arXiv:1703.09437

General Monogamy Relations for Multiqubit W-class States in terms of negativity and squared Rényi-$α$ entanglement

Yanying Liang, Xiufang Feng, Wei Chen

Published Mar 28, 2017• 0 citations

Abstract

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 $x$-th power of the dual of convex-roof extended negativity, namely CRENOA, for $x\geq2$ and $x\leq0$. As for The squared Rényi-$α$ entanglement (SR$α$E) with $α$ in the region $[(\sqrt 7 - 1)/2,(\sqrt {13} - 1)/2]$, we show the upper bound of SR$α$E.

DOI: 10.48550/arxiv.1703.09437

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arXiv:1703.09437

General Monogamy Relations for Multiqubit W-class States in terms of negativity and squared Rényi- $α$ entanglementGeneral Monogamy Relations for Multiqubit W-class States in terms of negativity and squared Rényi-$α$ entanglement

Yanying Liang, Xiufang Feng, Wei Chen

Published Mar 28, 2017• 0 citations

arXiv HTML PDF

Abstract

x

-th power of the dual of convex-roof extended negativity, namely CRENOA, for

x\geq2

and

x\leq0

. As for The squared Rényi-

α

entanglement (SR

α

E) with

α

in the region

[(\sqrt 7 - 1)/2,(\sqrt {13} - 1)/2]

, we show the upper bound of SR

α

DOI: 10.48550/arxiv.1703.09437

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Scientific ReportsNov 11, 201915 citations

General Monogamy Relations for Multiqubit W-class States in terms of negativity and squared Rényi-

α

entanglement

x

-th power of the dual of convex-roof extended negativity, namely CRENOA, for

x\geq2

and

x\leq0

. As for The squared Rényi-

α

entanglement (SR

α

E) with

α

in the region

[(\sqrt 7 - 1)/2,(\sqrt {13} - 1)/2]

, we show the upper bound of SR

α

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.

Monogamy of entanglement is an indispensable feature in multipartite quantum systems. In this paper we investigate monogamy and polygamy relations with respect to any partition for generalized

W

-class states using R\'enyi-$\ensuremath{\alpha}$ entropy. First, we present analytical formulas of R\'enyi-$\ensuremath{\alpha}$ entanglement $(\mathrm{R}\ensuremath{\alpha}\mathrm{E})$ and R\'enyi-$\ensuremath{\alpha}$ entanglement of assistance $(\mathrm{R}\ensuremath{\alpha}\mathrm{EoA})$ for a reduced density matrix of an

n

-qudit pure state in a superposition of generalized

W

-class states and vacuum. Based on the analytical formulas, we show monogamy and polygamy relations in terms of $\mathrm{R}\ensuremath{\alpha}\mathrm{E}$ and $\mathrm{R}\ensuremath{\alpha}\mathrm{EoA}$. Next a reciprocal relation of $\mathrm{R}\ensuremath{\alpha}\mathrm{EoA}$ in an arbitrary three-party quantum system is found. Later, we further develop tighter monogamy relations in terms of concurrence and convex-roof extended negativity than former ones. In order to show the usefulness of our results, two partition-dependent residual entanglements are established to get a comprehensive analysis of entanglement dynamics for generalized

W

-class states. Moreover, we apply our results to an interesting quantum game and find a bound of the difference between the quantum game and the classical game.

We investigate monogamy relations related to quantum entanglement for

n-

qubit quantum systems. General monogamy inequalities are presented to the

\beta

(\beta\in(0,2))

power of concurrence, negativity and the convex-roof extended negativity, as well as the

\beta

(\beta\in(0,\sqrt{2}))

power of entanglement of formation. These monogamy relations are complementary to the existing ones with different regions of parameter

\beta

. In additions, new monogamy relations are also derived which include the existing ones as special cases.

N

-qubit mixed state. Furthermore, for a class of R\'enyi-$\ensuremath{\alpha}$ entanglement, we prove that the monogamy relations of the $\mathrm{SR}\ensuremath{\alpha}\mathrm{E}$ have a hierarchical structure when the

N

-qubit system is divided into

k

parties. As a by-product, the analytical relation between the R\'enyi-$\ensuremath{\alpha}$ entanglement and the squared concurrence is derived for bipartite $2\ensuremath{\bigotimes}d$ systems. Based on the monogamy properties of $\mathrm{SR}\ensuremath{\alpha}\mathrm{E}$, we can construct the corresponding multipartite entanglement indicators, which still work well even when the indicators based on the squared concurrence and EOF lose their efficacy. In addition, the monogamy property of the $\ensuremath{\mu}\mathrm{th}$ power of R\'enyi-$\ensuremath{\alpha}$ entanglement is analyzed.