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arXiv:2503.14245 · Advanced Quantum Technologies

Parameterized Bipartite Entanglement Measures and Entanglement Constraints

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

Published Mar 17, 2025• 3 citations

Abstract

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.

DOI: 10.1002/qute.202400707

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Monogamy and polygamy relations are important properties of entanglement, which characterize the entanglement distribution of multipartite systems. We explore monogamy and polygamy relations of entanglement in multipartite systems by using two newly derived parameterized mathematical inequalities, and establish classes of parameterized monogamy and polygamy relations of multiqubit entanglement in terms of concurrence and entanglement of formation. We show that these new parameterized monogamy and poelygamy inequalities are tighter than the existing ones by detailed examples.

We prove exactly that the squared entanglement of formation, which quantifies the bipartite entanglement, obeys a general monogamy inequality in an arbitrary multiqubit mixed state. Based on this kind of exotic monogamy relation, we are able to construct two sets of useful entanglement indicators: the first one can detect all genuine multiqubit entangled states even in the case of the two-qubit concurrence and n-tangles being zero, while the second one can be calculated via quantum discord and applied to multipartite entanglement dynamics. Moreover, we give a computable and nontrivial lower bound for multiqubit entanglement of formation.

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)

-concurrence. Both the standard entanglement concurrence and the recently proposed

q

-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)

-concurrence in the cases of isotropic and Werner states under the constraint

q>1

and

qs\geq 1

. Furthermore, we explore the monogamy properties of the unified

(q,s)

-concurrence for

q\geq 2

,

0\leq s\leq 1

and

1\leq qs\leq 3

, in qubit systems. In addition, we derive an entanglement polygon inequality for the unified

(q,s)

-concurrence with

q\geq 1

and

qs\geq 1

, which manifests the relationship among all the marginal entanglements in any multipartite qudit system.

Tsallis-q entanglement is a bipartite entanglement measure which is the generalization of entanglement of formation for q tending to 1. We first expand the range of q for the analytic formula of Tsallis-q entanglement. For , we prove the monogamy relation in terms of the squared Tsallis-q entanglement for an arbitrary multi-qubit systems. It is shown that the multipartite entanglement indicator based on squared Tsallis-q entanglement still works well even when the indicator based on the squared concurrence loses its efficacy. We also show that the μ-th power of Tsallis-q entanglement satisfies the monogamy or polygamy inequalities for any three-qubit state.

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