We investigate possible generalizations of the Coffman-Kundu-Wootters monogamy inequality to four qubits, accounting for multipartite entanglement in addition to the bipartite terms. We show that the most natural extension of the inequality does not hold in general, and we describe the violations of this inequality in detail. We investigate alternative ways to extend the monogamy inequality to express a constraint on entanglement sharing valid for all four-qubit states, and perform an extensive numerical analysis of randomly generated four-qubit states to explore the properties of such extensions.
In this paper, we present some monogamy relations of multiqubit quantum entanglement in terms of the βth power of concurrence, entanglement of formation and convex-roof extended negativity. These monogamy relations are proved to be tighter than the existing ones, together with detailed examples showing the tightness.
Monogamy of quantum correlations provides a way to study restrictions on their sharability in multiparty systems. We find the critical exponent of these measures, above which randomly generated multiparty pure states satisfy the usual monogamy relation, and show that the critical power decreases with the increase in the number of parties. For three-qubit pure states, we detect that W-class states are more prone to being nonmonogamous as compared to the GHZ-class states. We also observe a different criticality in monogamy power up to which random pure states remain nonmonogamous. We prove that …
We investigate the distribution of bipartite and multipartite entanglement in multiqubit states. In particular we define a set of monogamy inequalities sharpening the conventional Coffman-Kundu-Wootters constraints, and we provide analytical proofs of their validity for relevant classes of states. We present extensive numerical evidence validating the conjectured strong monogamy inequalities for arbitrary pure states of four qubits.
The monogamy is a fundamental property of Bell nonlocality and contextuality. In this article, we studied the $n$-cycle noncontextual inequalities and generalized CHSH inequalities in detail and found the sufficient conditions for those inequalities to be hold. According to those conditions, we provide several kind of tradeoff relations: monogamy of generalized Bell inequalities in non-signaling framework, monogamy of cycle type noncontextual inequalities and monogamy between Bell inequality and noncontextual inequality in general no-disturbance framework. At last, some generic tradeoff relati…
We provide a fine-grained definition for monogamous measure of entanglement that does not invoke any particular monogamy relation. Our definition is given in terms an equality, as oppose to inequality, that we call the "disentangling condition". We relate our definition to the more traditional one, by showing that it generates standard monogamy relations. We then show that all quantum Markov states satisfy the disentangling condition for any entanglement monotone. In addition, we demonstrate that entanglement monotones that are given in terms of a convex roof extension are monogamous if they a…
Characterizing trade-offs between simultaneous violations of multiple Bell inequalities in a large network of qubits is computationally demanding. We propose a graph-theoretic approach to efficiently produce Bell monogamy relations in arbitrary arrangements of qubits. All the relations obtained for bipartite Bell inequalities are tight and leverage only a single Bell monogamy relation. This feature is unique to bipartite Bell inequalities, as we show that there is no finite set of such elementary monogamy relations for multipartite inequalities. Nevertheless, many tight monogamy relations for …
We derive fundamental constraints for the Schur complement of positive matrices, which provide an operator strengthening to recently established information inequalities for quantum covariance matrices, including strong subadditivity. This allows us to prove general results on the monogamy of entanglement and steering quantifiers in continuous variable systems with an arbitrary number of modes per party. A powerful hierarchical relation for correlation measures based on the log-determinant of covariance matrices is further established for all Gaussian states, which has no counterpart among qua…
Markovianity lies at the heart of communication problems. This in turn makes the information-theoretic characterization of Markov processes worthwhile. Data processing inequalities are ubiquitous in this sense, assigning necessary conditions for all Markov processes. We address here the problem of the information-theoretic analysis of constraints on Markov processes in the quantum regime. We show the existence of a novel class of quantum data processing inequalities called here quantum Markov monogamy inequalities. This new class of necessary conditions on quantum Markov processes is inspired …
In this paper, we present some generalized monogamy inequalities based on negativity and convex-roof extended negativity (CREN). These monogamy relations are satisfied by the negativity of $N$-qubit quantum systems $ABC_1\cdots C_{N-2}$, under the partitions $AB|C_1\cdots C_{N-2}$ and $ABC_1|C_2\cdots C_{N-2}$. Furthermore, the $W$-class states are used to test these generalized monogamy inequalities.
We examine the various properties of the three four-qubit monogamy relations, all of which introduce the power factors in the three-way entanglement to reduce the tripartite contributions. On the analytic ground as much as possible we try to find the minimal power factors, which make the monogamy relations hold if the power factors are larger than the minimal powers. Motivated to the three-qubit monogamy inequality we also examine whether those four-qubit monogamy relations provide the SLOCC-invariant four-way entanglement measures or not. Our analysis indicate that this is impossible provided…
Physical principles constraints the way nonlocal correlations can be distributed among distant parties in a Bell-type experiment. These constraints are usually expressed by monogamy relations that bound the amount of Bell inequality violation observed by a set of parties by the violation observed by a different set of parties. Here we show that the no-signaling principle yields a simple and tight monogamy relations for an important class of bipartite and multipartite Bell inequalities. We also link these trade-offs to the guessing probability--a key quantity in the device-independent informati…
We show that the perfect commutation graph is the sufficient tight condition for admitting the noncontextual description of each observable set satisfying it in the yes-no question scenario. With this condition, we propose a method for proving the monogamy relation between two informationtheoretic contextuality inequalities by decomposing the total commutation graph into perfect subgraphs. The results offer a powerful tool to investigate the contextuality and to understand quantum information theory. This theoretical work can be experimentally verified in current laboratorial technology.
We propose a method to detect genuine quantum correlation for multi-qubit pure states. We then derive a complementarity relations for pure quantum states of N qubits. We prove that in all many-qubit systems there exist strict monogamy laws for quantum correlations. On the other hand, it is known that the entanglement monogamy equality proposed by Coffman, Kundu, and Wootters is in general not true for multiqubit states. Inducing from the CKW equality, we find a proper form of entanglement monogamy equality for arbitrary quantum states. The total quantum correlation of qubit k with the remainin…
We investigate the monogamy relations related to the concurrence and the entanglement of formation. General monogamy inequalities given by the αth power of concurrence and entanglement of formation are presented for N-qubit states. The monogamy relation for entanglement of assistance is also established. Based on these general monogamy relations, the residual entanglement of concurrence and entanglement of formation are studied. Some relations among the residual entanglement, entanglement of assistance, and three tangle are also presented.
We provide a strong evidence for strong monogamy inequality of multi-qubit entanglement recently proposed in [B. Regula {\em et al.}, Phys. Rev. Lett. {\bf 113}, 110501 (2014)]. We consider a large class of multi-qubit generalized W-class states, and analytically show that the strong monogamy inequality of multi-qubit entanglement is saturated by this class of states.
In this paper, we investigate the generalised monogamy inequalities of convex-roof extended negativity (CREN) in multi-level systems. The generalised monogamy inequalities provide the upper and lower bounds of bipartite entanglement, which are obtained by using CREN and the CREN of assistance (CRENOA). Furthermore, we show that the CREN of multi-qubit pure states satisfies some monogamy relations. Additionally, we test the generalised monogamy inequalities for qudits by considering the partially coherent superposition of a generalised W-class state in a vacuum and show that the generalised mon…
Here we investigate the monogamy relations for the generalized W class. Monogamy inequalities for the generalized W class in terms of the $β$-th ($β\in (0,2)$) power of the concurrence, the concurrence of assistance and the negativity are presented. At last, under some restricted conditions of the generalized W class, we present stronger monogamy inequalities.
A striking result from nonrelativistic quantum mechanics is the monogamy of entanglement, which states that a particle can be maximally entangled only with one other party, not with several ones. While there is the exact quantitative relation for three qubits and also several inequalities describing monogamy properties it is not clear to what extent exact monogamy relations are a general feature of quantum mechanics. We prove that in all many-qubit systems there exist strict monogamy laws for quantum correlations. They come about through the curious relation between the nonrelativistic quantum…
We investigate monogamy relations related to quantum entanglement for $n-$qubit quantum systems. General monogamy inequalities are presented to the $β$th $(β\in(0,2))$ power of concurrence, negativity and the convex-roof extended negativity, as well as the $β$th $(β\in(0,\sqrt{2}))$ power of entanglement of formation. These monogamy relations are complementary to the existing ones with different regions of parameter $β$. In additions, new monogamy relations are also derived which include the existing ones as special cases.
We propose the square of convex-roof extended negativity(SCREN) as a powerful candidate to characterize strong monogamy of multi-party quantum entanglement. We first provide a strong monogamy inequality of multi-party entanglement using SCREN, and show that the tangle-based multi-qubit strong monogamy inequality can be rephrased by SCREN. We further show that SCREN strong monogamy inequality is still true for the counterexamples that violate tangle-based strong monogamy inequality in higher-dimensional quantum systems rather than qubits. We also analytically show that SCREN strong monogamy ine…
Monogamy relations characterize the distributions of entanglement in multipartite systems. We investigate monogamy relations related to the concurrence $C$ and the entanglement of formation $E$. We present new entanglement monogamy relations satisfied by the $α$-th power of concurrence for all $α\geq2$, and the $α$-th power of the entanglement of formation for all $α\geq\sqrt{2}$. These monogamy relations are shown to be tighter than the existing ones.
Monogamy relations characterize the distributions of entanglement in multipartite systems. We investigate monogamy relations for multiqubit generalized W-class states. We present new analytical monogamy inequalities for the concurrence of assistance, which are shown to be tighter than the existing ones. Furthermore, analytical monogamy inequalities are obtained for the negativity of assistance.
We strengthen the set of Bell-type inequalities presented by Sun & Fei [Phys. Rev. A 74, 032335 (2006)] that give a classification for biseparable correlations and entanglement in tripartite quantum systems. We will furthermore consider the restriction to local orthogonal spin observables and show that this strengthens all previously known such tripartite inequalities. The quadratic inequalities we find indicate a type of monogamy of maximal biseparable three-particle quantum correlations, although the nonmaximal ones can be shared. This is contrasted to recently found monogamy inequalities fo…
Nonclassical correlations have been found useful in many quantum information processing tasks, and various measures have been proposed to quantify these correlations. In this work, we mainly study one of nonclassical correlations, called measurement-induced nonlocality (MIN). First, we establish a close connection between this nonlocal effect and the Bell nonlocality for two-qubit states. Then, we derive a tight monogamy relation of MIN for any pure three-qubit state and provide an alternative way to obtain similar monogamy relations for other nonclassical correlation measures, including squar…
We consider multipartite states of qubits and prove that their bipartite quantum entanglement, as quantified by the concurrence, satisfies a monogamy inequality conjectured by Coffman, Kundu, and Wootters. We relate this monogamy inequality to the concept of frustration of correlations in quantum spin systems.
We investigate monogamy of correlations and entropy inequalities in the Bloch representation. Here, both can be understood as direct relations between different correlation tensor elements and thus appear intimately related. To that end we introduce the split Bloch basis, that is particularly useful for representing quantum states with low dimensional support and thus amenable to purification arguments. Furthermore, we find dimension dependent entropy inequalities for the Tsallis 2-entropy. In particular, we present an analogue of the strong subadditivity and a quadratic entropy inequality. Th…
The quantum steering ellipsoid can be used to visualise two-qubit states, and thus provides a generalisation of the Bloch picture for the single qubit. Recently, a monogamy relation for the volumes of steering ellipsoids has been derived for pure 3-qubit states and shown to be stronger than the celebrated Coffman-Kundu-Wootters (CKW) inequality. We first demonstrate the close connection between this volume monogamy relation and the classification of pure 3-qubit states under stochastic local operations and classical communication (SLOCC). We then show that this monogamy relation does not hold …
Using Rényi-$α$ entropy to quantify bipartite entanglement, we prove monogamy of entanglement in multi-qubit systems for $α\geq 2$. We also conjecture a polygamy inequality of multi-qubit entanglement with strong numerical evidence for $0.83-ε\leq α\leq 1.43+ε$ with $0<ε<0.01$.
In this paper, we study the $α$th power monogamy properties related to the entanglement measure in bipartite states. The monogamy relations related to the $α$th power of negativity and the Convex- Roof Extended Negativity are obtained for N-qubit states. We also give a tighter bound of hierarchical monogamy inequality for the entanglement of formation. We find that the GHZ state and W state can be used to distinguish the $α$th power the concurrence for $0<α<2$. Furthermore, we compare concurrence with negativity in terms of monogamy property and investigate the difference between them.
The monogamy inequality in terms of the concurrence, called the Coffman-Kundu-Wootters inequality [Phys. Rev. A {\bf 61}, 052306 (2000)], and its generalization [T.J. Osborne and F. Verstraete, Phys. Rev. Lett. {\bf 96}, 220503 (2006)] hold on general $n$-qubit states including mixed ones. In this paper, we consider the monogamy inequalities in terms of the fully entangled fraction and the teleportation fidelity. We show that the monogamy inequalities do not hold on general mixed states, while the inequalities hold on $n$-qubit pure states.
Monogamy inequalities for the way bipartite EPR steering can be distributed among N systems are derived. One set of inequalities is based on witnesses with two measurement settings, and may be used to demonstrate correlation of outcomes between two parties, that cannot be shared with more parties. It is shown that the monogamy for steering is directional. Two parties cannot independently demonstrate steering of a third system, using the same two-setting steering witness, but it is possible for one party to steer two independent systems. This result explains the monogamy of two-setting Bell ine…
For multipartite entangled states, entanglement monogamy is an important property. We investigate the monogamy relations for multiqubit generalized W-class states. We present new analytical monogamy inequalities satisfied by the $x$-th power of the dual of convex-roof extended negativity, namely CRENOA, for $x\geq2$ and $x\leq0$. As for The squared Rényi-$α$ entanglement (SR$α$E) with $α$ in the region $[(\sqrt 7 - 1)/2,(\sqrt {13} - 1)/2]$, we show the upper bound of SR$α$E.
The monogamy of entanglement means that entanglement cannot be freely shared. In 2014, Oliveira et al. [ Oliveira et al., Phys. Rev. A. 89, 034303 (2014)] proposed a monogamy relation in the linear version and considered it in terms of entanglement of formation. Here we generalize the above version and consider a multi-linear monogamy relation for a multi-qubit system in terms of entanglement of formation and concurrence. Based on the above results, we present an entanglement criterion for genuine entangled states, also we consider the absolutely maximally entangled states and present what an …
Unlike classical correlation, quantum entanglement cannot be freely shared among many parties. This restricted shareability of entanglement among multi-party systems is known as monogamy of entanglement, which is one of the most fundamental properties of entanglement. Here, we summarize recent theoretical progress in the field of monogamy of entanglement. We firstly review the standard CKW-type monogamy inequalities in terms of various entanglement measures. In particular, the squashed entanglement and one-way distillable entanglement are monogamous for arbitrary dimensional systems. We then i…
In this paper we demonstrate that the property of monogamy of Bell violations seen for no-signaling correlations in composite systems can be generalized to the monogamy of contextuality in single systems obeying the Gleason property of no-disturbance. We show how one can construct monogamies for contextual inequalities by using the graph-theoretic technique of vertex decomposition of a graph representing a set of measurements into subgraphs of suitable independence numbers that themselves admit a joint probability distribution. After establishing that all the subgraphs that are chordal graphs …
We investigate the monogamy relations related to the concurrence, the entanglement of formation, convex-roof extended negativity, Tsallis-q entanglement and R'enyi-α entanglement, the polygamy relations related to the entanglement of formation, Tsallis-q entanglement and R'enyi-α entanglement. Monogamy and polygamy inequalities are obtained for arbitrary multipartite qubit systems, which are proved to be tighter than the existing ones. Detailed examples are presented.
Our main result is a monogamy inequality satisfied by the entanglement of a focus qubit (one-tangle) in a four-qubit pure state and entanglement of subsystems. Analytical relations between three-tangles of three-qubit marginal states, two-tangles of two-qubit marginal states and unitary invariants of four-qubit pure state are used to obtain the inequality. The contribution of three-tangle to one-tangle is found to be half of that suggested by a simple extension of entanglement monogamy relation for three qubits. On the other hand, an additional contribution due to a two-qubit invariant which i…
We consider the Cavalcanti-Foster-Fuwa-Wiseman inequality~\cite{achsh} which is a necessary and sufficient steerability condition for two-qubit states with two measurement settings on each side. We derive the criterion which an arbitrary two-qubit state must satisfy in order to violate this inequality, and obtain its maximum attainable violation in quantum mechanics. The derived condition on the state parameters enables us to establish a tight monogamy relation for two-qubit steering.
Measurement-Induced NonLocality was introduced by Luo and Fu (Phys. Rev. Lett. \textbf{106}, 120401,(2011)) as a measure of nonlocality in a bipartite state. In this paper we will discuss monogamy property of measurement-induced nonlocality for some three- and four-qubit classes of states. Unlike discord, we find quite surprising results in this situation. Both the GHZ and W states satisfy monogamy relations in the three-qubit case, however, in general there are violations of monogamy relations in both the GHZ-class and W-class states. In case of four-qubit system, monogamy holds for most of t…
We discuss the possibility to interpret the residual entanglement for more than three qubits in terms of distributed multipartite entanglement, or, in other words, possible extensions of the Coffman-Kundu-Wootters monogamy equality to higher qubit numbers. Existing knowledge on entanglement in multipartite systems puts narrow constraints on the form of such extensions. We study various examples for families of pure four-qubit states for which the characterization of three-qubit and four-qubit entanglement in terms of polynomial invariants is known. These examples indicate that, although famili…
We investigate the generalized monogamy and polygamy relations $N$-qubit systems. We give a general upper bound of the $α$th ($0\leqα\leq2$) power of concurrence for $N$-qubit states. The monogamy relations satisfied by the $α$th ($0\leqα\leq2$) power of concurrence are presented for $N$-qubit pure states under the partition $AB$ and $C_1 . . . C_{N-2}$, as well as under the partition $ABC_1$ and $C_2\cdots C_{N-2}$. These inequalities give rise to the restrictions on entanglement distribution and the trade off of entanglement among the subsystems. Similar results are also derived for negativi…
We present a new kind of monogamous relations based on concurrence and concurrence of assistance. For $N$-qubit systems $ABC_1...C_{N-2}$, the monogamy relations satisfied by the concurrence of $N$-qubit pure states under the partition $AB$ and $C_1...C_{N-2}$, as well as under the partition $ABC_1$ and $C_2...C_{N-2}$ are established, which give rise to a kind of restrictions on the entanglement distribution and trade off among the subsystems.
We study the Bures measure of entanglement and the geometric measure of entanglement as special cases of entanglement measures based on fidelity, and find their tighter monogamy inequalities over tri-qubit systems as well as multi-qubit systems. Furthermore, we derive the monogamy inequality of concurrence for qudit quantum systems by projecting higher-dimensional states to qubit substates.
We study the monogamy and polygamy inequalities of unified entanglement in multipartite quantum systems. We first derive the monogamy inequality of unified-$(q, s)$ entanglement for multi-qubit states under arbitrary bipartition, and then obtain the monogamy inequalities of the $α$th ($0\leqα\leq\frac{r}{2}, r\geq\sqrt{2}$) power of entanglement of formation for tripartite states and their generalizations in multi-qubit quantum states. We also generalize the polygamy inequalities of unified-$(q, s)$ entanglement for multi-qubit states under arbitrary bipartition. Moreover, we investigate polyg…
\indent In this article, we consider the monogamy relations for the generalized W class states. Here we first present an analytical formula on Tsallis-$q$ entanglement and Tsallis-$q$ entanglement of assistance of a reduced density matrix of the generalized W class states, then we present a monogamy relation in terms of the T$q$EE for the GW state, and we also present generalized monogamy relations in terms of T$q$EE for the GW states. At last, we present tighter polygamy relations in terms of T$q$EEoA when $q=2$ for the GWV states.
We consider three modes $A$, $B$ and $C$ and derive continuous variable monogamy inequalities that constrain the distribution of bipartite entanglement amongst the three modes. The inequalities hold for all such tripartite states, without the assumption of Gaussian states, and are based on measurements of two conjugate quadrature phase amplitudes $X_{i}$ and $P_{i}$ at each mode $i=A,B$. The first monogamy inequality is $D_{BA}+D_{BC}\geq1$ where $D_{BA}<1$ is the widely used symmetric entanglement criterion, for which $D_{BA}$ is the sum of the variances of $(X_{A}-X_{B})/2$ and $(P_{A}+P_{B}…
We prove that the squared Rényi-$α$ entanglement (SR$α$E), which is the generalization of entanglement of formation (EOF), obeys a general monogamy inequality in an arbitrary $N$-qubit mixed state. Furthermore, for a class of Rényi-$α$ entanglement, we prove that the monogamy relations of the SR$α$E have a hierarchical structure when the $N$-qubit system is divided into $k$ parties. As a byproduct, the analytical relation between the Rényi-$α$ entanglement and the squared concurrence is derived for bipartite $2\otimes d$ systems. Based on the monogamy properties of SR$α$E, we can construct the…
One of the fundamental traits of quantum entanglement is the restricted shareability among multipartite quantum systems, namely monogamy of entanglement, while it is well known that monogamy inequalities are always satisfied by entanglement measures with convexity. Here we present a measure of entanglement, logarithmic convex-roof extended negativity (LCREN) satisfying important characteristics of an entanglement measure, and investigate the monogamy relation for logarithmic negativity and LCREN both without convexity. We show exactly that the $α$th power of logarithmic negativity, and a newly…
We introduce the concept of monogamy deficit for quantum correlation by combining together two types of monogamy inequalities depending on different measurement sides. For tripartite pure state, we demonstrate a relation which connects two types of monogamy inequalities for quantum discord and provide the difference between them. By using this relation, we obtain an unified physical interpretation for these two monogamy deficit. In addition, we find an interesting fact that there is a general monogamy condition for several quantum correlations for tripartite pure states. We then provide a nece…
We derive monogamy relations (tradeoffs) between strengths of violations of Bell's inequalities from the non-signaling condition. Our result applies to general Bell inequalities with an arbitrary large number of partners, outcomes and measurement settings. The method is simple, efficient and does not require linear programming. The results are used to derive optimal fidelity for asymmetric cloning in nonsignaling theories.
The set of all qubit states that can be steered to by measurements on a correlated qubit is predicted to form an ellipsoid---called the quantum steering ellipsoid---in the Bloch ball. This ellipsoid provides a simple visual characterisation of the initial 2-qubit state, and various aspects of entanglement are reflected in its geometric properties. We experimentally verify these properties via measurements on many different polarisation-entangled photonic qubit states. Moreover, for pure 3-qubit states, the volumes of the two quantum steering ellipsoids generated by measurements on the first qu…
The limitation on the shareability of quantum entanglement over several parties, the so-called monogamy of entanglement, is an issue that has caught considerable attention of quantum informa- tion community over the last decade. A natural question of interest in this connection is whether monogamy of correlations is true for correlations other than entanglement. This issue is examined here by choosing quantum deficit, proposed by Rajagopal and Rendell, an operational measure of correlations. In addition to establishing the polygamous nature of the class of three qubit sym- metric pure states c…
