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Communications in Theoretical Physics

Tighter Rényi- α entropy entanglement monogamy and polygamy relations of qubit systems

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

Published Feb 27, 2026• 0 citations
Publisher

Abstract

Abstract The monogamy and polygamy relations characterize the distributions of quantum entanglement in multipartite systems. We investigate the monogamy relations related to the Rényi- α entanglement (R α E) and polygamy relations related to the R α E of assistance (R α EoA). We present new entanglement monogamy relations satisfied by the η -th ( η ≥ 2) power of R α E for α ≥ 2, and by the γ -th ( γ ≥ 4) power of R α E for 7 − 1 2 ≤ α < 2 , respectively. Moreover, we present polygamy relations satisfied by the t th power of R α EoA with 7 − 1 2 ≤ α < 13 − 1 2 for 0 ≤ t ≤ 1. We also give relationships among the residual entanglements of R α E and R α EoA, and the three tangle based on our general monogamy relations.

DOI
10.1088/1572-9494/ae4b16

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Abstract The monogamy and polygamy relations characterize the distributions of quantum entanglement in multipartite systems. We investigate the monogamy relations related to the Rényi- α entanglement (R α E) and polygamy relations related to the R α E of assistance (R α EoA). We present new entanglement monogamy relations satisfied by the η -th ( η ≥ 2) power of R α E for α ≥ 2, and by the γ -th ( γ ≥ 4) power of R α E for 7 − 1 2 ≤ α < 2 , respectively. Moreover, we present polygamy relations satisfied by the t th power of R α EoA with 7 − 1 2 ≤ α < 13 − 1 2 for 0 ≤ t ≤ 1. We also give relationships among the residual entanglements of R α E and R α EoA, and the three tangle based on our general monogamy relations.
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.
Tighter monogamy and polygamy relations using Rényi-ααα entropy
We investigate monogamy relations related to the Rényi-ααα entanglement and polygamy relations related to the Rényi-ααα entanglement of assistance. We present new entanglement monogamy relations satisfied by the μμμ-th power of Rényi-ααα entanglement with α∈[7−1)/2,(13−1)/2]α\in[\sqrt{7}-1)/2,(\sqrt{13}-1)/2]α∈[7​−1)/2,(13​−1)/2] for μ⩾2μ\geqslant2μ⩾2, and polygamy relations satisfied by the μμμ-th power of Rényi-ααα entanglement of assistance with α∈[7−1)/2,(13−1)/2]α\in[\sqrt{7}-1)/2,(\sqrt{13}-1)/2]α∈[7​−1)/2,(13​−1)/2] for 0≤μ≤10\leqμ\leq10≤μ≤1. These relations are shown to be tighter than the existing ones.
Monogamy and polygamy relations are essential properties of quantum entanglement, which characterize the distributions of entanglement in multipartite systems. In this paper, we establish the general monogamy relations for γ-th (0 ≤ γ ≤ α, α ≥ 1) power of quantum entanglement based on unified-(q,s) entanglement and polygamy relations for δ-th (δ ≥ β, 0 ≤ β ≤ 1) power of entanglement of assistance based on unified-(q,s) entanglement of assistance, which provides a complement to the previous research in terms of different parameter regions of γ and δ. These results are then applied to specific quantum correlations, e.g., entanglement of formation, Renyi-q entanglement of assistance and Tsallis-q entanglement of assistance to get the corresponding monogamy and polygamy inequalities. Moreover, typical examples are presented for illustration.
Monogamy and polygamy relations characterize the distributions of entanglement in multipartite systems. We provide a characterization of multiqubit entanglement constraints in terms of unified-(q, s) entropy. A class of tighter monogamy inequalities of multiqubit entanglement based on the α-th power of unified-(q, s) entanglement for α ≥ 1 and a class of polygamy inequalities in terms of the β-th power of unified-(q, s) entanglement of assistance are established in this paper. Our results present a general class of the monogamy and polygamy relations for bipartite entanglement measures based on unified-(q, s) entropy, which are tighter than the existing ones. What is more, some usual monogamy and polygamy relations, such as monogamy and polygamy relations based on entanglement of formation, Renyi-q entanglement of assistance and Tsallis-q entanglement of assistance, can be obtained from these results by choosing appropriate parameters (q, s) in unified-(q, s) entropy entanglement. Typical examples are also presented for illustration.
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.