151 research outputs found
Entanglement and SU(n) symmetry in one-dimensional valence bond solid states
Here we evaluate the many-body entanglement properties of a generalized SU(n)
valence bond solid state on a chain. Our results follow from a derivation of
the transfer matrix of the system which, in combination with symmetry
properties, allows for a new, elegant and straightforward evaluation of
different entanglement measures. In particular, the geometric entanglement per
block, correlation length, von Neumann and R\'enyi entropies of a block,
localizable entanglement and entanglement length are obtained in a very simple
way. All our results are in agreement with previous derivations for the SU(2)
case.Comment: 4 pages, 2 figure
Visualizing elusive phase transitions with geometric entanglement
We show that by examining the global geometric entanglement it is possible to
identify "elusive" or hard to detect quantum phase transitions. We analyze
several one-dimensional quantum spin chains and demonstrate the existence of
non-analyticities in the geometric entanglement, in particular across a
Kosterlitz-Thouless transition and across a transition for a gapped deformed
Affleck-Kennedy-Lieb-Tasaki chain. The observed non-analyticities can be
understood and classified in connection to the nature of the transitions, and
are in sharp contrast to the analytic behavior of all the two-body reduced
density operators and their derived entanglement measures.Comment: 7 pages, 5 figures, revised version, accepted for publication in PR
Classical simulation of infinite-size quantum lattice systems in two spatial dimensions
We present an algorithm to simulate two-dimensional quantum lattice systems
in the thermodynamic limit. Our approach builds on the {\em projected
entangled-pair state} algorithm for finite lattice systems [F. Verstraete and
J.I. Cirac, cond-mat/0407066] and the infinite {\em time-evolving block
decimation} algorithm for infinite one-dimensional lattice systems [G. Vidal,
Phys. Rev. Lett. 98, 070201 (2007)]. The present algorithm allows for the
computation of the ground state and the simulation of time evolution in
infinite two-dimensional systems that are invariant under translations. We
demonstrate its performance by obtaining the ground state of the quantum Ising
model and analysing its second order quantum phase transition.Comment: 4 pages, 6 figures, 1 table. Revised version, with new diagrams and
plots. The results on classical systems can now be found at arXiv:0711.396
Entropy and Exact Matrix Product Representation of the Laughlin Wave Function
An analytical expression for the von Neumann entropy of the Laughlin wave
function is obtained for any possible bipartition between the particles
described by this wave function, for filling fraction nu=1. Also, for filling
fraction nu=1/m, where m is an odd integer, an upper bound on this entropy is
exhibited. These results yield a bound on the smallest possible size of the
matrices for an exact representation of the Laughlin ansatz in terms of a
matrix product state. An analytical matrix product state representation of this
state is proposed in terms of representations of the Clifford algebra. For
nu=1, this representation is shown to be asymptotically optimal in the limit of
a large number of particles
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
Equivalence of critical scaling laws for many-body entanglement in the Lipkin-Meshkov-Glick model
We establish a relation between several entanglement properties in the
Lipkin-Meshkov-Glick model, which is a system of mutually interacting spins
embedded in a magnetic field. We provide analytical proofs that the single-copy
entanglement and the global geometric entanglement of the ground state close to
and at criticality behave as the entanglement entropy. These results are in
deep contrast to what is found in one- dimensional spin systems where these
three entanglement measures behave differently.Comment: 4 pages, 2 figures, published versio
Exploring corner transfer matrices and corner tensors for the classical simulation of quantum lattice systems
In this paper we explore the practical use of the corner transfer matrix and
its higher-dimensional generalization, the corner tensor, to develop tensor
network algorithms for the classical simulation of quantum lattice systems of
infinite size. This exploration is done mainly in one and two spatial
dimensions (1d and 2d). We describe a number of numerical algorithms based on
corner matri- ces and tensors to approximate different ground state properties
of these systems. The proposed methods make also use of matrix product
operators and projected entangled pair operators, and naturally preserve
spatial symmetries of the system such as translation invariance. In order to
assess the validity of our algorithms, we provide preliminary benchmarking
calculations for the spin-1/2 quantum Ising model in a transverse field in both
1d and 2d. Our methods are a plausible alternative to other well-established
tensor network approaches such as iDMRG and iTEBD in 1d, and iPEPS and TERG in
2d. The computational complexity of the proposed algorithms is also considered
and, in 2d, important differences are found depending on the chosen simulation
scheme. We also discuss further possibilities, such as 3d quantum lattice
systems, periodic boundary conditions, and real time evolution. This discussion
leads us to reinterpret the standard iTEBD and iPEPS algorithms in terms of
corner transfer matrices and corner tensors. Our paper also offers a
perspective on many properties of the corner transfer matrix and its
higher-dimensional generalizations in the light of novel tensor network
methods.Comment: 25 pages, 32 figures, 2 tables. Revised version. Technical details on
some of the algorithms have been moved to appendices. To appear in PR
Matrix Product States Algorithms and Continuous Systems
A generic method to investigate many-body continuous-variable systems is
pedagogically presented. It is based on the notion of matrix product states
(so-called MPS) and the algorithms thereof. The method is quite versatile and
can be applied to a wide variety of situations. As a first test, we show how it
provides reliable results in the computation of fundamental properties of a
chain of quantum harmonic oscillators achieving off-critical and critical
relative errors of the order of 10^(-8) and 10^(-4) respectively. Next, we use
it to study the ground state properties of the quantum rotor model in one
spatial dimension, a model that can be mapped to the Mott insulator limit of
the 1-dimensional Bose-Hubbard model. At the quantum critical point, the
central charge associated to the underlying conformal field theory can be
computed with good accuracy by measuring the finite-size corrections of the
ground state energy. Examples of MPS-computations both in the finite-size
regime and in the thermodynamic limit are given. The precision of our results
are found to be comparable to those previously encountered in the MPS studies
of, for instance, quantum spin chains. Finally, we present a spin-off
application: an iterative technique to efficiently get numerical solutions of
partial differential equations of many variables. We illustrate this technique
by solving Poisson-like equations with precisions of the order of 10^(-7).Comment: 22 pages, 14 figures, final versio
Simulation of many-qubit quantum computation with matrix product states
Matrix product states provide a natural entanglement basis to represent a
quantum register and operate quantum gates on it. This scheme can be
materialized to simulate a quantum adiabatic algorithm solving hard instances
of a NP-Complete problem. Errors inherent to truncations of the exact action of
interacting gates are controlled by the size of the matrices in the
representation. The property of finding the right solution for an instance and
the expected value of the energy are found to be remarkably robust against
these errors. As a symbolic example, we simulate the algorithm solving a
100-qubit hard instance, that is, finding the correct product state out of ~
10^30 possibilities. Accumulated statistics for up to 60 qubits point at a slow
growth of the average minimum time to solve hard instances with
highly-truncated simulations of adiabatic quantum evolution.Comment: 5 pages, 4 figures, final versio
Universal geometric entanglement close to quantum phase transitions
Under successive Renormalization Group transformations applied to a quantum
state of finite correlation length , there is typically a
loss of entanglement after each iteration. How good it is then to replace
by a product state at every step of the process? In this paper we
give a quantitative answer to this question by providing first analytical and
general proofs that, for translationally invariant quantum systems in one
spatial dimension, the global geometric entanglement per region of size diverges with the correlation length as
close to a quantum critical point with central charge , where is
a cut-off at short distances. Moreover, the situation at criticality is also
discussed and an upper bound on the critical global geometric entanglement is
provided in terms of a logarithmic function of .Comment: 4 pages, 3 figure
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