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arXiv:1807.08101 · Physical Review A

Monogamy relations for the generalized W -class states beyond qubits

Xian Shi, Lin Chen

Published Mar 25, 2020• 19 citations
arXivHTMLPDF

Abstract

In this article, we consider the monogamy relations for the generalized $W$-class states. Here we first present an analytical formula on Tsallis-$q$ entanglement ($\mathrm{T}q\mathrm{EE}$) and Tsallis-$q$ entanglement of assistance ($\mathrm{T}q\mathrm{EEoA}$) of a reduced density matrix for a generalized $W$-class state. By using the analytical formula, we present a monogamy relation in terms of the squared $\mathrm{T}q\mathrm{EE}$ for a generalized $W$-class state. Then we present generalized monogamy relations in terms of $\mathrm{T}q\mathrm{EE}$ for a generalized $W$-class state. We also present a monogamy relation in terms of an arbitrary power of $\mathrm{T}q\mathrm{EE}$ for a generalized $W$-class state. Last, we apply our results to quantum games and present the bound of the nonclassicality of the quantum games when restricting to the generalized $W$-class states.

DOI
10.1103/physreva.101.032344

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In this article, we consider the monogamy relations for the generalized WWW-class states. Here we first present an analytical formula on Tsallis-qqq entanglement (TqEE\mathrm{T}q\mathrm{EE}TqEE) and Tsallis-qqq entanglement of assistance (TqEEoA\mathrm{T}q\mathrm{EEoA}TqEEoA) of a reduced density matrix for a generalized WWW-class state. By using the analytical formula, we present a monogamy relation in terms of the squared TqEE\mathrm{T}q\mathrm{EE}TqEE for a generalized WWW-class state. Then we present generalized monogamy relations in terms of TqEE\mathrm{T}q\mathrm{EE}TqEE for a generalized WWW-class state. We also present a monogamy relation in terms of an arbitrary power of TqEE\mathrm{T}q\mathrm{EE}TqEE for a generalized WWW-class state. Last, we apply our results to quantum games and present the bound of the nonclassicality of the quantum games when restricting to the generalized WWW-class states.
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 WWW-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 nnn-qudit pure state in a superposition of generalized WWW-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 WWW-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.
Abstract The monogamy of entanglement stands as an indispensable feature within multipartite quantum systems. We study monogamy relations with respect to any partitions for the generalized W-class (GW) states based on the unified-(q, s) entanglement (UE). We provide the monogamy relation based on the squared UE for a reduced density matrix of a qudit GW state, as well as tighter monogamy relations based on the αth (α ≥ 2) power of UE. Furthermore, for an n-qudit system ABC 1...C n−2, a generalized monogamy relation and an upper bound satisfied by the βth (0 ≤ β ≤ 1) power of the UE for the GW states under the partition AB and C 1...C n−2 are established. In particular, two partition-dependent residual entanglements for the GW states are analyzed in detail.
Unified-(q, s) entanglement [Formula: see text] is a generalized bipartite entanglement measure, which encompasses Tsallis-q entanglement, Rényi-q entanglement, and entanglement of formation as its special cases. We first provide the extended (q; s) region of the generalized analytic formula of [Formula: see text]. Then, the monogamy relation based on the squared [Formula: see text] for arbitrary multiqubit mixed states is proved. The monogamy relation proved in this paper enables us to construct an entanglement indicator that can be utilized to identify all genuine multiqubit entangled states even the cases where three tangle of concurrence loses its efficiency. It is shown that this monogamy relation also holds true for the generalized W-class state. The αth power [Formula: see text] based general monogamy and polygamy inequalities are established for tripartite qubit states.
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.
Tsallis-q entanglement is a bipartite entanglement measure which is the generalization of entanglement of formation for q tending to 1. We first expand the range of q for the analytic formula of Tsallis-q entanglement. For , we prove the monogamy relation in terms of the squared Tsallis-q entanglement for an arbitrary multi-qubit systems. It is shown that the multipartite entanglement indicator based on squared Tsallis-q entanglement still works well even when the indicator based on the squared concurrence loses its efficacy. We also show that the μ-th power of Tsallis-q entanglement satisfies the monogamy or polygamy inequalities for any three-qubit state.
Abstract We present optimized generalized monogamy relations and upper bounds derived from concurrence and concurrence of assistance. We first establish a tighter general upper bound of the αth ( 0 ⩽ α ⩽ 2 ) power of concurrence for N-qubit states. Then for N-qubit systems A B C 1 … C N − 2 , the optimized monogamy relations and upper bounds satisfied by the αth ( 0 ⩽ α ⩽ 2 ) power of concurrence of N-qubit pure states under the partition AB and C 1 … C N − 2 , as well as under the partition ABC 1 and C 2 … C N − 2 are established, which give rise to restrictions on the entanglement distribution and trade offs among the subsystems. Moreover, the utilization of the W-class states demonstrates that our results are tighter compared with the existing results. Similar results are also obtained for negativity.