The recent interest in aspects common to quantum information and condensed matter has prompted a flory of activity at the border of these disciplines that were far distant untill few years ago. Numerous interesting questions have been addressed so far. Here we review an important part of this field, the properties of the entanglement in many-body systems. We discuss the zero and finite temperature properties of entanglement in interacting spin, fermion and boson model systems. Both bipartite and multipartite entanglement will be considered. In equilibrium we show how entanglement is tightly connected to the characteristics of the phase diagram. The behavior of entanglement can be related, via certain witnesses, to thermodynamic quantities thus offering interesting possibilities for an experimental test. Out of equilibrium we discuss how to generate and manipulate entangled states by means of many-body Hamiltonians.

In this Letter we discuss the entanglement near a quantum phase transition by analyzing the properties of the concurrence for a class of exactly solvable models in one dimension. We find that entanglement can be classified in the framework of scaling theory. Further, we reveal a profound difference between classical correlations and the non-local quantum correlation, entanglement: the correlation length diverges at the phase transition, whereas entanglement in general remains short ranged.Classical phase transitions occur when a physical system reaches a state below a critical temperature characterized by a macroscopic order [1]. Quantum phase transitions occur at absolute zero; they are induced by the change of an external parameter or coupling constant [2], and are driven by fluctuations. Examples include transitions in quantum Hall systems [3], localization in Si-MOSFETs (metal oxide silicon field-effect transistors; Ref. [4]) and the superconductor-insulator transition in two-dimensional systems [5,6]. Both classical and quantum critical points are governed by a diverging correlation length, although quantum systems possess additional correlations that do not have a classical counterpart. This phenomenon, known as entanglement [7], is the resource that enables quantum computation and communication [8]. The role of the entanglement at a phase transition is not captured by statistical mechanics -a complete classification of the critical many-body state requires the introduction of concepts from quantum information theory [9]. Here we connect the theory of critical phenomena with quantum information by exploring the entangling resources of a system close to its quantum critical point. We demonstrate, for a class of one-dimensional magnetic systems, that entanglement shows scaling behaviour in the vicinity of the transition point.There are various questions that emerge in the study of this problem. Since the ground state wave-function undergoes qualitative changes at a quantum phase transition, it is important to understand how its genuine quantum aspects evolve throughout the transition. Will entanglement between distant subsystems be extended over macroscopic regions, as correlations are? Will it carry distinct features of the transition itself and show scaling behaviour? Answering these questions is important for a deeper understanding of quantum phase transitions and also from the perspective of quantum information theory. So results that bridge these two areas of research are of great relevance. We study a set of localized spins coupled through exchange interaction and subject to an external magnetic field (we consider only spin-1/2 particles), a model central both to condensed matter and information theory and subject to intense study [10]. FIG. 1. The change in the ground state wave-function in the critical region is analyzed considering ∂ λ C(1) as a function of the reduced coupling strength λ. The curves correspond to different lattice sizes N = 11, 41, 101, 251, 401, ∞. We choose N odd to avoid the sub...

We study the dynamics of quantum correlations in a class of exactly solvable Ising-type models. We analyze in particular the time evolution of initial Bell states created in a fully polarized background and on the ground state. We find that the pairwise entanglement propagates with a velocity proportional to the reduced interaction for all the four Bell states. Singlet-like states are favored during the propagation, in the sense that triplet-like states change their character during the propagation under certain circumstances. Characteristic for the anisotropic models is the instantaneous creation of pairwise entanglement from a fully polarized state; furthermore, the propagation of pairwise entanglement is suppressed in favor of a creation of different types of entanglement. The "entanglement wave" evolving from a Bell state on the ground state turns out to be very localized in space-time. Further support to a recently formulated conjecture on entanglement sharing is given.

We present a method to construct entanglement measures for pure states of multipartite qubit systems. The key element of our approach is an antilinear operator that we call comb in reference to the hairy-ball theorem. For qubits (or spin 1/2) the combs are automatically invariant under SL(2, C). This implies that the filters obtained from the combs are entanglement monotones by construction. We give alternative formulae for the concurrence and the 3-tangle as expectation values of certain antilinear operators. As an application we discuss inequivalent types of genuine four-qubit entanglement.Entanglement is one the most striking features of quantum mechanics, but it is also one of its most counterintuitive consequences of which we still have rather incomplete knowledge [1]. Although the concentrated effort during the past decade has produced an impressive progress, there is no general qualitative and quantitative theory of entanglement.A pure quantum-mechanical state of distinguishable particles is called disentangled with respect to a given partition P of the system iff it can be written as a tensor product of the parts of this partition. In the opposite case, the state must contain some finite amount of entanglement. The question then is to characterize and quantify this entanglement.As to measuring the amount of entanglement in a given pure multipartite state, the first major step was made by Bennett et al.[2] who discovered that the partial entropy of a party in a bipartite quantum state is a measure of entanglement. It coincides (asymptotically) with the entanglement of formation (i.e., the number of Einstein-Podolsky-Rosen pairs required to prepare a given state). Subsequently, the entanglement of formation of a two-qubit state was related to the concurrence [3,4]. Interestingly, by exploiting the knowledge of the mixed-state concurrence, a measure for three-partite pure states could be derived, the so-called 3-tangle τ 3 [5].

We provide a complete analysis of mixed three-qubit states composed of a Greenberger-Horne-Zeilinger 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 that, while the amounts of inequivalent entanglement types strictly add up for pure states, this "monogamy" can be lifted for mixed states by virtue of vanishing tangle measures.