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arXiv:2507.12680 · Physical Review A

Multiqubit monogamy relations beyond shadow inequalities

Eduardo Serrano-Ensástiga, Olivier Giraud, John Martin

Published Nov 18, 2025• 0 citations

Abstract

Multipartite quantum systems are subject to monogamy relations that impose fundamental constraints on the distribution of quantum correlations between subsystems. These constraints can be studied quantitatively through sector lengths, defined as the average value of $m$-body correlations, which have applications in quantum information theory and coding theory. In this work, we derive a set of monogamy inequalities that complement the shadow inequalities, enabling a complete characterization of the numerical range of sector lengths for systems with $N\leq 5$ qubits in a pure state. This range forms a convex polytope, facilitating the efficient extremization of key physical quantities, such as the linear entropy of entanglement and the quantum shadow enumerators, by a simple evaluation at the polytope vertices. For larger systems ($N\geq 6$), we highlight a significant increase in complexity that neither our inequalities nor the shadow inequalities can fully capture.

DOI: 10.1103/9fkf-hm8l

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Multipartite quantum systems are subject to monogamy relations that impose fundamental constraints on the distribution of quantum correlations between subsystems. These constraints can be studied quantitatively through sector lengths, defined as the average value of

m

-body correlations, which have applications in quantum information theory and coding theory. In this work, we derive a set of monogamy inequalities that complement the shadow inequalities, enabling a complete characterization of the numerical range of sector lengths for systems with

N\leq 5

qubits in a pure state. This range forms a convex polytope, facilitating the efficient extremization of key physical quantities, such as the linear entropy of entanglement and the quantum shadow enumerators, by a simple evaluation at the polytope vertices. For larger systems (

N\geq 6

), we highlight a significant increase in complexity that neither our inequalities nor the shadow inequalities can fully capture.

Abstract Correlations in multiparticle systems are constrained by restrictions from quantum mechanics. A prominent example for these restrictions are monogamy relations, limiting the amount of entanglement between pairs of particles in a three-particle system. A powerful tool to study correlation constraints is the notion of sector lengths. These quantify, for different k, the amount of k-partite correlations in a quantum state in a basis-independent manner. We derive tight bounds on the sector lengths in multi-qubit states and highlight applications of these bounds to entanglement detection, monogamy relations and the n-representability problem. For the case of two- and three qubits we characterize the possible sector lengths completely and prove a symmetrized version of strong subadditivity for the linear entropy.

Abstract Quantum entanglement in multipartite systems exhibits a unique restriction on its distribution among subsystems, a fundamental property known as the monogamy of entanglement. Traditionally, monogamy has been quantified for measures such as concurrence through summation-form inequalities. Recently, it has been discovered that concurrence also obeys monogamy constraints expressed via product-form inequalities. In this work, we establish enhanced monogamy inequalities for the ν th power of concurrence (with ν ⩾ 2 ). We demonstrate, through explicit examples, that these new inequalities provide tighter bounds than existing product-form results. Furthermore, we show that our findings remain applicable to monogamy relations in high-dimensional systems.

It is a well-known fact that both quantum entanglement and nonlocality (implied by the violation of Bell inequalities) constitute quantum correlations that cannot be arbitrarily shared among subsystems. They are both monogamous, albeit in a different fashion. In the present contribution we focus on nonlocality monogamy relations such as the Toner-Verstraete, the Seevinck, and a derived monogamy inequality for three parties and compare them with multipartite nonlocality measures for the whole set of pure states distributed according to the Haar measure. In this numerical endeavor, we also see that, although monogamy relations for nonlocality cannot exist for more than three parties, in practice the exploration of the whole set of states for different numbers of qubits will return effective bounds on the maximum value of all bipartite Bell violations among subsystems. Hence, we shed light on the effective nonlocality monogamy bounds in the multiqubit case.

We study the distribution of quantum correlations characterized by monogamy relations in multipartite systems. By using the Hamming weight of the binary vectors associated with the subsystems, we establish a class of monogamy inequalities for multiqubit entanglement based on the

\alpha

th (

\alpha\geq 2

) power of concurrence, and a class of polygamy inequalities for multiqubit entanglement in terms of the

\beta

th (

0\leq \beta\leq2

) power of concurrence and concurrence of assistance. Moveover, we give the monogamy and polygamy inequalities for general quantum correlations. Application of these results to quantum correlations like squared convex-roof extended negativity (SCREN), entanglement of formation and Tsallis-

q

entanglement gives rise to either tighter inequalities than the existing ones for some classes of quantum states or less restrictions on the quantum states. Detailed examples are presented.

The monogamy and polygamy properties of quantum entanglement characterize fundamental constraints on the distribution of entanglement in multipartite quantum systems. In this paper, we investigate tighter monogamy and polygamy relations for multipartite entanglement. By establishing a new mathematical inequality, we derive a family of improved monogamy and polygamy inequalities for tripartite quantum systems and further extend these results to general multipartite systems. Comparisons with existing results show that the obtained bounds are tighter. Illustrative examples are provided to demonstrate the effectiveness of the proposed relations.