Quantum phase transitions occur at zero temperature and involve the appearance of long-range correlations. These correlations are not due to thermal fluctuations but to the intricate structure of a strongly entangled ground state of the system. We present a microscopic computation of the scaling properties of the ground-state entanglement in several 1D spin chain models both near and at the quantum critical regimes. We quantify entanglement by using the entropy of the ground state when the system is traced down to L spins. This entropy is seen to scale logarithmically with L, with a coefficient that corresponds to the central charge associated to the conformal theory that describes the universal properties of the quantum phase transition. Thus we show that entanglement, a key concept of quantum information science, obeys universal scaling laws as dictated by the representations of the conformal group and its classification motivated by string theory. This connection unveils a monotonicity law for ground-state entanglement along the renormalization group flow. We also identify a majorization rule possibly associated to conformal invariance and apply the present results to interpret the breakdown of density matrix renormalization group techniques near a critical point. The study of entanglement in composite systems is one of the major goals of quantum information science [1,2], where entangled states are regarded as a valuable resource for processing information in novel ways. For instance, the entanglement between systems A and B in a joint pure state |Ψ AB can be used, together with a classical channel, to teleport or send quantum information [3]. From this resource-oriented perspective, the entropy of entanglement E(Ψ AB ) measures the entanglement contained in |Ψ AB [4]. It is defined as the von Neumann entropy of the reduced density matrix ρ A (equivalently ρ B ),and directly determines, among other aspects, how much quantum information can be teleported by using |Ψ AB . On the other hand, entanglement is appointed to play a central role in the study of strongly correlated quantum systems [5,6,7], since a highly entangled ground state is at the heart of a large variety of collective quantum phenomena. Milestone examples are the entangled ground states used to explain superconductivity and the fractional quantum Hall effect, namely the BCS ansatz [8] and the Laughlin ansatz [9]. Ground-state entanglement is, most promisingly, also a key factor to understand quantum phase transitions [10,11], where it is directly responsible for the appearance of long-range correlations. Consequently, a gain of insight into phenomena including, among others, Mott insulator-superfluid transitions, quantum magnet-paramagnet transitions and phase transitions in a Fermi liquid is expected by studying the structure of entanglement in the corresponding underlying ground states.In the following we analyze the ground-state entanglement near and at a quantum critical point in a series of 1D spin-1/2 chain models. In particular, we consi...

superconducting circuits, semiconductor quantum wells, and other hybrid quantum systems. Finally, anticipated applications are highlighted utilizing USC and DSC regimes, including novel quantum optical phenomena, quantum simulation, and quantum computation. CONTENTSCaltech Caltech Caltech LKB Paris LKB Paris LKB Paris LKB Paris LKB Paris Harvard Harvard Würzburg U Tokyo ETH U Tokyo Stanford Princeton (MW) ETH (MW) Yale Delft, NTT IMS WMI, Delft WMI NICT ETH ETH UPD ISSP UPD UPD U Reg IMS U Tokyo CNRS CNRS NCU CNR ICL Caltech Caltech Caltech LKB Paris LKB Paris LKB Paris Harvard Harvard Würzburg LKB Paris LKB Paris NICT Yale Delft, NTT WMI, Delft WMI UPD IMS UPD UPD IMS ETH ETH

Robust edge states and non-Abelian excitations are the trademark of topological states of matter, with promising applications such as "topologically protected" quantum memory and computing. While so far topological phases have been exclusively discussed in a Hamiltonian context, we show that such phases and the associated topological protection and phenomena also emerge in open quantum systems with engineered dissipation. The specific system studied here is a quantum wire of spinless atomic fermions in an optical lattice coupled to a bath. The key feature of the dissipative dynamics described by a Lindblad master equation is the existence of Majorana edge modes, representing a non-local decoherence free subspace. The isolation of the edge states is enforced by a dissipative gap in the p-wave paired bulk of the wire. We describe dissipative non-Abelian braiding operations within the Majorana subspace, and we illustrate the insensitivity to imperfections. Topological protection is granted by a nontrivial winding number of the system density matrix.Topological properties can protect quantum systems from microscopic details and imperfections. In condensed matter physics this is illustrated by the seminal examples of the quantum Hall effect and the recently discovered topological insulators [1][2][3][4][5][6]. The ground state of the Hamiltonian of such systems is characterized by nonzero values of topological invariants which imply the existence of robust edge states in interfaces to topologically trivial phases. Due to their topological origin, these modes are immune against a wide class of perturbations.The conceptually simplest example illustrating these phenomena is Kitaev's quantum wire representing a topological superconducting state supporting Majorana fermions as edge states [7]. The pair of Majorana edge modes represents a nonlocal fermion which is a promising building block to encode topological qubits [8][9][10]. Similar to the Majorana excitations near vortices of a p x +ip y superconductor [11,12], they show nonabelian exchange statistics when braided in 1D wire networks [10]. Remarkably, the above described topological features and phenomena not only occur as properties of Hamiltonians, but appear also in driven dissipative quantum systems. Below we will develop such a topological program for a dissipative many-body system parallel to the Hamiltonian case. We will do this for a dissipative version of Kitaev's quantum wire. This represents the simplest instance exhibiting the key features such as dissipation induced Majorana edge modes, decoupled from the dynamically created p-wave superfluid bulk by a dissipative gap. The dissipation induced topological order is generated in stationary states far away from thermodynamic equilibrium which are not necessarily pure, i.e. described in terms of a wave function only, and is reached exponentially fast from arbitrary initial states. This is in marked contrast to recent ideas of topological order in Hamiltonian systems under non-equilibrium periodic driving con...