We present a continuous time generalization of a random walk with complete memory of its

We study entanglement between the spin components of the Bardeen-Cooper-Schrieffer (BCS) ground state by calculating the full entanglement spectrum and the corresponding von Neumann entanglement entropy. The entanglement spectrum is effectively modeled by a generalized Gibbs ensemble (GGE) of non-interacting electrons, which may be approximated by a canonical ensemble at the BCS critical temperature. We further demonstrate that the entanglement entropy is jointly proportional to the pairing energy and to the number of electrons about the Fermi surface (an area law). Furthermore, the entanglement entropy is also proportional to the number fluctuations of either spin component in the BCS state.PACS numbers: 74.20.Fg, 03.65.Ud Bipartite entanglement in a pure state, say ρ AB = |ψ ψ|, arises from quantum correlations between subsystem partitions A and B. Due to these correlations measurements performed on one partition, say A, exhibits fluctuations of purely quantum character. Complete information on these subsystem fluctuations is contained in the reduced density operator ρ A = tr B ρ AB , which is obtained by averaging over a complete set of states belonging to B. Quantifying entanglement in ρ AB involves measuring the degree of uncertainty of the underlying probability distribution over projections onto the Schmidt states of ρ A (that is, the eigenvalues and eigenvectors of the reduced density operator). A popular scalar measure used for this purpose is the von Neumann entanglement entropywhich is identical to the Gibbs entropy associated with the probability distribution {p i } = spec ρ A . Alternatively, the full eigenvalue spectrum of ρ A may be used as a measure of entanglement in pure states, because comparisons with effective thermal distributions can sometimes provide additional physical insight. 1-3Many recent studies of entanglement entropy in manyparticle systems focus on correlations between spatial partitions.4 This emphasis may be based on some current designs of quantum computers that manipulate entangled qubits that are separated in space. 5,6 However, the more general idea of entanglement as a manifestation of quantum correlations makes studies of entanglement under other partitioning schemes valuable in the understanding of interacting systems. For instance, a general scheme for the computation of modewise entanglement entropy that is relevant to the system discussed here has been derived for bosonic 7-10 and fermionic 11,12 Gaussian states. One of the main conclusions in these papers is that the analysis of mode entanglement in such Gaussian states can be reduced to an analysis of two-mode (pair-wise) entanglement, which greatly simplifies the theoretical study of entanglement in these many-body systems. Also, mode entanglement has been studied previously in the context of examining single-particle nonlocal quantum effects (Bell inequalities)13-17 and extractable entanglement from assemblies of identical particles for quantum information processing tasks (entanglement of particles). 18-23In t...

We study the statistics of the work done in a zero temperature quench of the coupling constant in the Dicke model describing the interaction between an ensemble of two level systems and a single bosonic mode. When either the final or the initial coupling constants approach the critical coupling lambdac that separates the normal and superradiant phases of the system, the probability distribution of the work done displays singular behavior. The average work tends to diverge as the initial coupling parameter is brought closer to the critical value lambdac. In contrast, for quenches ending close to criticality, the distribution of work has finite moments but displays a sequence of edge singularities. This contrasting behavior is related to the difference between the processes of compression and expansion of a particle subject to a sudden change in its confining potential. We confirm this by studying in detail the time-dependent statistics of other observables, such as the quadratures of the photons and the total occupation of the bosonic modes.

We give an exact solution to the generalized Langevin equation of motion of a charged Brownian particle in a uniform magnetic field that is driven internally by an exponentially correlated stochastic force. A strong dissipation regime is described in which the ensemble-averaged fluctuations of the velocity exhibit transient oscillations that arise from memory effects. Also, we calculate generalized diffusion coefficients describing the transport of these particles and briefly discuss how they are affected by the magnetic field strength and correlation time. Our asymptotic results are extended to the general case of internal driving by correlated Gaussian stochastic forces with finite autocorrelation times.

We derive exact analytic expressions for the average work done and work fluctuations in instantaneous quenches of the ground and thermal states of a one-dimensional anisotropic XY model. The average work and a quantum fluctuation relation is used to determine the amount of irreversible entropy produced during the quench, eventually revealing how the closing of the excitation gap leads to increased dissipated work. The work fluctuation is calculated and shown to exhibit nonanalytic behavior as the prequench anisotropy parameter and transverse field are tuned across quantum critical points. Exact compact formulas for the average work and work fluctuation in ground state quenches of the transverse field Ising model allow us to calculate the first singular field derivative at the critical field values.

We calculate the reduced density matrix of a block of integer spin-S's in a q-deformed valence-bond-solid (VBS) state. This matrix is diagonalized exactly for an infinitely long block in an infinitely long chain. We construct an effective Hamiltonian with the same spectrum as the logarithm of the density matrix. We also derive analytic expressions for the von Neumann and Rényi entanglement entropies. For blocks of finite length, we calculate the eigenvalues of the reduced density matrix by perturbation theory and numerical diagonalization. These results enable us to describe the effects of finite-size corrections on the entanglement spectrum and entropy in this generalized VBS model.

We study a one-dimensional multicomponent anyon model that reduces to a multicomponent Lieb-Liniger gas of impenetrable bosons (Tonks-Girardeau gas) for vanishing statistics parameter. At fixed component densities, the coordinate Bethe ansatz gives a family of quantum phase transitions at special values of the statistics parameter. We show that the ground state energy changes extensively between different phases. Special regimes are studied and a general classification for the transition points is given. An interpretation in terms of statistics of composite particles is proposed.PACS numbers: 05.30. Pr, 02.30.Ik, 73.43.Nq Introduction-Anyons play an important role in topological quantum computation. 1,2 They are the natural quasiparticle excitations of fractional quantum Hall states. 3 While the fractional quantum Hall effect occurs in a two-dimensional electron liquid, its edge excitations are described by one-dimensional anyons. 4,5 These edge states are essential for performing quantum computation operations by topological quantum gates. 6,7 A model for one-dimensional anyons is the Lieb-Liniger model 8 with fractional exchange statistics. 9 An important feature of this model is that it is solvable by the Bethe ansatz. 10 In this model the statistics parameter (exchange phase) does not modify the bulk ground state energy in large systems (correction terms vanish in the thermodynamic limit). 11 However, the effect of anyon statistics on other nonlocal physical quantities are more prominent. For example, momentum distributions, reduced density matrices, and other correlation functions have a nontrivial dependence on the statistics parameter. 11-22 Experiments involving photons 23,24 and ultracold atoms 25 that have anyonic exchange statistics in one-dimension are accessible with current technology.In this paper, we generalize the anyonic Lieb-Liniger model to the multicomponent case. As known for the integrable multicomponent Bose gas, 26-28 the additional internal degrees of freedom lead to diverse physical effects. [29][30][31][32][33][34][35][36][37][38] In particular, we consider a gas of two mutually impenetrable anyonic species. This exactly solvable model may prove useful in studies of the edge states of bilayer fractional quantum Hall systems. [39][40][41][42] We encounter an interesting quantum phase transition 43 (QPT) that is controlled by the statistics parameters of the model. This QPT distinguishes between two phases with ground states that are described by either one or two Dirac seas (one-dimensional Fermi spheres). The ground state energies of these phases can therefore differ significantly by an extensive amount.We first motivate this work by describing a similar QPT that is controlled by the dynamical interparticle coupling in a multicomponent boson gas. We then proceed with a formal description of our multicomponent anyonic model. This model is formulated in terms of quantum fields satisfying generalized commutation relations. Next, we calculate exact eigenfunctions and the

PACS 75.10.Pq -Spin chain models PACS 75.10.Jm -Quantized spin models, including quantum spin frustration PACS 03.65.Ud -QM Entanglement and quantum nonlocality Abstract -We exactly calculate the reduced density matrix of matrix product states (MPS). Our compact result enables one to perform analytic studies of entanglement in MPS. In particular, we consider the MPS ground states of two anisotropic spin chains. One is a q-deformed Affleck-Kennedy-Lieb-Tasaki (AKLT) model and the other is a general spin-1 quantum antiferromagnet with nearest-neighbor interactions. Our analysis shows how anisotropy affects entanglement on different continuous parameter manifolds. We also construct an effective boundary spin model that describes a block of spins in the ground state of the q-deformed AKLT Hamiltonian. The temperature of this effective model is given in terms of the deformation parameter q.