Entanglement renormalization in fermionic systems

We demonstrate, in the context of quadratic fermion lattice models in one and two spatial dimensions, the potential of entanglement renormalization (ER) to define a proper real-space renormalization group transformation. Our results show, for the first time, the validity of the multi-scale entanglement renormalization ansatz (MERA) to describe ground states in two dimensions, even at a quantum critical point. They also unveil a connection between the performance of ER and the logarithmic violations of the boundary law for entanglement in systems with a one-dimensional Fermi surface. ER is recast in the language of creation/annihilation operators and correlation matrices.Comment: 5 pages, 4 figures Second appendix adde

Entanglement entropy in one-dimensional disordered interacting system: The role of localization

The properties of the entanglement entropy (EE) in one-dimensional disordered interacting systems are studied. Anderson localization leaves a clear signature on the average EE, as it saturates on length scale exceeding the localization length. This is verified by numerically calculating the EE for an ensemble of disordered realizations using density matrix renormalization group (DMRG). A heuristic expression describing the dependence of the EE on the localization length, which takes into account finite size effects, is proposed. This is used to extract the localization length as function of the interaction strength. The localization length dependence on the interaction fits nicely with the expectations.Comment: 5 pages, 4 figures, accepted for publication in Physical Review Letter

Boundary and impurity effects on entanglement of Heisenberg chains

We study entanglement of a pair of qubits and the bipartite entanglement between the pair and the rest within open-ended Heisenberg $XXX$ and XY models. The open boundary condition leads to strong oscillations of entanglements with a two-site period, and the two kinds of entanglements are 180 degree out of phase with each other. The mean pairwise entanglement and ground-state energy per site in the $XXX$ model are found to be proportional to each other. We study the effects of a single bulk impurity on entanglement, and find that there exists threshold values of the relative coupling strength between the impurity and its nearest neighbours, after which the impurity becomes pairwise entangled with its nearest neighbours.Comment: 6 pages and 6 figure

Renormalization group transformations on quantum states

We construct a general renormalization group transformation on quantum states, independent of any Hamiltonian dynamics of the system. We illustrate this procedure for translational invariant matrix product states in one dimension and show that product, GHZ, W and domain wall states are special cases of an emerging classification of the fixed points of this coarse--graining transformation.Comment: 5 pages, 2 figur

Quantum Phase Transitions and Bipartite Entanglement

We develop a general theory of the relation between quantum phase transitions (QPTs) characterized by nonanalyticities in the energy and bipartite entanglement. We derive a functional relation between the matrix elements of two-particle reduced density matrices and the eigenvalues of general two-body Hamiltonians of $d$-level systems. The ground state energy eigenvalue and its derivatives, whose non-analyticity characterizes a QPT, are directly tied to bipartite entanglement measures. We show that first-order QPTs are signalled by density matrix elements themselves and second-order QPTs by the first derivative of density matrix elements. Our general conclusions are illustrated via several quantum spin models.Comment: 5 pages, incl. 2 figures. v3: The version published in PRL, including a few extra comments and clarifications for which there was no space in the PR

Aplicaciones empresariales de data mining

El Data Mining, o extracción de información útil y no evidente de grandes bases de datos, es una tecnología con un gran potencial para ayudar a las empresas a focalizar sus esfuerzos alrededor de la información importante contenida en sus "data warehouses". En este artículo analizaremos las ideas básicas que sustentan el Data Mining y, más concretamente, la utilización de redes neuronales como herramienta estadística avanzada. Presentaremos también dos ejemplos reales de la aplicación de estas técnicas: predicciones bursátiles y predicción de propagación de fuego en cables eléctrico

Density of defects and the scaling law of the entanglement entropy in quantum phase transition of one dimensional spin systems induced by a quench

We have studied quantum phase transition induced by a quench in different one dimensional spin systems. Our analysis is based on the dynamical mechanism which envisages nonadiabaticity in the vicinity of the critical point. This causes spin fluctuation which leads to the random fluctuation of the Berry phase factor acquired by a spin state when the ground state of the system evolves in a closed path. The two-point correlation of this phase factor is associated with the probability of the formation of defects. In this framework, we have estimated the density of defects produced in several one dimensional spin chains. At the critical region, the entanglement entropy of a block of $L$ spins with the rest of the system is also estimated which is found to increase logarithmically with $L$. The dependence on the quench time puts a constraint on the block size $L$. It is also pointed out that the Lipkin-Meshkov-Glick model in point-splitting regularized form appears as a combination of the XXX model and Ising model with magnetic field in the negative z-axis. This unveils the underlying conformal symmetry at criticality which is lost in the sharp point limit. Our analysis shows that the density of defects as well as the scaling behavior of the entanglement entropy follows a universal behavior in all these systems.Comment: 4 figures, Accepted in Phys. Rev.

Neural network determination of the non-singlet quark distribution

We summarize the main features of our approach to parton fitting, and we show a preliminary result for the non-singlet structure function. When comparing our result to other PDF sets, we find a better description of large x data and larger error bands in the extrapolation regions.Comment: 4 pages, 1 eps figure. Presented at the XIV International Workshop on Deep Inelastic Scattering (DIS2006), Tsukuba, Japan, 20-24 April 200

Simulation of two-dimensional quantum systems using a tree tensor network that exploits the entropic area law

This work explores the use of a tree tensor network ansatz to simulate the ground state of a local Hamiltonian on a two-dimensional lattice. By exploiting the entropic area law, the tree tensor network ansatz seems to produce quasi-exact results in systems with sizes well beyond the reach of exact diagonalisation techniques. We describe an algorithm to approximate the ground state of a local Hamiltonian on a L times L lattice with the topology of a torus. Accurate results are obtained for L={4,6,8}, whereas approximate results are obtained for larger lattices. As an application of the approach, we analyse the scaling of the ground state entanglement entropy at the quantum critical point of the model. We confirm the presence of a positive additive constant to the area law for half a torus. We also find a logarithmic additive correction to the entropic area law for a square block. The single copy entanglement for half a torus reveals similar corrections to the area law with a further term proportional to 1/L.Comment: Major rewrite, new version published in Phys. Rev. B with highly improved numerical results for the scaling of the entropies and several new sections. The manuscript has now 19 pages and 30 Figure

Adiabatic quantum computation and quantum phase transitions

We analyze the ground state entanglement in a quantum adiabatic evolution algorithm designed to solve the NP-complete Exact Cover problem. The entropy of entanglement seems to obey linear and universal scaling at the point where the mass gap becomes small, suggesting that the system passes near a quantum phase transition. Such a large scaling of entanglement suggests that the effective connectivity of the system diverges as the number of qubits goes to infinity and that this algorithm cannot be efficiently simulated by classical means. On the other hand, entanglement in Grover's algorithm is bounded by a constant.Comment: 5 pages, 4 figures, accepted for publication in PR