Tensor network states and algorithms in the presence of a global U(1) symmetry

Tensor network decompositions offer an efficient description of certain many-body states of a lattice system and are the basis of a wealth of numerical simulation algorithms. In a recent paper [arXiv:0907.2994v1] we discussed how to incorporate a global internal symmetry, given by a compact, completely reducible group G, into tensor network decompositions and algorithms. Here we specialize to the case of Abelian groups and, for concreteness, to a U(1) symmetry, often associated with particle number conservation. We consider tensor networks made of tensors that are invariant (or covariant) under the symmetry, and explain how to decompose and manipulate such tensors in order to exploit their symmetry. In numerical calculations, the use of U(1) symmetric tensors allows selection of a specific number of particles, ensures the exact preservation of particle number, and significantly reduces computational costs. We illustrate all these points in the context of the multi-scale entanglement renormalization ansatz.Comment: 22 pages, 25 figures, RevTeX

Fast convergence of imaginary time evolution tensor network algorithms by recycling the environment

We propose an environment recycling scheme to speed up a class of tensor network algorithms that produce an approximation to the ground state of a local Hamiltonian by simulating an evolution in imaginary time. Specifically, we consider the time-evolving block decimation (TEBD) algorithm applied to infinite systems in 1D and 2D, where the ground state is encoded, respectively, in a matrix product state (MPS) and in a projected entangled-pair state (PEPS). An important ingredient of the TEBD algorithm (and a main computational bottleneck, especially with PEPS in 2D) is the computation of the so-called environment, which is used to determine how to optimally truncate the bond indices of the tensor network so that their dimension is kept constant. In current algorithms, the environment is computed at each step of the imaginary time evolution, to account for the changes that the time evolution introduces in the many-body state represented by the tensor network. Our key insight is that close to convergence, most of the changes in the environment are due to a change in the choice of gauge in the bond indices of the tensor network, and not in the many-body state. Indeed, a consistent choice of gauge in the bond indices confirms that the environment is essentially the same over many time steps and can thus be re-used, leading to very substantial computational savings. We demonstrate the resulting approach in 1D and 2D by computing the ground state of the quantum Ising model in a transverse magnetic field.Comment: 17 pages, 28 figure

Optimal distillation of a GHZ state

We present the optimal local protocol to distill a Greenberger-Horne-Zeilinger (GHZ) state from a single copy of any pure state of three qubits.Comment: RevTex, 4 pages, 2 figures. Published version, some references adde

Nonlocal Entanglement Transformations Achievable by Separable Operations

For manipulations of multipartite quantum systems, it was well known that all local operations assisted by classical communication (LOCC) constitute a proper subset of the class of separable operations. Recently, Gheorghiu and Griffiths found that LOCC and general separable operations are equally powerful for transformations between bipartite pure states. In this letter we extend this comparison to mixed states and show that in general separable operations are strictly stronger than LOCC when transforming a mixed state to a pure entangled state. A remarkable consequence of our finding is the existence of entanglement monotone which may increase under separable operations.Comment: v2 has rephrased Theorem 1 and corrected Kraus operators in Theorem 2. Additional comments are welcome

Dynamical windows for real-time evolution with matrix product states

We propose the use of a dynamical window to investigate the real-time evolution of quantum many-body systems in a one-dimensional lattice. In a recent paper [H. Phien et al, arxiv:????.????], we introduced infinite boundary conditions (IBC) in order to investigate real-time evolution of an infinite system under a local perturbation. This was accomplished by restricting the update of the tensors in the matrix product state to a finite window, with left and right boundaries held at fixed positions. Here we consider instead the use of a dynamical window, namely a window where the positions of left and right boundaries are allowed to change in time. In this way, all simulation efforts can be devoted to the space-time region of interest, which leads to a remarkable reduction in computational costs. For illustrative purposes, we consider two applications in the context of the spin-1 antiferromagnetic Heisenberg model in an infinite spin chain: one is an expanding window, with boundaries that are adjusted to capture the expansion in time of a local perturbation of the system; the other is a moving window of fixed size, where the position of the window follows the front of a propagating wave

Three qubits can be entangled in two inequivalent ways

Invertible local transformations of a multipartite system are used to define equivalence classes in the set of entangled states. This classification concerns the entanglement properties of a single copy of the state. Accordingly, we say that two states have the same kind of entanglement if both of them can be obtained from the other by means of local operations and classical communcication (LOCC) with nonzero probability. When applied to pure states of a three-qubit system, this approach reveals the existence of two inequivalent kinds of genuine tripartite entanglement, for which the GHZ state and a W state appear as remarkable representatives. In particular, we show that the W state retains maximally bipartite entanglement when any one of the three qubits is traced out. We generalize our results both to the case of higher dimensional subsystems and also to more than three subsystems, for all of which we show that, typically, two randomly chosen pure states cannot be converted into each other by means of LOCC, not even with a small probability of success.Comment: 12 pages, 1 figure; replaced with revised version; terminology adapted to earlier work; reference added; results unchange

Electrochemical and photo-electrochemical processes of Methylene blue oxidation by Ti/TiO2 electrodes modified with Fe-allophane

Indexación: Scopus.This work reports the degradation of methylene blue (MB) on Ti/TiO2 and Ti/TiO2/Fe-allophane electrodes in a pH 3 using 0.1 M Na2SO4 as support electrolyte. SEM micrographs show a homogeneous distribution of TiO2 over the whole electrode surface forming nanotubes and nanopores. Fe-allophane modified electrode shows the formation of large-grains agglomerate on the electrode surface due to allophane, which provides a greater surface area to the electrode due to meso and micropore structures. Preliminary cyclic voltammetry show that Ti/TiO2 has the typical voltammetric response due to Ti(III)/Ti(IV) pair. Diffusional problems were observed through of the film when the electrode is modified with Fe-allophane modifying the quasi-reversible process Ti(III)/Ti(IV). Different kind of methodologies in the degradation process were used: Electrochemistry (EC), Photochemistry (PC), Photoelectrochemistry (PEC) and Adsorption (Ads). These methods were developing to discard any reaction or interaction that is not of interest. On Ti/TiO2 with PC and Ads methodologies was not observed any activity to MB degradation showing that is not photosensitive and that the interaction between this and surface electrode is low. But with EC and PEC degradation to 55% is reached after 3 hours of electrolysis. With Ti/TiO2-Fe-allophane electrodes are observed a higher activity for all methodologies. The PC and Ads methods show that the MB degradation reaches to ∼20 % of the initial concentration. As mentioned above, the PC and Ads processes no show degradation on Ti/TiO2, therefore the degradation it only due to the adsorption of MB in/on allophane coat behaving as concentrator matrix. A lower improvement is observed with EC process when is incorporated Ti/TiO2-Fe-allophane is due to the barrier of the electrode surface by oxidation products. With PEC is reached the higher degradation value of ∼88 %, showing an improvement of the degradation with the presence of Fe-allophane. The results indicate that the main role of Fe-allophane on the electrode is similar to a concentrator matrix.http://ref.scielo.org/shz7t