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 present a new kind of monogamous relations based on concurrence and concurrence of assistance. For N -qubit systems ABC1...CN−2, the monogamy relations satisfied by the concurrence of N -qubit pure states under the partition AB and C1...CN−2, as well as under the partition ABC1 and C2...CN−2 are established, which give rise to a kind of restrictions on the entanglement distribution and trade off among the subsystems.PACS numbers: 03.67.Mn,03.65.UdQuantum entanglement [1][2][3][4][5][6] is an essential feature of quantum mechanics, which distinguishes the quantum from classical world. As one of the fundamental differences between quantum entanglement and classical correlations, a key property of entanglement is that a quantum system entangled with one of other systems limits its entanglement with the remaining others. In multipartite quantum systems, there can be several inequivalent types entanglement among the subsystems and the amount of entanglement with different types might not be directly comparable to each other. The monogamy relation of entanglement is a way to characterize the different types of entanglement distribution. The monogamy relations give rise to the structures of entanglement in the multipartite setting. Monogamy is also an essential feature allowing for security in quantum key distribution [7]. Monogamy relations are not always satisfied by entanglement measures. Although the concurrence and entanglement of formation do not satisfy such monogamy inequality, it has been shown that the αth (α ≥ 2) power of concurrence and αth (α ≥ √ 2) power entanglement of formation for N -qubit states do satisfy the monogamy relations [8].In this paper, we study the general monogamy inequalities satisfied by the concurrence and concurrence of assistance. We show that the concurrence of multi-qubit pure states satisfies some generalized monogamy inequalities.The concurrence for a bipartite pure state |ψ AB is given by [10][11][12] where ρ A is the reduced density matrix by tracing over the subsystem B, ρ A = T r B (|ψ AB ψ|). The concurrence is extended to mixed states ρ = i p i |ψ i ψ i |, 0 ≤ p i ≤ 1, i p i = 1, by the convex roof extension,where the minimum is taken over all possible pure state decompositions of ρ AB . For a tripartite state |ψ ABC , the concurrence of assistance is defined by [13] C a (|ψ ABC ) ≡ C a (ρ AB ) = max {pi,|ψi } ifor all possible ensemble realizations of ρ AB = T r C (|ψ ABC ψ|) = i p i |ψ i AB ψ i |. When ρ AB = |ψ AB ψ| is a pure state, then one has C(|ψ AB

We investigate monogamy relations related to quantum entanglement for n−qubit quantum systems. General monogamy inequalities are presented to the βth (β ∈ (0, 2)) power of concurrence, negativity and the convex-roof extended negativity, as well as the βth (β ∈ (0, √ 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.PACS numbers:

Abstract:We study the concurrence of four-qubit quantum states and provide analytical lower bounds of concurrence in terms of the monogamy inequality of concurrence for qubit systems. It is shown that these lower bounds are able to improve the existing bounds and detect entanglement better. The approach is generalized to arbitrary qubit systems.

We present a lower bound of concurrence for arbitrary-dimensional bipartite quantum states. This lower bound may be used to improve all the known lower bounds of concurrence. Moreover, the lower bound gives rise to an operational sufficient criterion of distillability of quantum entanglement. The significance of our result is illustrated by quantitative evaluation of entanglement for entangled states that fail to be identified by the usual concurrence estimation method and by showing the distillability of mixed states that cannot be recognized by other distillability criteria.