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arXiv:2605.22150

Unified entropy entanglement

Wenxue Ren, Binghao Li, Ruiqun Niu, Yu Guo, Shuanping Du

quant-phSubmitted May 21, 2026• Announced May 22, 2026• 0 citations
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Abstract

The unified entropy as a promotion of the von Neumann entropy exhibits distinct diversity which contains the Tsallis entropy, the Rényi entropy, the von Neumann entropy as special cases. The unified-($r,t$) entropy entanglement with $0<r<1$ and $0< t\leq 1$ was shown to be an entanglement monotone in literature. In this paper, we explore unified-($q,s$) entropy entanglement with $q>1$ and $qs\geq1$ and show that it is also an entanglement monotone and that both of them are monogamous. Going further, we present two kinds of global multipartite entanglement measures (GlMEMs) based on the unified entropy and each kind has two subclasses which are classified by the parameters $(q,s)$ and $(r,t)$. Consequently, from the view of the complete multipartite entanglement measure theory, we show that one of them is a complete multipartite entanglement monotone and is not only completely monogamous but also tightly completely monogamous, but the other three are even not complete. We also explore the genuine entanglement measures induced by the unified entropy and the relations with the bipartite entanglement and the global entanglement are discussed, respectively.

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12 pages, 1 figure

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The unified entropy as a promotion of the von Neumann entropy exhibits distinct diversity which contains the Tsallis entropy, the Rényi entropy, the von Neumann entropy as special cases. The unified-(r,tr,tr,t) entropy entanglement with 0<r<10<r<10<r<1 and 0<t≤10< t\leq 10<t≤1 was shown to be an entanglement monotone in literature. In this paper, we explore unified-(q,sq,sq,s) entropy entanglement with q>1q>1q>1 and qs≥1qs\geq1qs≥1 and show that it is also an entanglement monotone and that both of them are monogamous. Going further, we present two kinds of global multipartite entanglement measures (GlMEMs) based on the unified entropy and each kind has two subclasses which are classified by the parameters (q,s)(q,s)(q,s) and (r,t)(r,t)(r,t). Consequently, from the view of the complete multipartite entanglement measure theory, we show that one of them is a complete multipartite entanglement monotone and is not only completely monogamous but also tightly completely monogamous, but the other three are even not complete. We also explore the genuine entanglement measures induced by the unified entropy and the relations with the bipartite entanglement and the global entanglement are discussed, respectively.
Although many different entanglement measures have been proposed so far, much less is known in the multipartite case, which leads to the previous monogamy relations in the literature being not complete. We establish here a strict framework for defining the multipartite entanglement measure (MEM): apart from the postulates of the bipartite measure, i.e., vanishing on separable measures and nonincreasing under local operations and classical communication, a complete MEM should additionally satisfy the unification condition and the hierarchy condition. We then come up with a complete monogamy formula for the unified MEM (an MEM is called a unified MEM if it satisfies the unification condition) and a tightly complete monogamy relation for the complete MEM (an MEM is called a complete MEM if it satisfies both the unification condition and the hierarchy condition). Consequently, we propose MEMs which are multipartite extensions of entanglement of formation (EoF), concurrence, tangle, Tsallis qqq entropy of entanglement, R\'enyi $\ensuremath{\alpha}$ entropy of entanglement, the convex-roof extension of negativity, and negativity. We show that (i) the extensions of EoF, concurrence, tangle, and Tsallis qqq entropy of entanglement are complete MEMs; (ii) multipartite extensions of R\'enyi $\ensuremath{\alpha}$ entropy of entanglement, negativity, and the convex-roof extension of negativity are unified MEMs but not complete MEMs; and (iii) all these multipartite extensions are completely monogamous, and the ones which are defined by the convex-roof structure (except for the R\'enyi $\ensuremath{\alpha}$ entropy of entanglement and the convex-roof extension of negativity) are not only completely monogamous but also tightly completely monogamous. In addition, as a byproduct, we find a class of states that satisfy the additivity of EoF. We also find a class of tripartite states one part of which can be maximally entangled with the other two parts simultaneously according to the definition of mixed maximally entangled state (MMES) in Li et al. [Z. Li, M. Zhao, S. Fei, H. Fan, and W. Liu, Quantum Inf. Comput. 12, 0063 (2012)]. Consequently, we improve the definition of maximally entangled state (MES) and prove that the only MES is the pure MES.
In this paper, we investigate the characterization of multipartite entanglement based on the unified (q,s)(q,s)(q,s)-entropy framework. For bipartite quantum states, we propose an entanglement measure Eq,sA∣B(ρ)E_{q,s}^{A|B}(ρ)Eq,sA∣B​(ρ) based on unified entropy and derive several analytical lower bounds for it using local orthonormal observables. We demonstrate with concrete examples that our lower bounds provide tighter estimates of quantum entanglement than several existing analytical bounds. For multipartite quantum states, based on unified entropy, we propose two measures Aq,sk(ρ)A^k_{q,s}(ρ)Aq,sk​(ρ) and Gq,sk(ρ)G^k_{q,s}(ρ)Gq,sk​(ρ) for quantifying kkk-nonseparability, and prove that these measures satisfy several desirable properties, such as faithfulness, local unitary invariance, and convexity. Moreover, we rigorously prove that when the parameters qqq and sss are in certain ranges, Aq,sk(ρ)A^k_{q,s}(ρ)Aq,sk​(ρ) and Gq,sk(ρ)G^k_{q,s}(ρ)Gq,sk​(ρ) also satisfy monotonicity and strong monotonicity. Furthermore, we establish the order relations of the measures Aq,sk(ρ)A^k_{q,s}(ρ)Aq,sk​(ρ) and Gq,sk(ρ)G^k_{q,s}(ρ)Gq,sk​(ρ) with respect to the parameters qqq and sss for fixed quantum states. In addition, for fixed qqq and sss, we also give their order relations for any two pure states.
A complete characterization and quantification of entanglement, particularly the multipartite entanglement, remains an unfinished long-term goal in quantum information theory. As long as the multipartite system is concerned, the relation between the entanglement contained in different partitions or different subsystems need to take into account. The complete multipartite entanglement measure and the complete monogamy relation is a framework that just deals with such a issue. In this paper, we put forward conditions to justify whether the multipartite entanglement monotone (MEM) and genuine multipartite entanglement monotone (GMEM) are complete, completely monogamous, and tightly complete monogamous according to the feature of the reduced function. Especially, we proposed a class of complete MEMs and a class of complete GMEMs via the maximal reduced function for the first time. By comparison, it is shown that, for the tripartite case, this class of GMEMs is better than the one defined from the minimal bipartite entanglement in literature under the framework of complete MEM and complete monogamy relation. In addition, the relation between monogamy, complete monogamy, and the tightly complete monogamy are revealed in light of different kinds of MEMs and GMEMs.
A crucial issue in quantum communication tasks is characterizing how quantum resources can be quantified and distributed over many parties. Consequently, entanglement has been explored extensively. However, there are few genuine multipartite entanglement measures and whether it is monogamous is so far unknown. In this work, we explore the complete monogamy of genuine multipartite entanglement measure (GMEM) for which, at first, we investigate a framework for unified/complete GMEM according to the unified/complete multipartite entanglement measure we proposed in 2020. We find a way of inducing unified/complete GMEM from any given unified/complete multipartite entanglement measure. It is shown that any unified GMEM is completely monogamous, and any complete GMEM that is induced by given complete multipartite entanglement measure is completely monogamous. In addition, the previous GMEMs are checked under this framework. It turns out that the genuinely multipartite concurrence is not as good of a candidate as GMEM.
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.
Multipartite entanglement includes not only the genuine entanglement but also the kkk-entanglement $(k\ensuremath{\geqslant}2)$. It is known that the most bipartite entanglement measures have been shown to be monogamous, but the monogamy relation is involved in the bipartite entanglement measures rather than the multipartite ones. So how we can explore the monogamy relation for kkk-entanglement becomes a basic open problem. In this paper we establish an axiomatic definition of the monogamy relation for the kkk-entanglement measure based on the coarser relation of the system partition. We also present the axiomatic definition of the complete kkk-entanglement measure and the associated complete monogamy relation according to the framework of the complete multipartite entanglement measure we established in [Phys. Rev. A 101, 032301 (2020)], which is shown to be an efficient tool for characterizing the multipartite quantum correlation as complementary to the monogamy relation associated with the bipartite measure. Consequently, the relation and the difference between monogamy, complete monogamy, and tightly complete monogamy are clearly depicted in light of the three types of coarser relation of the system partition. We then illustrate our approach with two classes of kkk-entanglement measures in detail. We find that all these kkk-entanglement measures are monogamous if the reduced function is strictly concave, they are not completely monogamous, and they are not complete generally.