We obtain a simplified and obvious expression of "concurrence" in Wootters' measure of entanglement of a pair of qubits having no more than two non-zero eigenvalues in terms of concurrences of eigenstates and their simple combinations. It not only simplifies the calculation of Wootters' measure of entanglement, but also reveals some its general and important features. Our conclusions are helpful to understand and use quantum entanglement further.
We present an extension of the Wootters concurrence for the case of two qutrits in mixed states. The reduction of our extension to the case of two levels shows complete agreement with Wootters concurrence for two qubits. As an explicit example, we compute the concurrence for a family of symmetric states and we obtain the bounds on the limit for separability. Our results are compared with those of the negativity.
Entanglement monotones, such as the concurrence, are useful tools to characterize quantum correlations in various physical systems. The computation of the concurrence involves, however, difficult optimizations and only for the simplest case of two qubits a closed formula was found by Wootters [Phys. Rev. Lett. 80, 2245 (1998)]. We show how this approach can be generalized, resulting in lower bounds on the concurrence for higher dimensional systems as well as for multipartite systems. We demonstrate that for certain families of states our results constitute the strongest bipartite entanglement …
We obtain Lewenstein-Sanpera decomposition of iso-concurrence decomposable states of $2\otimes 2$ quantum systems. It is shown that in these systems average concurrence of the decomposition is equal to the concurrence of the state and also it is equal to the amount of violation of positive partial transpose criterion. It is also shown that the product states introduced by Wootters in [W. K. Wootters, Phys. Rev. Lett. {\bf 80} 2245 (1998)] form the best separable approximation ensemble for these states.
A new entanglement measure, which is called D-concurrence, is proposed. Then the upper and lower bounds for D-concurrence are obtained and the relationship between D-concurrence and the usual concurrence of Wootters was established. In addition, comparing with the usual concurrence, D-concurrence has some special merits.
Wootters [Phys. Rev. Lett. 80, 2245 (1998)] has given an explicit formula for the entanglement of formation of two qubits in terms of what he calls the concurrence of the joint density operator. Wootters's concurrence is defined with the help of the superoperator that flips the spin of a qubit. We generalize the spin-flip superoperator to a "universal inverter," which acts on quantum systems of arbitrary dimension, and we introduce the corresponding concurrence for joint pure states of (D1 X D2) bipartite quantum systems. The universal inverter, which is a positive, but not completely positive…
Concurrence, introduced by Hill and Wootters [Phys. Rev. Lett. 78, 5022 (1997)], provides an important measure of entanglement for a general pair of qubits that is faithful: strictly positive for entangled states and vanishing for all separable states. Such a measure captures the entire content of entanglement, providing necessary and sufficient conditions for separability. We present an extension of concurrence to multiparticle pure states in arbitrary dimensions by a new framework using the Lagrange's identity and wedge product representation of separability conditions, which coincides with …
We define the concurrence hierarchy as d-1 independent invariants under local unitary transformations in d-level quantum system. The first one is the original concurrence defined by Wootters et al in 2-level quantum system and generalized to d-level pure quantum states case. We propose to use this concurrence hierarchy as measurement of entanglement. This measurement does not increase under local quantum operations and classical communication.
We refine the notion of concurrence in this paper by a redefinition of the concept. The new definition is simpler, computationally straight forward, and allows the concurrence to be directly read off from the state. It has all the positive features of the definiton given by Wootters over and above which it can discriminate between different systems to which the Wootters prescription would assign the same value. Finally, the definiton leads to a natural extension of the notion to multiqubit concurrence, which we illustrate with examples from quantum error correction codes.
By considering the decomposition of a generic two qubit density matrix presented by Wootters [W. K. Wootters, Phys. Rev. Lett. {\bf 80} 2245 (1998)], the robustness of entanglement for any mixed state of two qubit systems is obtained algebraically. It is shown that the robustness of entanglement is proportional to concurrence and in Bell decomposable density matrices it is equal to the concurrence. We also give an analytic expression for two separable states which wipe out all entanglement of these states. Since thus obtained robustness is function of the norm of the vectors in the decompositi…
We obtain an analytical lower bound of entanglement quantified by concurrence for arbitrary bipartite quantum states. It is shown that our bound is tight for some mixed states and is complementary to the previous known lower bounds. On the other hand, it is known that the entanglement monogamy inequality proposed by Coffman, Kundu, and Wootters is in general not true for higher dimensional quantum states. Inducing from the new lower bound of concurrence, we find a proper form of entanglement monogamy inequality for arbitrary quantum states.
We propose two classes of the generalized concurrence vectors of the multipartite systems consisting of qubits. Making use of them, we are able to, respectively, describe and quantify GHZ-class and W-class entanglement both in total and between arbitrary two partite in the multipartite system consisting of qubits. In the case of pure state of three qubits that one partite is separable, it is shown to exactly back to the usual Wootters' concurrence after introduce a whole concurrence vector. In principle, our method is applicable to any $N$-partite systems consisting of $N$ qubits.
In this paper, we show a direct method of deriving the Peres-Horodecki criterion for the two-qubit states from the Hill-Wootters formula for the entanglement of formation. Although the Peres-Horodecki criterion and the Hill-Wootters formula are established results in the field of quantum information theory, they are proved independently and connections between them are not discussed precisely. In this paper, we clarify these connections. First, we replace the original Peres-Horodecki criterion with its another equivalent statement found by Augusiak et al. Second, we obtain an analytical form o…
We consider the maximum bipartite entanglement that can be distilled from a single copy of a multipartite mixed entangled state, where we focus mostly on $d\times d\times n$-dimensional tripartite mixed states. We show that this {\em assisted entanglement}, when measured in terms of the generalized concurrence (named G-concurrence) is (tightly) bounded by an entanglement monotone, which we call the G-concurrence of assistance. The G-concurrence is one of the possible generalizations of the concurrence to higher dimensions, and for pure bipartite states it measures the {\em geometric mean} of t…
It has been shown that, for the two-qubit Heisenberg XY model, anisotropy and magnetic field may together be used to produce entanglement for any finite temperature by adjusting the external magnetic field beyond some finite critical strength. This interesting result arises from an analysis employing the Wootters concurrence, a computable measure of entanglement for two-qubit states. Recently, Mintert {\em et al.} proposed generalizations of Wootters concurrence for multipartite states. These MKB concurrences possess a mathematical property that enables one to understand the origin of this cha…
In this report, a scheme different from the PT and Wootters concurrence is developed to acquire a criterion to investigate the bipartite separability of the Werner state.
We derive an inequality for three fermions with six single particle states which reduces to the sum of the famous Coffman-Kundu-Wootters inequalities when an embedded three qubit system is considered. We identify the quantities which are playing the role of the concurrence, the three-tangle and the invariant $\det ρ_A+\det ρ_B+\det ρ_C$ for this tripartite system. We show that this latter one is almost interchangeable with the von Neumann entropy and conjecture that it measures the entanglement of one fermion with the rest of the system. We prove that the vanishing of the fermionic "concurrenc…
We study the ordering of two-qubit states with respect to the degree of bipartite entanglement using the Wootters concurrence -- a measure of the entanglement of formation, and the negativity -- a measure of the entanglement cost under the positive-partial-transpose-preserving operations. For two-qubit pure states, the negativity is the same as the concurrence. However, we demonstrate analytically on simple examples of various mixtures of Bell and separable states that the entanglement measures can impose different orderings on the states. We show which states, in general, give the maximally d…
We provide a complete analysis of mixed three-qubit states composed of a GHZ state and a W state orthogonal to the former. We present optimal decompositions and convex roofs for the three-tangle. Further, we provide an analytical method to decide whether or not an arbitrary rank-2 state of three qubits has vanishing three-tangle. These results highlight intriguing differences compared to the properties of two-qubit mixed states, and may serve as a quantitative reference for future studies of entanglement in multipartite mixed states. By studying the Coffman-Kundu-Wootters inequality we find th…
The entanglement of the two movable mirrors in a single mode cavity field is studied in an ideal situation without thermal noise. When gn/w <<1 where g is the coupling coefficient between the mirror and the field, w is the mirror' vibration frequency and n is photon number of the cavity field, we work out Wootters concurrence of the system initially with the field in Fock number state and the two mirrors in vacuum states, and find that the states of the mirrors are entangled into space spanned by |0>,|1> via radiation pressure, though the concurrence is very small.
Bell-inequality violation and entanglement, measured by Wootters' concurrence and negativity, of two qubits initially in Werner or Werner-like states coupled to thermal reservoirs are analyzed within the master equation approach. It is shown how this simple decoherence process leads to generation of states manifesting the relativity of two-qubit entanglement measures.
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.
It is well known that a particle cannot freely share entanglement with two or more particles. This restriction is generally called monogamy. However the formal quantification of such restriction is only known for some measures of entanglement and for two-level systems. The first and broadly known monogamy relation was established by Coffman, Kundu, and Wootters for the square of the concurrence. Since then, it is usually said that the entanglement of formation is not monogamous, as it does not obey the same relation. We show here that despite that, the entanglement of formation cannot be freel…
We investigate the entanglement dynamics of bipartite quantum system between two qubits with the dissipative environment. We begin with the standard Markovian master equation in the Lindblad form and the initial state which is prepared in the extended Werner-like state: $ρ^Φ_{AB}(0)$. We examine the conditions for entanglement sudden death (ESD) and calculate the corresponding ESD time by the Wootters' concurrence. We observe that ESD is determined by the parameters like the mean occupation number of the environment $N$, amount of initial entanglement $α$, and the purity $r$. For N=0, we get t…
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.
We examine the unique spectroscopic features which give rise to quantum information processing attributes of one-dimensional J-aggregate systems, and as revealed by entanglement measures such as the von Neumann entropy, Wootters concurrence and Wei-Goldbart geometric measure of entanglement. The effect of dispersion and resonance terms in the exciton-phonon interaction are analyzed using Green function formalism and present J-aggregate systems as robust channels for large scale energy propagation for a select range of parameters. We show that scaling of the third order optical response $χ^{(3)…
We investigate entanglement of spin pairs in alternating open spin chains in the equilibrium state in an external magnetic field. We calculate the reduced density matrix of spin pairs and estimate the concurrence with Wootter's criteria. The obtained results demonstrate the dependence of the entanglement on the temperature, chain's length, the positions of the spins, and the ratio of the spin-spin interaction constants.
We propose replacing concurrence by convex-roof extended negativity (CREN) for studying monogamy of entanglement (MoE). We show that all proven MoE relations using concurrence can be rephrased in terms of CREN. Furthermore we show that higher-dimensional (qudit) extensions of MoE in terms of CREN are not disproven by any of the counterexamples used to disprove qudit extensions of MoE in terms of concurrence. We further test the CREN version of MoE for qudits by considering fully or partially coherent mixtures of a qudit W-class state with the vacuum and show that the CREN version of MoE for qu…
For a two-qubit system under local depolarizing channels, the most robust and most fragile states are derived for a given concurrence or negativity. For the one-sided channel, the pure states are proved to be the most robust ones, with the aid of the evolution equation for entanglement given by Konrad et al. [Nat. Phys. 4, 99 (2008)]. Based on a generalization of the evolution equation for entanglement, we classify the ansatz states in our investigation by the amount of robustness, and consequently derive the most fragile states. For the two-sided channel, the pure states are the most robust f…
Quantum entanglement and its paradoxical properties hold the key to an information processing revolution. Much attention has focused recently on the challenging problem of characterizing entanglement. Entanglement for a two qubit system is reasonably well understood, however, the nature and properties of multiple qubit systems are largely unexplored. Motivated by the importance of such systems in quantum computing, we show that typical pure states of N qubits are highly entangled but have decreasing amounts of pairwise entanglement (measured using the Wootter's concurrence formula) as N increa…
Concurrence introduced by Hill and Wootters [Phys. Rev. Lett. 78, 5022 (1997)], provides an important measure of entanglement for a general pair of qubits that is strictly positive entangled states and vanishes for all separable states. We present an extension of entanglement measure to general continuous variable states of multiple degrees of freedom by generalizing the Lagrange's identity and wedge product framework proposed by Bhaskara et al. [Quantum Inf. Process. 16, 118 (2017)] for pure discrete variable systems in arbitrary dimensions. A family of faithful entanglement measures is const…
We study 12 parameter families of two qubit density matrices, arising from a special class of two-fermion systems with four single particle states or alternatively from a four-qubit state with amplitudes arranged in an antisymmetric matrix. We calculate the Wooters concurrences and the negativities in a closed form and study their behavior. We use these results to show that the relevant entanglement measures satisfy the generalized Coffman-Kundu-Wootters formula of distributed entanglement. An explicit formula for the residual tangle is also given. The geometry of such density matrices is elab…
A new formulation called as entanglement measure for simplification, is presented to characterize genuine tripartite entanglement of $(2\times 2\times n)-$dimensional quantum pure states. The formulation shows that the genuine tripartite entanglement can be described only on the basis of the local $(2\times 2)-$dimensional reduced density matrix. In particular, the two exactly solvable models of spin system studied by Yang (Phys. Rev. A \textbf{71}, 030302(R) (2005)) is reconsidered by employing the entanglement measure. The results show that a discontinuity in the first derivative of the enta…
We study special relativistic effects on the entanglement between either spins or momenta of composite quantum systems of two spin-1/2 massive particles, either indistinguishable or distinguishable, in inertial reference frames in relative motion. For the case of indistinguishable particles, we consider a balanced scenario where the momenta of the pair are well-defined but not maximally entangled in the rest frame while the spins of the pair are described by a one-parameter ($η$) family of entangled bipartite states. For the case of distinguishable particles, we consider an unbalanced scenario…
