Abstract This study provides a generalized definition of polygamy relations for any quantum correlation measures. Instead of the usual polygamy inequality, a polygamy relation with equality is given by introducing the polygamy weight. From the polygamy relation with equality, it presents polygamy inequalities satisfied by the βth power of the quantum correlation measures. Taking concurrence of assistance as an example, it further illustrates the significance and advantages of these relations. It also obtains a polygamy relation with equality by considering the one‐to‐group entanglements for any quantum entanglement measures that do not satisfy the polygamy relations. It demonstrates that such relations for tripartite states can be generalized to multipartite systems.
Abstract This work proposes a novel class of parameterized entanglement measures that are named as ‐concurrence () (), and demonstrates comprehensively that they satisfy all the necessary axiomatic conditions required for an entanglement measure. Furthermore, an analytical formula relating to concurrence for the range of within two‐qubit systems is derived. Additionally, a new polygamy relation of multiqubit quantum entanglement in terms of ‐concurrence of assistance () is proved. However, it fails to obey the monogamy relation, but it is demonstrated that the squared ‐concurrence () does obey a general monogamy relation in an arbitrary ‐qubit mixed state. Utilizing the monogamy properties of , the corresponding multipartite entanglement indicators can be constructed, which can detect all genuine multiqubit entangled states even in the case of ‐tangle vanishes. In addition, for multipartite higher‐dimensional systems, it is illustrated that still has the applicability of the monogamy relation.
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.
For a bipartite entanglement measure
E that satisfies the
γth-power monogamy inequality (Eq.~\eqref{e:chap1-ineq1}), and for its assisted counterpart
Ea that obeys the
δth-power polygamy inequality (Eq.~\eqref{e:chap1-ineq2}), we introduce a unified, tunable framework indexed by a parameter
m≥1. Within this framework, we derive two hierarchical families of refined inequalities:
a tightened
α-power monogamy relation for
E, valid for all
α≥mγ;
a tightened
β-power polygamy relation for
Ea, applicable for
(m−1)δ<β≤mδ.
As
m increases, the bounds become progressively tighter, recovering known results at
m=1. Notably, the optimal monogamy bound emerges as a piecewise function of
α, with additional correction terms activated as
α crosses successive integer thresholds, thereby offering a sharper characterization of entanglement distribution.
We demonstrate that our results generalize and strengthen existing monogamy and polygamy relations through analytical comparisons and numerical evaluations using concurrence and concurrence of assistance. This hierarchical, parameterized approach offers enhanced and flexible tools for applications in quantum communication, quantum networks, and multipartite quantum information processing.
The monogamy of entanglement is generally discussed using a bipartite entanglement measure as an upper bound. Here we discuss a new kind of monogamous relation where the upper bound is given by a multipartite measure of entanglement, the generalized concurrence. We show a monogamous equality involving the multipartite concurrence, all the bipartite concurrences, and the genuine tripartite entanglement for pure three-qubit systems. The result extends to mixed states in an inequality involving the generalized concurrence and all the bipartite concurrences. We provide a counterexample showing that the result cannot be extended for systems with more than three qubits.
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.