Quantum entanglement has been recognized as the most important resource in many quantum information processing tasks [1–3]. One of the essential differences between quantum entanglement and classical correlation is that quantum entanglement cannot be freely shared among many parties. For example, in a multi-party state, if two parties are maximally entangled, then none of them can share entanglement with any part of the rest of the system. This restriction of entanglement shareability among multi-party systems is known as the monogamy of entanglement (MOE) [4, 5].

Since the MOE restricts on the amount of information that an eavesdropper could potentially obtain the secret key extraction, it is a crucial property that guarantees quantum key distribution secure [3, 4, 6]. MOE also has many fundamental applications in other areas of physics, including classification of quantum states [7–9], no-signaling theories [10], condensed-matter physics [11–13], statistical physics [14] and even black-hole physics [15].

An important basic question in the study of MOE is to determine whether a given entanglement measure is monogamous. Usually, there are several ways to define the monogamy property of entanglement measure. Originally, a monogamy relation of entanglement measure E is quantitatively displayed as an inequality of the following form E ρ A | B C ≥ E ρ A | B + E ρ A | C (1) where E(ρ _A|BC ) is an entanglement measure quantifying the degree of entanglement between subsystems A and BC, and E(ρ _A|B ) (E(ρ _A|C )) is the bipartite entanglement between A and B (A and C) (See Figure 1 for a graphical representation). This inequality means that the sum of entanglement between A and each of the other parties B or C cannot exceed the entanglement between A and BC. Using squared concurrence (SC) as an entanglement measure, Coffman, Kundu and Wootters (CKW) proved the first monogamy inequality for three qubit states [16] which we shall refer to as the CKW inequality. The CKW inequality was later generalized by Osborne and Verstraete for arbitrary multi-qubit system. It should be noticed that the entanglement of formation (EOF), when not squared, does not obey the monogamy relation given by Eq. 1 [17]. Besides SC, it was further proven that similar monogamy inequality can be established for the squared entanglement of formation (SEF) [18, 19], Rényi-α entanglement (RαE) [20], the squared Rényi-α entanglement (SRαE) [21], Tsallis-q entanglement (TqE) [22], the squared Tsallis-q entanglement (STqE) [23, 24], and unified-(q, s) entanglement [25]. The establishment of these inequalities depends on monogamy inequality of SC. In this sense, these inequalities can be classified into concurrence-based monogamy relations. For high-dimensional systems, it has been shown that monogamy inequality of SC can be violated due to the existence of counterexamples [26, 27]. At present, it is still unclear whether other concurrence-based monogamy relations hold in high-dimensional systems.

Another way to generalize the CKW inequality is using negativity [28] or convex-roof extended negativity (CREN) [26], and CREN is a good candidate for MOE without any known example violating its CKW-type inequality even in higher-dimensional systems [26]. More recently, Gao et al [29] established a class of CKW-type monogamy inequalities based on the μ-th power of logarithmic negativity and logarithmic convex-roof extended negativity (LCREN). The CKW-type inequality was also generalized to other entanglement measures, such as squashed entanglement [30, 31], one-way distillable entanglement [30] and continuous-variable entanglement [32–34]. Among them, the squashed entanglement and one-way distillable entanglement fulfill Eq. 1 for arbitrary dimensional systems. Furthermore, other types of monogamy relations were presented in Refs. [35–51]. In particular, Regula et al [52, 53] have proposed a set of strong monogamy (SM) inequalities sharpening the conventional CKW-type inequality. For the validity of SM inequality, an extensive numerical evidence was presented for four qubit pure states together with analytical proof for some cases of multi-qubit systems.

On the other hand, the polygamous property can be regarded as another kind of entanglement constraints in multi-qubit systems, and Gour et al [54] established the first dual polygamy inequality for multi-qubit systems using concurrence of Assistance (CoA). Subsequently, polygamy inequalities was generalized into various entanglement measures [55–64].

However, the main problem with the definition of monogamy in Eq. 1 is that their validity is not universal, but depends on the specific choice of E. Moreover, several important measures of entanglement do not satisfy the relation (1). Therefore, the summation in the right-hand sides of Eq. 1 is only a convenient choice and not a necessity. To overcome this problem, one attempt is to replace Eq. 1 with a family of monogamy relations of the form E(ρ _A|BC ) ≥ f (E (ρ _A|B ), E (ρ _A|C )), where f is some function of two variables that satisfies certain conditions [65]. Another approach is based on the definition of monogamy relations without inequalities introduced in Ref. [66]. According to this definition, we can reproduce the traditional monogamy relations similar to Eq. 1 by replacing E with E ^α for some α > 0.

In this review, we focus on introducing theoretical advances on monogamy of quantum entanglement but not include the topic of quantum correlations, see Ref. [67] for the summary of recent advances in monogamy of quantum correlations. In Sec.II, we firstly review the standard CKW-type monogamy inequalities in terms of various entanglement measures. In Sec.III, we then introduce some other types of monogamy inequalities which extend and sharpen the existing ones. In Sec.IV, we focus on reviewing the dual polygamy inequalities for multi-qubit systems. The new definitions of MOE are discussed in Sec.V. Finally, in Sec. VI, we give some concluding remarks.

In this section we briefly review the CKW-type monogamy inequality and we divide them into three categories according to different entanglement measures.

In this section we focus on reviewing some generalized version of monogamy relation. It is well known that tightening the monogamy inequalities can provide a precise characterization of the entanglement sharing and distribution in multipartite systems, thus it is important to find tight monogamy inequalities for various entanglement measure. We first consider the strong monogamy (SM) inequality introduced by Regula et al [52]. For an n-qubit pure state |ψ⟩, it was conjectured that the following inequality holds: τ | ψ 〉 A 1 | A 2 … A n ≥ ∑ m = 2 n − 1 ∑ j ⃗ m τ ρ A 1 | A j 1 m | … | A j m − 1 m m 2 (10) where the index vector j ⃗ m = ( j 1 m , … , j m − 1 m ) spans all the ordered subsets of the index set {2, …, n} with m − 1 distinct elements, and τ ( ρ A 1 | A j 1 m | … | A j m − 1 m ) is defined as τ ρ A 1 | A j 1 m | … | A j m − 1 m = min p h , | ψ h 〉 ∑ h p h τ | ψ h 〉 A 1 | A j 1 m | … | A j m − 1 m 2 (11) with the minimization over all possible pure state decompositions ρ A 1 A j 1 m … A j m − 1 m = ∑ h p h | ψ h 〉 A 1 A j 1 m … A j m − 1 m 〈 ψ h | . The right side of Eq. 10 appears in between the both side of the n-qubit CKW inequality, therefore it is a stronger inequality. The difference between left and right hand side of Eq. 10 is defined as n-tangle which is a quantifier of genuinely entanglement shared among n-partites. In fact, this inequality comes from the strong monogamy inequality of continuous variable Gaussian states introduced in Ref. [34]. Eq.10 reduces to normal three-qubit CKW inequality for n = 3. For a four-qubit state |ψ⟩, the SM inequality can be written as: τ A 1 | A 2 A 3 A 4 ≥ τ A 1 | A 2 ( 2 ) + τ A 1 | A 3 ( 2 ) + τ A 1 | A 4 ( 2 ) + [ τ A 1 | A 2 | A 3 ( 3 ) ] 3 / 2 + [ τ A 1 | A 3 | A 4 ( 3 ) ] 3 / 2 + [ τ A 1 | A 2 | A 4 ( 3 ) ] 3 / 2 .

For the validity of SM inequality, an extensive numerical evidence has been presented for four-qubit state together with analytical proof for some cases of multi-qubit state. Another generalization of SM inequality in terms of squared convex roof extended negativity (SCREN) has been presented by Choi and Kim [84], and it is shown that the superposition of the generalized W-class states and vacuum (GWV) states satisfy the SM inequality based on SCREN. In Ref. [85], Kim further proved that SM inequality holds good even in a class of higher dimensional state where the original SM inequality fails.

Next we present some other generalized version of monogamy relation. In Ref. [42], Jin et al have investigated tighter entanglement monogamy relations related to C ^μ and E ^μ for μ ≥ 2 and μ ≥ 2 , respectively. Using the Hamming weight of the binary vector related with the distribution of subsystems, Kim [44, 45] established a class of monogamy inequalities of multi-qubit entanglement based on the μ-th power of unified-(q, s) entanglement. Other approaches to construct tighter monogamy inequalities in terms of various entanglement measures were also proposed in Ref. [43]. Moreover, Oliveira et al [38] proposed a monogamy relation in the linear version for a three-qubit system, which was proved by Liu et al [41]. In. Reference [49], Shi et al generalized the multi-linear monogamy relation for a multi-qubit system in terms of EOF and concurrence.

In previous section, we have reviewed MOE which reveals the limited shareability of multiparty quantum entanglement, the assisted entanglement was shown to have a dually monogamous property in multi-party quantum systems, i.e., polygamy of entanglement (PoE). PoE is mathematically characterized as the polygamy inequality E a ρ A | B C ≤ E a ρ A | B + E a ρ A | C (12) for a three-party quantum state and E _a (ρ _A|BC ) denotes the bipartite assisted entanglement in the partition A|BC. In contrast to monogamy inequality, which provides an upper bound on the bipartite shareability of entanglement in multi-party systems, the polygamy inequality in Eq. 12 provides a lower bound for distribution of bipartite entanglement in multi-party systems.

The polygamy inequality in (12) was first proposed in three-qubit systems. For a three-qubit pure state |ψ⟩ _ABC , the following inequality holds. τ | ψ 〉 A | B C ≤ τ a ρ A | B + τ a ρ A | C (13) where τ(|ψ⟩ _A|BC ) is the tangle of the pure state |ψ⟩ _A|BC between A and BC, and τ _a (ρ _AB ) = max ∑ _i p _i τ(|ψ⟩ _AB ) is the tangle of assistance of ρ _AB = Tr _C |ψ⟩ _ABC ⟨ψ| with the maximum taken over all possible pure-state decomposition ρ _AB = ∑ _i p _i |ψ _i ⟩ _AB ⟨ψ _i |. This inequality was generalized into multi-qubit system τ a ( ρ A 1 | A 2 … A n ) ≤ ∑ i = 2 n τ a ( ρ A 1 | A i ) for an arbitrary multi-qubit mixed state ρ A 1 A 2 … A n and its reduced density matrices ρ A 1 A i with i = 2, …, n. In Refs. [57–63], polygamy inequalities were also established for other entanglement measures.

For polygamy inequality beyond qubits, it was shown that von Neumann entropy can be used to establish a polygamy inequality of three-party quantum system [55]. We have E(|ψ⟩ _A|BC ) ≤ E _a (ρ _A|B ) + E _a (ρ _A|C ) for any three-party pure state |ψ⟩ _A|BC , where E(|ψ⟩ _A|BC ) = S (ρ _A ) = − Tr ρ _A ln ρ _A is the von Neumann entropy of entanglement between A and BC, and E _a (ρ _AB ) is the entanglement of assistance of ρ _AB defined by E _a (ρ _AB ) = max ∑ _i p _i E (|ψ _i ⟩ _AB ), where the maximization is taken over all possible pure state decompositions of ρ _AB . In Ref. [57], a general polygamy inequality of multipartite quantum entanglement was established for arbitrary-dimensional quantum states ρ A 1 A 2 … A n . Recently, Kim [64] further proposed a class of weighted polygamy inequalities of multipartite entanglement in arbitrary-dimensional quantum systems.

In this section we present some alternative methods to define MOE. The main problem with CKW inequalities is that their validity is not universal since several important measures of entanglement do not satisfy Eq. 1. In Ref. [65], Lancien et al raise the following question: Should any entanglement measure be monogamous in a CKW-type sense? In fact, the summation in the right-hand side of Eq. 1 is only a convenient choice but not a necessity. For example, it has been shown that if E does not satisfy the CKW monogamy inequalities, it is still possible to find a positive μ such that E ^μ satisfies the Eq. 1. Inspired by this idea, one attempt is to replace Eq. 1 with the following generalized monogamy relation: E ρ A | B C ≥ f E ρ A | B , E ρ A | C (14) where f : R + × R + → R + is a function independent on the dimension of the underlying Hilbert space, and it is continuous, and satisfies the condition f (x, y) ≥ max (x, y). This requirement comes from the fact that E is an entanglement monotone which is nonincreasing under partial traces. The CKW-type monogamy inequality can be recovered for the particular choice f (x, y) = x + y. It has been proved that the entanglement of formation E _F and the relative entropy of entanglement E _R , as well as their regularizations, cannot satisfy the new definition in Eq. 14. In addition, any additive entanglement measure which is geometrically faithful in the sense of being-lower bounded by a quantity with a sub-polynomial dimensional dependence on the antisymmetric state, cannot be monogamous. Nevertheless, we can recover the monogamy relation (14) if we allow the function to be dimension-dependent. For example, it has been shown that the non-trivial dimension-dependent monogamy relations can be established for E _F and E R ∞ in any finite dimension.

Another approach to define MOE is given in terms of an equality, as opposed to the traditional monogamy inequality. According to the definition in Ref. [66], a measure of entanglement E is monogamous if for any ρ _ABC ∈ S _ABC that satisfies E ρ A | B C = E ρ A B (15) we have that E (ρ _AC ) = 0. With respect to this definition, if the entanglement between system A and the composite system BC is as much as the entanglement that system A shares with subsystem B, then it is left with no entanglement to share with C. If E satisfies Eq.1, then any state ρ _A|BC that satisfies the definition (15) must E (ρ _AC ) = 0. Therefore, the condition in Eq. 1 is stronger than the definition in Eq. 15. This new definition is consistent with Eq.1 and it has been shown that they are equivalent if and only if there exists 0 ≤ μ ≤ ∞ such that E μ ρ A | B C ≥ E μ ρ A B + E μ ρ A C (16) for all ρ _ABC ∈ S _ABC with fixed dim H ABC = d < ∞ . It is to be noted that Eq. 16 is not a special case of Eq. 14 because the exponent factor μ depends on the dimension d, whereas the function f defined in Eq. 14 is universal and does not depend on the dimension. By adopting the new definition of monogamy without inequalities, Guo and Gour [86] further proved the monogamy of EOF on mixed tripartite states.

The subject of MOE has attracted extensive research interest in the past two decades. In this review, we present the theoretical developments in the field of MOE, as well as some new definitions of MOE. Despite the rapid progress in recent years, there are still many challenging problems to be solved and we briefly list them as follows.

First, most previous studies of MOE are focussed on the multi-qubit systems. But our knowledge of MOE in the high-dimensional case is still very limited and there are few results on MOE for high-dimensional systems [84, 87–90]. In [91] Kim et al. proved that the n-qudit generalized W-class (GW) states satisfy the monogamy inequality in terms of the SC. Recently, Shi et al. presented in [47] new monogamy and polygamy relations for n-qudit generalized W-class states and vacuum (GWV) states in terms of the TqE. In [59, 92] the authors investigated the monogamy and polygamy relations for the GWV states in high-dimensional systems in terms of the RαE. Except for squashed entanglement and one-way distillable entanglement, monogamy relations for various entanglement measures only hold for some special high-dimensional states. The difficulties are caused by the entanglement properties in higher-dimensional systems are hardly known so far and there is no analytical formula for calculating the high-dimensional entanglement measure. Thus, it is important to explore monogamy inequality for general high-dimensional states in terms of various entanglement measures.

Second, the validity of the traditional monogamy inequality is not universal and several important measures of entanglement do not satisfy Eq. 1. However, MOE has been mathematically proven to be a valid property of entanglement in the n-shareability sense [5]. In order to solve this problem, two new definitions of MOE have been proposed. One definition is to replace the right-hand side of Eq. 1 with a universal function f independent of dimension d [65]. This definition [65] is somewhat artificial and some important entanglement measure such as EOF and relative entropy of entanglement cannot satisfy the new monogamy inequality. Another approach is to define MOE with an equality rather than inequality. It was shown that this definition is consistent with the traditional notion of MOE if the measure E is replaced by E ^α for some exponent α > 0. According to this new definition, EOF are monogamous on mixed tripartite systems. It supports that monogamy is a property of entanglement and not of some particular functions quantifying entanglement. Although the second definition of MOE seems more natural in physical, there is no mathematical proof of which definition is better, and we do not know whether there are entangled states that violate the second definition. Therefore, extensive efforts are still needed to investigate the relationship between these two definitions. Moreover, by adopting these new definitions, it is necessary to explore whether many important measures of entanglement are monogamous.

Third, different attempts have been made to construct a sharper version of monogamy inequality. In particular, Regula et al [52] have proposed a set of SM inequalities in terms of concurrence. Although an extensive numerical evidence has been presented for four qubit systems, an analytical proof of SM conjecture is still desired. It would also be interesting to answer whether there are counterexamples that violate the SM inequality for more qubits. This conjecture can be further extended to negativity and SCREN for some classes of states. Future directions may include the study of SM inequalities for other entanglement monotones such as squashed entanglement.

In summary, we have reviewed the mathematical foundation of MOE but not include many problems concerning real physical phenomena, and monogamy is being considered in the study of these problems. For example, it was argued that the black hole evaporation is incompatible with our understanding of MOE [93]. Thus, it is desirable for us to have a sufficient understanding of monogamy further.