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arXiv:1608.06413 · Physical Review A

Generalized monogamy inequalities and upper bounds of negativity for multiqubit systems

Yanmin Yang, Wei Chen, Gang Li, Zhu‐Jun Zheng

Published Jan 29, 2018• 9 citations
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Abstract

In this paper, we present some generalized monogamy inequalities and upper bounds of negativity based on convex-roof extended negativity (CREN) and CREN of assistance (CRENOA). These monogamy relations are satisfied by the negativity of $N$-qubit quantum systems $AB{C}_{1}\ensuremath{\cdots}{C}_{N\ensuremath{-}2}$, under the partitions $AB|{C}_{1}\ensuremath{\cdots}{C}_{N\ensuremath{-}2}$ and $AB{C}_{1}|{C}_{2}\ensuremath{\cdots}{C}_{N\ensuremath{-}2}$. Furthermore, the $W$-class states are used to test these generalized monogamy inequalities.

DOI
10.1103/physreva.97.012336

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In this paper, we present some generalized monogamy inequalities and upper bounds of negativity based on convex-roof extended negativity (CREN) and CREN of assistance (CRENOA). These monogamy relations are satisfied by the negativity of NNN-qubit quantum systems ABC1\ensuremath⋯CN\ensuremath−2AB{C}_{1}\ensuremath{\cdots}{C}_{N\ensuremath{-}2}ABC1​\ensuremath⋯CN\ensuremath−2​, under the partitions AB∣C1\ensuremath⋯CN\ensuremath−2AB|{C}_{1}\ensuremath{\cdots}{C}_{N\ensuremath{-}2}AB∣C1​\ensuremath⋯CN\ensuremath−2​ and ABC1∣C2\ensuremath⋯CN\ensuremath−2AB{C}_{1}|{C}_{2}\ensuremath{\cdots}{C}_{N\ensuremath{-}2}ABC1​∣C2​\ensuremath⋯CN\ensuremath−2​. Furthermore, the WWW-class states are used to test these generalized monogamy inequalities.
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.
In this paper, we present some generalized monogamy inequalities based on negativity and convex-roof extended negativity (CREN). The monogamy relations are satisfied by the negativity of N−N-N−qubit quantum systems ABC1⋯CN−2,ABC_1\cdots C_{N-2},ABC1​⋯CN−2​, under the partition ABABAB and C1C2⋯CN−2.C_1C_2\cdots C_{N-2}.C1​C2​⋯CN−2​. This result can be generalized to the N−N-N−qubit pure states under the partition ABC1ABC_1ABC1​ and C2⋯CN−2.C_2\cdots C_{N-2}.C2​⋯CN−2​. Furthermore, we give some examples to test the generalized monogamy inequalities.
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.
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.
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 $0\ensuremath{\le}\ensuremath{\beta}\ensuremath{\le}1$.
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.