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arXiv:2411.10740 · Communications in Theoretical Physics

Unified monogamy relations for the generalized W-class states beyond qubits

Zhong‐Xi Shen, Wen Zhou, Dong‐Ping Xuan, Zhi‐Xi Wang, Shao-Ming Fei

Published Sep 2, 2024• 0 citations
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Abstract

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.

DOI
10.1088/1572-9494/ad766d

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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.
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 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.
Monogamy relation is one of the essential properties of quantum entanglement, which characterizes the distribution of entanglement in a multipartite system. By virtual of the unified-(q,s) entropy, we obtain some novel monogamy and polygamy inequalities in general class of entanglement measures. For the multiqubit system, a class of tighter monogamy relations are established in term of the α-th power of unified-(q,s) entanglement for α ≥ 1. We also obtain a class of tighter polygamy relations in the β-th (0 ≤ β ≤ 1) power of unified-(q,s) entanglement of assistance. Applying these results to specific quantum correlations, e.g., entanglement of formation, Renyi-q entanglement of assistance, and Tsallis-q entanglement of assistance, we obtain the corresponding monogamy and polygamy relations. Typical examples are presented for illustration. Furthermore, the complementary monogamy and polygamy relations are investigated for the α-th (0 ≤ α ≤ 1) and β-th (β ≥ 1) powers of unified entropy, respectively, and the corresponding monogamy and polygamy inequalities are obtained.
General Monogamy Relations for Multiqubit W-class States in terms of negativity and squared Rényi-ααα entanglement
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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.
Abstract Monogamy and polygamy relations characterize the distributions of entanglement in multipartite systems. We investigate the monogamy and polygamy relations with respect to any partitions for a superposition of the generalized W -class state and vacuum in terms of the Tsallis- q entanglement and the Rényi- α entanglement. By using the Hamming weight of the binary vectors related to the partitions of the subsystems, new classes of monogamy and polygamy inequalities are derived, which are shown to be tighter than the existing ones. Detailed examples are presented to illustrate the finer characterization of entanglement distributions.