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arXiv:2512.24057 · Advanced Quantum Technologies

A new entanglement measure based on the total concurrence

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

quant-phSubmitted Dec 30, 2025• Announced Dec 31, 2025• 4 citations
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Abstract

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 $q$-concurrence ($q\geq 2$) we introduce the total concurrence. A bona fide measure of quantum entanglement is introduced, the $\mathcal{C}^{t}_q$-concurrence ($q \geq 2$), which is based on the total concurrence. Analytical lower bounds for the $\mathcal{C}^{t}_q$-concurrence are derived. In addition, an analytical expression is derived for the $\mathcal{C}^{t}_q$-concurrence in the cases of isotropic and Werner states. Furthermore, the monogamy relations that the $\mathcal{C}^{t}_q$-concurrence satisfies for qubit systems are examined. Additionally, based on the parameterized $α$-concurrence and its complementary dual, the $\mathcal{C}^{t}_α$-concurrence $(0\leqα\leq\frac{1}{2})$ is also proposed.

Journal reference
Adv Quantum Technol.8,no.10(2025):2500149
DOI
10.1002/qute.202500149
Comments
19pages, 12 figures

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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.
Although numerous measures of entanglement have been proposed so far, the calculation of a given faithful entanglement measure is a hard work since it is always involved in some optimization process. It is, therefore, important to estimate the lower bound of a given entanglement measure for an arbitrary quantum state. This results in a subject of intensive mathematical research. In particular, along this line, the lower bounds of concurrence or other measures that are induced from concurrence have been explored a lot. Here, we investigate the lower bounds of two kinds of entanglement monotones, i.e., qqq-concurrence (q>1q>1q>1) and ααα-concurrence (0<α<10<α<10<α<1), or termed the parameterized entanglement monotone together. We obtain, in the light of the informationally complete (NNN, MMM)-positive operator-valued measure [(NNN, MMM)-POVM], the lower bounds for the case of 12<α<1\frac12<α<121​<α<1, 1<q<21<q<21<q<2 for two-qudit states, and the case of 2⩽q<32\leqslant q<32⩽q<3 for two-qubit states. We list several examples which show that the lower bounds based on (NNN, MMM)-POVM outperform that of GSIC-POVM and SIC-POVM, and all these measurement based bounds are better then the ones induced by positive partial transpose (PPT) and realignment criteria in literature. In addition, we obtain an analytical formula of the parameterized entanglement monotone with 12<α<1\frac12<α<121​<α<1 and 1<q<21<q<21<q<2 for the isotropic state.
Abstract It is well known that entanglement is one of the distinctive aspects of the quantum world. The measurement and quantification of entanglement is one of the major problems in understanding and applying this resource. This research aims to investigate entanglement measures and their properties in high-dimensional systems. One such property is entanglement monogamy, viz. quantum correlations are not shareable, different from the classical ones. On the other hand, the study of entanglement depends on the definitions of its measures. Here we examine the applicability of concurrence, an entanglement measure known for its effectiveness in the analysis of qubit entanglement. While the entanglement of two-level systems or qubits is widely understood, its extension to higher dimensions remains challenging. Our research establishes equivalence between different concurrence definitions for bipartite qubit pure states and extends them to qudits. Subsequently, we investigate the monogamous relations of these measurements to explore their characteristics. This approach might enhance our comprehension of quantum traits in qudit systems and advance our understanding of entanglement in broader quantum contexts.
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.
The entanglement content of superpositions of quantum states is investigated based on a measure called concurrence. Given a bipartite pure state in arbitrary dimension written as the quantum superposition of two other such states, we find simple inequalities relating the concurrence of the state to that of its components. We derive an exact expression for the concurrence when the component states are biorthogonal and provide elegant upper and lower bounds in all other cases. For quantum bits, our upper bound is tighter than the previously derived bound [N. Linden et al., Phys. Rev. Lett. 97, 100502 (2006)].