We give an explicit tight lower bound for the entanglement of formation for arbitrary bipartite mixed states by using the convex hull construction of a certain function. This is achieved by revealing a novel connection among the entanglement of formation, the well-known Peres-Horodecki and realignment criteria. The bound gives a quite simple and efficiently computable way to evaluate quantitatively the degree of entanglement for any bipartite quantum state.PACS numbers: 03.67. Mn, 03.65.Ud, 89.70.+c Quantum entangled states are used as key resources in quantum information processing and communication, such as in quantum cryptography, quantum teleportation, dense coding, error correction and quantum computation [1]. A fundamental problem in quantum information theory is how to quantify the degree of entanglement in a practical and operational way [2,3]. One of the most meaningful and physically motivated measures is the entanglement of formation (EOF) [2,3], which quantifies the minimal cost needed to prepare a certain quantum state in terms of EPR pairs. Related to the EOF the behavior of entanglement has recently been shown to play important roles in quantum phase transition for various interacting quantum many-body systems [4] and may significantly affect macroscopic properties of solids [5]. Moreover, it has been shown that there is a remarkable connection between entanglement and the capacity of quantum channels [6]. A quantitative evaluation of EOF is thus of great significance both theoretically and experimentally.Considerable efforts have been spent on deriving EOF or its lower bound through analytical and numerical approaches, for some limited sets of mixed states [7,8,9,10,11,12,13,14,15,16,17]. Among them, the most noteworthy results are an elegant analytical formula for two qubits [7,8] [18,19] in giving analytic lower bounds [that can be optimized further numerically [18]] for the concurrence, which permits to furnish a lower bound of EOF for a generic mixed state. However, this lower bound is not explicit except for the case of 2 ⊗ n systems [19]. For low dimensional systems, numerical methods [10] can be used to estimate EOF. Nevertheless, they are generally time-consuming and often not very efficient. The notorious difficulty of evaluation for EOF is due to the complexity to solve a high dimensional optimization problem, which becomes a formidable task, as the dimensionality of the Hilbert space grows.In this Letter, we present the first analytical calculation of a tight lower bound of EOF for arbitrary bipartite quantum states. An explicit expression for the bound is obtained from the convex hull of a simple function, based on a known result in [9]. This is achieved by establishing a key connection between EOF and two strong separability criteria, the Peres-Horodecki criterion [20,21] and the realignment criterion [22,23]. The bound is shown to be exact for some special states such as isotropic states [9,24] and permits to provide EOF estimations for many bound entangled states (BES). It provi...

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.