Monogamy Inequality Research Articles

Strong monogamy inequalities for four qubits

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.

Comparison of qubit and qutrit like entangled squeezed and coherent states of light

Squeezed state of light is one of the important subjects in quantum optics which is generated by optical nonlinear interactions. In this paper, we especially focus on qubit like entangled squeezed states (ESS's) generated by beam splitters, phase-shifter and cross Kerr nonlinearity. Moreover the Wigner function of two-mode qubit and qutrit like ESS are investigated. We will show that the distances of peaks of Wigner functions for two-mode ESS are entanglement sensitive and can be a witness for entanglement. Like the qubit cases, monogamy inequality is fulfilled for qutrit like ESS. These trends are compared with those obtained for qubit and qutrit like entangled coherent states (ECS).

Strong monogamy of multiparty quantum entanglement for partially coherently superposed states

We provide an evidence for the validity of strong monogamy inequality of multi-party quantum entanglement using square of convex-roof extended negativity(SCREN). We first consider a large class of multi-qudit mixed state that are in a partially coherent superposition of a generalized W-class state and the vacuum, and provide some useful properties about this class of states. We show that monogamy inequality of multi-qudit entanglement in terms of SCREN holds for this class of states. We further show that SCREN strong monogamy inequality of multi-qudit entanglement also holds for this class of states. Thus SCREN is a good alternative to characterize the monogamous and strongly monogamous properties of multi-qudit entanglement.

General monogamy relation of multiqubit systems in terms of squared Rényi-αentanglement

We prove that the squared R\'{e}nyi-$\alpha$ entanglement (SR$\alpha$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 R\'{e}nyi-$\alpha$ entanglement, we prove that the monogamy relations of the SR$\alpha$E have a hierarchical structure when the $N$-qubit system is divided into $k$ parties. As a byproduct, the analytical relation between the R\'{e}nyi-$\alpha$ entanglement and the squared concurrence is derived for bipartite $2\otimes d$ systems. Based on the monogamy properties of SR$\alpha$E, we can construct the corresponding multipartite entanglement indicators which still work well even when the indicators based on the squared concurrence and EOF lose their efficacy. In addition, the monogamy property of the $\mu$-th power of R\'{e}nyi-$\alpha$ entanglement is analyzed.

Strong monogamy conjecture in a four-qubit system

Monogamy is a defining feature of entanglement, having far reaching applications. Recently, Regula \textit{et.al.} in Phys. Rev. Lett. \textbf{113}, 110501(2014) have proposed a stronger version of monogamy relation for concurrence. We have extended the strong monogamy inequality for another entanglement measure, viz., negativity. In particular, we have concentrated on four-qubit system and provided a detail study on the status of strong monogamy on pure states. Further, we have analytically provided some classes of states for which negativity and squared negativity satisfy strong monogamy. Numerical evidences have also been shown in proper places. Our analysis also provides cases where strong monogamy is violated.

Noise Effects on Entangled Coherent State Generated via Atom-Field Interaction and Beam Splitter

In this paper, we introduce a controllable method for producing two and three-mode entangled coherent states (ECS's) using atom-field interaction in cavity QED and beam splitter. The generated states play central roles in linear optics, quantum computation and teleportation. We especially focus on qubit, qutrit and qufit like ECS's and investigate their entanglement by concurrence measure. Moreover, we illustrate decoherence properties of ECS's due to noisy channels, using negativity measure. At the end the effect of noise on monogamy inequality is discussed.

Negativity and strong monogamy of multiparty quantum entanglement beyond qubits

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 inequality is true for a large class of multi-qudit states, a superposition of multi-qudit generalized W-class states and vacuums. Thus SCREN is a good alternative to characterize the strong monogamy of entanglement even in multi-qudit systems.

Entanglement of Multi-qudit States Constructed by Linearly Independent Coherent States: Balanced Case

Multi-mode entangled coherent states are important resources for linear optics quantum computation and teleportation. Here we introduce the generalized balanced N-mode coherent states which recast in the multi-qudit case. The necessary and sufficient condition for bi-separability of such balanced N-mode coherent states is found. We particularly focus on pure and mixed multi-qubit and multi-qutrit like states and examine the degree of bipartite as well as tripartite entanglement using the concurrence measure. Unlike the N-qubit case, it is shown that there are qutrit states violating monogamy inequality. Using parity, displacement operator and beam splitters, we will propose a scheme for generating balanced N-mode entangled coherent states for even number of terms in superposition.

Monogamy of [formula omitted]th power entanglement measurement in qubit systems

In this paper, we study the αth power monogamy properties related to the entanglement measure in bipartite states. The monogamy relations related to the αth 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 both the αth power of the concurrence for 0<α<2 and the αth power of the entanglement of formation for 0<α≤12. Furthermore, we compare concurrence with negativity in terms of monogamy property and investigate the difference between them.

Unveiling $${\pi }$$ π -tangle and quantum phase transition in the one-dimensional anisotropic XY model

In this paper, the relationship between $${\pi }$$?-tangle and quantum phase transition (QPT) is investigated by employing the quantum renormalization-group method in the one-dimensional anisotropic XY model. The results show that all the 1-tangles increase firstly and then decrease with the anisotropy parameter $$\gamma $$? increasing, and the Coffman---Kundu---Wootters monogamy inequality is always tenable for this system. The entanglement's status of subsystems depends on its site position, and this proposition can be generalized to a multipartite system. Meanwhile, with the increasing of the size of the system, the $${\pi }$$?-tangle decreases slowly and tends to a fixed value finally. Additionally, it exhibits a QPT and a maximum value for the next-nearest-neighbor entanglement at the critical point in our model, which is different from the case of two-body system. After several iterations of the renormalization, the quantum entanglement measure can develop two saturated values, which are associated with two different phases: spin-fluid phase and the Neel phase. To gain further insight, the nonanalytic and scaling behaviors of $${\pi }$$?-tangle have also been analyzed in detail.

Strong monogamy of quantum entanglement for multiqubitW-class states

We provide strong evidence for the strong monogamy inequality of multiqubit entanglement recently proposed [B. Regula et al., Phys. Rev. Lett. 113, 110501 (2014)]. We consider a large class of multiqubit generalized $W$-class states and analytically show that the strong monogamy inequality of multiqubit entanglement is saturated by this class of states.

Monogamy of Measurement-Induced Nonlocality Based on Relative Entropy

In this paper, using relative entropy, we study monogamous properties of measurement-induced nonlocality based on relative entropy. Depending on different measurement sides, we provide necessary and sufficient conditions for two types of monogamy inequalities. By the concept of nonlocality monogamy score, we find a necessary condition of the vanished nonlocality monogamy score for arbitrary three-party states. In addition, two types of necessary and sufficient conditions of the vanished nonlocality monogamy scores are obtained for any pure states. As an application, we show that measurement-induced nonlocality based on relative entropy can be viewed as a “nonlocality witness” to distinguish generalized GHZ states from the generalized W states.

Strong monogamy conjecture for multiqubit entanglement: the four-qubit case.

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.

General monogamy relation for the entanglement of formation in multiqubit systems.

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 monogamy relations of qubit systems

We investigate the monogamy relations related to the concurrence and the entanglement of formation. General monogamy inequalities given by the {\alpha}th 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.

General monogamy property of global quantum discord and the application

We provide a family of general monogamy inequalities for global quantum discord (GQD), which can be considered as an extension of the usual discord monogamy inequality. It can be shown that those inequalities are satisfied under the similar condition for the holding of usual monogamy relation. We find that there is an intrinsic connection among them. Furthermore, we present a different type of monogamy inequality and prove that it holds under the condition that the bipartite GQDs do not increase when tracing out some subsystems. We also study the residual GQD based on the second type of monogamy inequality. As applications of those quantities, we investigate the GQDs and residual GQD in characterizing the quantum phase transition in the transverse field Ising model.

How the Hawking effect affects multipartite entanglement of Dirac particles in the background of a Schwarzschild black hole

In this paper, the effect of Hawking radiation on the multipartite entanglement of Dirac particles in the background of a Schwarzschild black hole is investigated. It is shown that all the 1-tangles decrease as the Hawking temperature grows, and the Coffman--Kundu--Wootters monogamy inequality is always saturated in this system. It is also shown that the entanglement's status of subsystems, which have the same quantum state, depend on its detector's position, and this proposition can be generalized to a multipartite system. Then, we find that the Hawking effect does not change the entanglement structure of the quantum state in Schwarzschild space-time. In addition, we discuss the distributions of quantum information and find that not only is the entanglement redistributed, but also the mutual information is distributed to the physical inaccessible region. At last, we calculate the mutual information of a $N$-qubit multipartite system, ${|\mathrm{\ensuremath{\Psi}}⟩}_{123\dots{}N}={(\ensuremath{\beta}{|0⟩}^{\ensuremath{\bigotimes}N}+\sqrt{1\ensuremath{-}{\ensuremath{\beta}}^{2}}{|1⟩}^{\ensuremath{\bigotimes}N})}_{123\dots{}N}$, and draw a conclusion that when the black hole approximates to evaporate completely the mutual information of this system will be close to $(N\ensuremath{-}1)/N$ of the original if a particle freely falls in toward a Schwarzschild black hole and locates near the event horizon, and the other particles remain in the asymptotically flat region.

What Does Monogamy in Higher Powers of a Correlation Measure Mean?

We examine here the proposition that all multiparty quantum states can be made monogamous by considering positive integral powers of any quantum correlation measure. With Rajagopal-Rendell quantum deficit as the measure of quantum correlations for symmetric 3-qubit pure states, we illustrate that monogamy inequality is satisfied for higher powers of quantum deficit. We discuss the drawbacks of this inequality in quantification of correlations in the state. We also prove a monogamy inequality in higher powers of classical mutual information and bring out the fact that such inequality need not necessarily imply restricted shareability of correlations. We thus disprove the utility of higher powers of any correlation measure in establishing monogamous nature in multiparty quantum states.

Monogamy inequalities for the Einstein-Podolsky-Rosen paradox and quantum steering

Monogamy inequalities for the way bipartite Einstein-Podolsky-Rosen (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 inequality violations and the sensitivity of the continuous variable (CV) EPR criterion to losses on the steering party. We generalize to $m$ settings. A second type of monogamy relation gives the quantitative amount of sharing possible, when the number of parties is less than or equal to $m$, and takes a form similar to the Coffman-Kundu-Wootters relation for entanglement. The results enable characterization of the tripartite steering for CV Gaussian systems and qubit Greenberger-Horne-Zeilinger and $W$ states.

Unification of Bell, Leggett-Garg and Kochen-Specker inequalities: Hybrid spatio-temporal inequalities

The Bell-type (spatial), Kochen-Specker (contextuality) or Leggett-Garg (temporal) inequalities are based on classically plausible but otherwise quite distinct assumptions. For any of these inequalities, satisfaction is equivalent to a joint probability distribution for all observables in the experiment. This implies a joint distribution for all pairs of observables, and is indifferent to whether or not they commute in the theory. This indifference underpins a unification of the above inequalities into a general framework of correlation inequalities. When the physical scenario is such that the correlated pairs are all compatible, the resulting correlation is nonsignaling, which may be local or multi-particle, corresponding to contextuality or Bell-type inequalities. If the pairs are incompatible, the resulting correlation corresponds to Leggett-Garg (LG) inequalities. That quantum mechanics (QM) violates all these inequalities suggests a close connection between the local, spatial and temporal properties of the theory. As a concrete manifestation of the unification, we extend the method due to Roy and Singh (J. Phys. A, 11 (1978) L167) to derive and study a new class of hybrid spatio-temporal inequalities, where the correlated pairs in the experiment are both compatible or incompatible. The implications for cryptography and monogamy inequalities of the unification are briefly touched upon.