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arXiv:2402.00457 · Entropy

Tighter Constraints of Multi-Qubit Entanglement in Terms of Nonconvex Entanglement Measures LCREN and LCRENoA

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

Published Jan 31, 2024• 2 citations

Abstract

The monogamy property of entanglement is an intriguing feature of multipartite quantum entanglement. Most entanglement measures satisfying the monogamy inequality have turned out to be convex. Whether nonconvex entanglement measures obey the monogamy inequalities remains less known at present. As a well-known measure of entanglement, the logarithmic negativity is not convex. We elucidate the constraints of multi-qubit entanglement based on the logarithmic convex-roof extended negativity (LCREN) and the logarithmic convex-roof extended negativity of assistance (LCRENoA). Using the Hamming weight derived from the binary vector associated with the distribution of subsystems, we establish monogamy inequalities for multi-qubit entanglement in terms of the αth-power (α≥4ln2) of LCREN, and polygamy inequalities utilizing the αth-power (0≤α≤2) of LCRENoA. We demonstrate that these inequalities give rise to tighter constraints than the existing ones. Furthermore, our monogamy inequalities are shown to remain valid for the high-dimensional states that violate the CKW monogamy inequality. Detailed examples are presented to illustrate the effectiveness of our results in characterizing the multipartite entanglement distributions.

DOI: 10.3390/e26020127

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We establish a characterization of multi-qubit entanglement constraints in terms of non-negative power of entanglement measures based on unified-(q, s) entropy. Using the Hamming weight of the binary vector related with the distribution of subsystems, we establish a class of tight monogamy inequalities of multi-qubit entanglement based on the αth-power of unified-(q, s) entanglement for α ≥ 1. For 0 ≤ β ≤ 1, we establish a class of tight polygamy inequalities of multi-qubit entanglement in terms of the βth-power of unified-(q, s) entanglement of assistance. Thus our results characterize the monogamy and polygamy of multi-qubit entanglement for the full range of non-negative power of unified entanglement.

We establish a characterization of multi-qubit entanglement constraints in terms of non-negative power of entanglement measures based on unified-

(q,s)

entropy. Using the Hamming weight of the binary vector related with the distribution of subsystems, we establish a class of tight monogamy inequalities of multi-qubit entanglement based on the

α

th-power of unified-

(q,s)

entanglement for

α\geq 1

. For

0 \leq β\leq 1

, we establish a class of tight polygamy inequalities of multi-qubit entanglement in terms of the

β

th-power of unified-

(q,s)

entanglement of assistance. Thus our results characterize the monogamy and polygamy of multi-qubit entanglement for the full range of non-negative power of unified entanglement.

Monogamy and polygamy relations of quantum entanglement characterize the sharing and distribution of entanglement in a multipartite system. Multiqubit entanglement can be characterized entirely with bipartite combinations by saturating the monogamy and polygamy inequalities. In this paper, we tighten monogamy and polygamy constraints for the squared convex-roof extended negativity and its dual measure by employing a genetic algorithm. This evolutionary algorithm optimizes inequality residual functions to improve the monogamy and polygamy relations of these entanglement measures.

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.