In the quest to completely describe entanglement in the general case of a finite number of parties sharing a physical system of finite-dimensional Hilbert space an entanglement magnitude is introduced for its pure and mixed states: robustness. It corresponds to the minimal amount of mixing with locally prepared states which washes out all entanglement. It quantifies in a sense the endurance of entanglement against noise and jamming. Its properties are studied comprehensively. Analytical expressions for the robustness are given for pure states of two-party systems, and analytical bounds for mixed states of two-party systems. Specific results are obtained mainly for the qubit-qubit system ͑qubit denotes quantum bit͒. As by-products local pseudomixtures are generalized, a lower bound for the relative volume of separable states is deduced, and arguments for considering convexity a necessary condition of any entanglement measure are put forward.

We prove for any pure three-quantum-bit state the existence of local bases which allow one to build a set of five orthogonal product states in terms of which the state can be written in a unique form. This leads to a canonical form which generalizes the two-quantum-bit Schmidt decomposition. It is uniquely characterized by the five entanglement parameters. It leads to a complete classification of the threequantum-bit states. It shows that the right outcome of an adequate local measurement always erases all entanglement between the other two parties. The Schmidt decomposition [1,2] allows one to write any pure state of a bipartite system as a linear combination of biorthogonal product states or, equivalently, of a nonsuperfluous set of product states built from local bases. For two quantum bits (qubits) it readsHere jii͘ ϵ ji͘ A ≠ ji͘ B , both local bases ͕ji͖͘ A,B depend on the state jC͘, the relative phase has been absorbed into any of the local bases, and the state j00͘ has been defined by carrying the larger (or equal) coefficient. A larger value of u means more entanglement. The only entanglement parameter, u, plus the hidden relative phase, plus the two parameters which define each of the two local bases are the six parameters of any two-qubit pure state, once normalization and global phase have been disposed of. Very many results in quantum information theory have been obtained with the help of the Schmidt decomposition: its simplicity reflects the simplicity of bipartite systems as compared to N-partite systems. Much of its usefulness comes from it not being superfluous: to carry one entanglement parameter one needs only two orthogonal product states built from local bases states, no more, no less.The aim of this work is to generalize the Schmidt decomposition of (1) to three qubits. It is well known [2] that its straightforward generalization, that is, in terms of triorthogonal product states, is not possible (see also [3]). Nevertheless, having a minimal canonical form in which to cast any pure state, by performing local unitary transformations, will provide a new tool for quantifying entanglement for three qubits, a notoriously difficult problem. It will lead to a complete classification of exceptional states which, as we will see, is much more complex than in the two-qubit case. The generalization to N quantum dits (d-state systems) is not completely straightforward and will be given elsewhere.Linden and Popescu [4] and Schlienz [5] showed that for any pure three-qubit state the number of entanglement parameters is five and, using repeatedly the two-qubit

We present a family of 3-qubit states to which any arbitrary state can be depolarized. We fully classify those states with respect to their separability and distillability properties. This provides a sufficient condition for nonseparability and distillability for arbitrary states. We generalize our results to N-particle states.PACS numbers: 03.67. Hk, 03.65.Bz, 03.65.Ca Entanglement is an essential ingredient in most applications of quantum information. It arises when the state of a multiparticle system is nonseparable; that is, when it cannot be prepared locally by acting on the particles individually. Although in recent years there have been important steps towards the understanding of this feature of quantum mechanics, we do not know yet how to classify and quantify entanglement.Several years ago, entanglement was thought to be directly connected to the violation of Bell-type inequalities [1]. However, Werner [2] introduced a family of mixed states describing a pair of two-level systems (qubits), the so-called Werner states (WS) which, despite being nonseparable, do not violate any of those inequalities [3]. This family is characterized by a single parameter, the fidelity F, which measures the overlap of WS with a Bell (maximally entangled) state. A WS with F . 1͞2 is nonseparable, whereas if F # 1͞2, it is separable. WS have played an essential role in our understanding of the quantum properties of two-qubit states [4]. First of all, any state of two qubits can be reduced to a WS by acting locally on each qubit (the so-called depolarization process) [5]. This automatically provides a sufficient criterion to determine if a given state is nonseparable [6,7]. On the other hand, Bennett et al. [5] showed how one can obtain WS of arbitrarily high fidelity out of many pairs with F . 1͞2 by using local operations and classical communication. This process, called distillation (or purification), is one of the most important concepts in quantum information theory. When combined with teleportation [8], it allows one to convey secret information via quantum privacy amplification [9] or to send quantum information over noisy channels [8,10]. In the case of two qubits, the partial transposition [11] turned out to be a central tool in the classification of such systems, providing necessary and sufficient conditions for separability [6,7] and distillability [12].The description of the entanglement and distillability properties of systems with more than two particles is still almost unexplored (see Refs. [13,14], however). In this Letter we provide a complete classification of a family of states of three-particle systems. These states are characterized in terms of four parameters and play the role of WS in such systems. In order to classify 3-qubit states with respect to their entanglement, we define five different classes. To display the distillability properties, we introduce a powerful purification procedure. We also generalize our results to systems of N qubits. Among other things, this allows us to give the necessary...

No abstract