of multiparty quantum entanglement. We first provide a strong monogamy inequality of multiparty entanglement using SCREN and show that the tangle-based multiqubit strong-monogamy inequality can be rephrased by SCREN. We further show that the SCREN strong-monogamy inequality is still true for the counterexamples that violate tangle-based the strong-monogamy inequality in higher-dimensional quantum systems other than qubits. We also analytically show that SCREN strong-monogamy inequality is true for a large class of multiqudit states, a superposition of multiqudit generalized $W$-class states and vacuums. Thus SCREN is a good alternative to characterize the strong monogamy of entanglement even in multiqudit systems.

DOI
10.1103/physreva.92.042307

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We propose the square of convex-roof extended negativity (SCREN) as a powerful candidate to characterize strong monogamy of multiparty quantum entanglement. We first provide a strong monogamy inequality of multiparty entanglement using SCREN and show that the tangle-based multiqubit strong-monogamy inequality can be rephrased by SCREN. We further show that the SCREN strong-monogamy inequality is still true for the counterexamples that violate tangle-based the strong-monogamy inequality in higher-dimensional quantum systems other than qubits. We also analytically show that SCREN strong-monogamy inequality is true for a large class of multiqudit states, a superposition of multiqudit generalized WWW-class states and vacuums. Thus SCREN is a good alternative to characterize the strong monogamy of entanglement even in multiqudit systems.
We provide evidence for the validity of strong monogamy inequality of multiparty quantum entanglement using the square of convex-roof extended negativity (SCREN). We first consider a large class of multiqudit mixed states that are in a partially coherent superposition of a generalized WWW-class state and the vacuum, and provide some useful properties about this class of states. We show that monogamy inequality of multiqudit entanglement in terms of SCREN holds for this class of states. We further show that SCREN strong monogamy inequality of multiqudit entanglement also holds for this class of states. Thus SCREN is a good alternative for characterizing the monogamous and strongly monogamous properties of multiqudit entanglement.
We provide a characterization of multiqubit entanglement constraints in terms of negativity. By using the square of convex-roof extended negativity (SCREN) and the Hamming weight of the binary vector related with the distribution of subsystems, we show that the $\ensuremath{\alpha}\mathrm{th}$ power of SCREN provides a class of monogamy inequalities of multiqubit entanglement in a tight way for $\ensuremath{\alpha}\ensuremath{\ge}1$. We further show that the $\ensuremath{\beta}\mathrm{th}$ power of SCREN also provides a class of tight polygamy inequalities for $0\ensuremath{\le}\ensuremath{\beta}\ensuremath{\le}1$.
One of the fundamental traits of quantum entanglement is the restricted shareability among multipartite quantum systems, namely monogamy of entanglement, while it is well known that monogamy inequalities are always satisfied by entanglement measures with convexity. Here we present a measure of entanglement, logarithmic convex-roof extended negativity (LCREN) satisfying important characteristics of an entanglement measure, and investigate the monogamy relation for logarithmic negativity and LCREN both without convexity. We show exactly that the αth power of logarithmic negativity, and a newly defined good measure of entanglement, LCREN, obey a class of general monogamy inequalities in multiqubit systems, 2⊗2⊗3 systems and 2⊗2⊗2n systems for α≥4ln2. We provide a class of general polygamy inequalities of multiqubit systems in terms of logarithmic convex-roof extended negativity of assistance (LCRENoA) for 0≤β≤2. Given that the logarithmic negativity and LCREN are not convex, these results are surprising. Using the power of the logarithmic negativity and LCREN, we further establish a class of tight monogamy inequalities of multiqubit systems, 2⊗2⊗3 systems and 2⊗2⊗2n systems in terms of the αth power of logarithmic negativity and LCREN for α≥4ln2. We also show that the βth power of LCRENoA obeys a class of tight polygamy inequalities of multiqubit systems for 0≤β≤2.
The monogamy property of entanglement is an intriguing feature of multipartite quantum entanglement. Most entanglement measures satisfying the monogamy inequality have turned out to be convex. Whether nonconvex entanglement measures obey the monogamy inequalities remains less known at present. As a well-known measure of entanglement, the logarithmic negativity is not convex. We elucidate the constraints of multi-qubit entanglement based on the logarithmic convex-roof extended negativity (LCREN) and the logarithmic convex-roof extended negativity of assistance (LCRENoA). Using the Hamming weight derived from the binary vector associated with the distribution of subsystems, we establish monogamy inequalities for multi-qubit entanglement in terms of the αth-power (α≥4ln2) of LCREN, and polygamy inequalities utilizing the αth-power (0≤α≤2) of LCRENoA. We demonstrate that these inequalities give rise to tighter constraints than the existing ones. Furthermore, our monogamy inequalities are shown to remain valid for the high-dimensional states that violate the CKW monogamy inequality. Detailed examples are presented to illustrate the effectiveness of our results in characterizing the multipartite entanglement distributions.
General Monogamy Relations for Multiqubit W-class States in terms of negativity and squared Rényi-ααα entanglement
For multipartite entangled states, entanglement monogamy is an important property. We investigate the monogamy relations for multiqubit generalized W-class states. We present new analytical monogamy inequalities satisfied by the xxx-th power of the dual of convex-roof extended negativity, namely CRENOA, for x≥2x\geq2x≥2 and x≤0x\leq0x≤0. As for The squared Rényi-ααα entanglement (SRαααE) with ααα in the region [(7−1)/2,(13−1)/2][(\sqrt 7 - 1)/2,(\sqrt {13} - 1)/2][(7​−1)/2,(13​−1)/2], we show the upper bound of SRαααE.
In this paper, we investigate the generalised monogamy inequalities of convex-roof extended negativity (CREN) in multi-level systems. The generalised monogamy inequalities provide the upper and lower bounds of bipartite entanglement, which are obtained by using CREN and the CREN of assistance (CRENOA). Furthermore, we show that the CREN of multi-qubit pure states satisfies some monogamy relations. Additionally, we test the generalised monogamy inequalities for qudits by considering the partially coherent superposition of a generalised W-class state in a vacuum and show that the generalised monogamy inequalities are satisfied in this case as well.