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arXiv:2406.19080

G_q-concurrence and entanglement constraints in multiqubit systems

Hui Li, Ting Gao, Fengli Yan

Published Jun 27, 2024• 0 citations
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Abstract

In this paper, we introduce a category of one-parameter bipartite entanglement quantifiers, termed $G_q$-concurrence ($q>1$), and show rigorously that they satisfy all the axiomatic conditions of an entanglement measure and can be considered as a generalization of concurrence. In addition, we establish an analytic formula relating $G_q$-concurrence to concurrence for $1

DOI
10.48550/arxiv.2406.19080

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In this paper, we introduce a category of one-parameter bipartite entanglement quantifiers, termed GqG_qGq​-concurrence (q>1q>1q>1), and show rigorously that they satisfy all the axiomatic conditions of an entanglement measure and can be considered as a generalization of concurrence. In addition, we establish an analytic formula relating GqG_qGq​-concurrence to concurrence for $1
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.
Entanglement concurrence is an important bipartite entanglement measure that has found wide applications in quantum technologies. In this work, inspired by unified entropy, we introduce a two-parameter family of entanglement measures, referred to as the unified (q,s)(q,s)(q,s)-concurrence. Both the standard entanglement concurrence and the recently proposed qqq-concurrence emerge as special cases within this family. By combining the positive partial transposition and realignment criteria, we derive an analytical lower bound for this measure for arbitrary bipartite mixed states, revealing a connection to strong separability criteria. Explicit expressions are obtained for the unified (q,s)(q,s)(q,s)-concurrence in the cases of isotropic and Werner states under the constraint q>1q>1q>1 and qs≥1qs\geq 1qs≥1. Furthermore, we explore the monogamy properties of the unified (q,s)(q,s)(q,s)-concurrence for q≥2q\geq 2q≥2, 0≤s≤10\leq s\leq 10≤s≤1 and 1≤qs≤31\leq qs\leq 31≤qs≤3, in qubit systems. In addition, we derive an entanglement polygon inequality for the unified (q,s)(q,s)(q,s)-concurrence with q≥1q\geq 1q≥1 and qs≥1qs\geq 1qs≥1, which manifests the relationship among all the marginal entanglements in any multipartite qudit system.
Characterizing entanglement shareability and distribution in NNN-partite systems
Exploring the shareability and distribution of entanglement possesses fundamental significance in quantum information tasks. In this paper, we demonstrate that the square of bipartite entanglement measures GqG_qGq​-concurrence, which is the generalization of concurrence, follows a set of hierarchical monogamy relations for any NNN-qubit quantum state. On the basis of these monogamy inequalities, we render two kinds of hierarchical indicators that exhibit evident advantages in the capacity of witnessing entanglement. Moreover, we show an analytical relation between GqG_qGq​-concurrence and concurrence in 2⊗d2\otimes d2⊗d systems. Furthermore, we rigorously prove that the monogamy property of squared GqG_qGq​-concurrence is superior to that of squared concurrence in 2⊗d2⊗d3⊗⋯⊗dN2\otimes d_2\otimes d_3\otimes\cdots\otimes d_N2⊗d2​⊗d3​⊗⋯⊗dN​ systems. In addition, several concrete examples are provided to illustrate that for multilevel systems, the squared GqG_qGq​-concurrence satisfies the monogamy relation, even if the squared concurrence does not. These results better reveal the intriguing characteristic of multilevel entanglement and provide critical insights into the entanglement distribution within multipartite quantum systems.
We show that the restricted shareability and distribution of multiqubit entanglement can be characterized by Tsallis-qqq entropy. We first provide a class of bipartite entanglement measures named Tsallis-qqq entanglement, and provide its analytic formula in two-qubit systems for $1\ensuremath{\leqslant}q\ensuremath{\leqslant}4$. For $2\ensuremath{\leqslant}q\ensuremath{\leqslant}3$, we show a monogamy inequality of multiqubit entanglement in terms of Tsallis-qqq entanglement, and we also provide a polygamy inequality using Tsallis-qqq entropy for $1\ensuremath{\leqslant}q\ensuremath{\leqslant}2$ and $3\ensuremath{\leqslant}q\ensuremath{\leqslant}4$.
Quantum entanglement is a crucial resource in quantum information processing, advancing quantum technologies. The greater the uncertainty in subsystems' pure states, the stronger the quantum entanglement between them. From the dual form of qqq-concurrence (q≥2q\geq 2q≥2) we introduce the total concurrence. A bona fide measure of quantum entanglement is introduced, the Cqt\mathcal{C}^{t}_qCqt​-concurrence (q≥2q \geq 2q≥2), which is based on the total concurrence. Analytical lower bounds for the Cqt\mathcal{C}^{t}_qCqt​-concurrence are derived. In addition, an analytical expression is derived for the Cqt\mathcal{C}^{t}_qCqt​-concurrence in the cases of isotropic and Werner states. Furthermore, the monogamy relations that the Cqt\mathcal{C}^{t}_qCqt​-concurrence satisfies for qubit systems are examined. Additionally, based on the parameterized ααα-concurrence and its complementary dual, the Cαt\mathcal{C}^{t}_αCαt​-concurrence (0≤α≤12)(0\leqα\leq\frac{1}{2})(0≤α≤21​) is also proposed.
We investigate parametrized multipartite entanglement measures from the perspective of kkk nonseparability in this paper. We present two types of entanglement measures in nnn-partite systems, $q\text{\ensuremath{-}}k\text{\ensuremath{-}}\mathrm{ME}$ concurrence $(q>1,\phantom{\rule{0.28em}{0ex}}2\ensuremath{\le}k\ensuremath{\le}n)$ and $\ensuremath{\alpha}\text{\ensuremath{-}}k\text{\ensuremath{-}}\mathrm{ME}$ concurrence $(0\ensuremath{\le}\ensuremath{\alpha}<1,\phantom{\rule{0.28em}{0ex}}2\ensuremath{\le}k\ensuremath{\le}n)$, which unambiguously detect all kkk-nonseparable states in arbitrary nnn-partite systems. Rigorous proofs show that the proposed kkk-nonseparable measures satisfy all the requirements for being an entanglement measure including the entanglement monotone, strong monotone, convexity, vanishing on all kkk-separable states, and being strictly greater than zero for all kkk-nonseparable states. In particular, the $q\text{\ensuremath{-}}\mathrm{2}\text{\ensuremath{-}}\mathrm{ME}$ concurrence and $\ensuremath{\alpha}\text{\ensuremath{-}}2\text{\ensuremath{-}}\mathrm{ME}$ concurrence, renamed as $q\text{\ensuremath{-}}\mathrm{GME}$ concurrence and $\ensuremath{\alpha}\text{\ensuremath{-}}\mathrm{GME}$ concurrence, respectively, are two kinds of genuine entanglement measures corresponding to the case where the systems are divided into bipartition (k=2)(k=2)(k=2). The lower bounds of the two classes kkk-nonseparable measures are obtained by employing the approach that takes into account the permutationally invariant part of a quantum state. Furthermore, the relations between $q\text{\ensuremath{-}}n\text{\ensuremath{-}}\mathrm{ME}$ concurrence $(\ensuremath{\alpha}\text{\ensuremath{-}}n\text{\ensuremath{-}}\mathrm{ME}$ concurrence) and global negativity are established. In addition, we discuss the degree of separability and elaborate on an effective detection method with concrete examples. Moreover, we compare the $q\text{\ensuremath{-}}\mathrm{GME}$ concurrence defined by us to other genuine entanglement measures.