30 research outputs found
Entanglement and Frustration in Ordered Systems
This article reviews and extends recent results concerning entanglement and
frustration in multipartite systems which have some symmetry with respect to
the ordering of the particles. Starting point of the discussion are Bell
inequalities: their relation to frustration in classical systems and their
satisfaction for quantum states which have a symmetric extension. It is then
discussed how more general global symmetries of multipartite systems constrain
the entanglement between two neighboring particles. We prove that maximal
entanglement (measured in terms of the entanglement of formation) is always
attained for the ground state of a certain nearest neighbor interaction
Hamiltonian having the considered symmetry with the achievable amount of
entanglement being a function of the ground state energy. Systems of Gaussian
states, i.e. quantum harmonic oscillators, are investigated in more detail and
the results are compared to what is known about ordered qubit systems.Comment: 13 pages, for the Proceedings of QIT-EQIS'0
Matrix Product State Representations
This work gives a detailed investigation of matrix product state (MPS)
representations for pure multipartite quantum states. We determine the freedom
in representations with and without translation symmetry, derive respective
canonical forms and provide efficient methods for obtaining them. Results on
frustration free Hamiltonians and the generation of MPS are extended, and the
use of the MPS-representation for classical simulations of quantum systems is
discussed.Comment: Minor changes. To appear in QI
Entanglement frustration for Gaussian states on symmetric graphs
We investigate the entanglement properties of multi-mode Gaussian states,
which have some symmetry with respect to the ordering of the modes. We show how
the symmetry constraints the entanglement between two modes of the system. In
particular, we determine the maximal entanglement of formation that can be
achieved in symmetric graphs like chains, 2d and 3d lattices, mean field models
and the platonic solids. The maximal entanglement is always attained for the
ground state of a particular quadratic Hamiltonian. The latter thus yields the
maximal entanglement among all quadratic Hamiltonians having the considered
symmetry.Comment: 5 pages, 1 figur
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
Sequential generation of entangled multi-qubit states
We consider the deterministic generation of entangled multi-qubit states by
the sequential coupling of an ancillary system to initially uncorrelated
qubits. We characterize all achievable states in terms of classes of matrix
product states and give a recipe for the generation on demand of any
multi-qubit state. The proposed methods are suitable for any sequential
generation-scheme, though we focus on streams of single photon time-bin qubits
emitted by an atom coupled to an optical cavity. We show, in particular, how to
generate familiar quantum information states such as W, GHZ, and cluster
states, within such a framework.Comment: 4 pages and 2 figures, submitted for publicatio
Criticality, the Area Law, and the Computational Power of Projected Entangled Pair States
The projected entangled pair state (PEPS) representation of quantum states on two-dimensional lattices induces an entanglement based hierarchy in state space. We show that the lowest levels of this hierarchy exhibit a very rich structure including states with critical and topological properties. We prove, in particular, that coherent versions of thermal states of any local 2D classical spin model correspond to such PEPS, which are in turn ground states of local 2D quantum Hamiltonians. This correspondence maps thermal onto quantum fluctuations, and it allows us to analytically construct critical quantum models exhibiting a strict area law scaling of the entanglement entropy in the face of power law decaying correlations. Moreover, it enables us to show that there exist PEPS which can serve as computational resources for the solution of NP-hard problems
String order and symmetries in quantum spin lattices
We show that the existence of string order in a given quantum state is
intimately related to the presence of a local symmetry by proving that both
concepts are equivalent within the framework of finitely correlated states.
Once this connection is established, we provide a complete characterization of
local symmetries in these states. The results allow to understand in a
straightforward way many of the properties of string order parameters, like
their robustness/fragility under perturbations and their typical disappearance
beyond strictly one-dimensional lattices. We propose and discuss an alternative
definition, ideally suited for detecting phase transitions, and generalizations
to two and more spatial dimensions.Comment: 5 pages, 1 figur
Quantum kinetic Ising models
We introduce a quantum generalization of classical kinetic Ising models,
described by a certain class of quantum many body master equations. Similarly
to kinetic Ising models with detailed balance that are equivalent to certain
Hamiltonian systems, our models reduce to a set of Hamiltonian systems
determining the dynamics of the elements of the many body density matrix. The
ground states of these Hamiltonians are well described by matrix product, or
pair entangled projected states. We discuss critical properties of such
Hamiltonians, as well as entanglement properties of their low energy states.Comment: 20 pages, 4 figures, minor improvements, accepted in New Journal of
Physic
A variational method based on weighted graph states
In a recent article [Phys. Rev. Lett. 97 (2006), 107206], we have presented a
class of states which is suitable as a variational set to find ground states in
spin systems of arbitrary spatial dimension and with long-range entanglement.
Here, we continue the exposition of our technique, extend from spin 1/2 to
higher spins and use the boson Hubbard model as a non-trivial example to
demonstrate our scheme.Comment: 36 pages, 13 figure