Separability and Distillability of Multiparticle Quantum Systems

Abstract

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...

Separability properties of three-mode Gaussian states

“…We will use the scheme introduced in [9], to classify three-mode tripartite Gaussian states. The important point is that from the extension of Theorem 2 we can derive a simple criterion that allows to determine which class a given state belongs to.…”

“…In this paper we provide a complete classification of tri-mode entanglement (according to the scheme [9]) and obtain -in contrast to the finite dimensional case -a simple, directly computable criterion that allows to determine to which class a given state belongs. We show that none of these classes are empty and in particular provide examples of genuine tripartite bound entangled states, i.e.…”

“…Pure multipartite entanglement was first considered in [8]. A classification of Npartite mixed states according to their separability properties has been given [9]. But even for three qubits there is currently no general way to decide to which of these classes a given state belongs [10].…”

Nonclassicality and information exchange in deterministic entanglement formation

“…The resulting state is a 1-qubit biseparable (class 2) entangled state [29]. More specifically, when the modes B and C are prepared in vacuum state and with Φ = 0, the state of Eq.…”

“…Before we move on with our discussion about the initial condition role for deterministic entanglement formation we give a brief review of three partite entanglement classification as given by Dür et al [29], which we use extensively in what follows. Let us write down four possible state configuration to which we will address later…”