24 research outputs found
Matrix Product Density Operators: Renormalization Fixed Points and Boundary Theories
We consider the tensors generating matrix product states and density
operators in a spin chain. For pure states, we revise the renormalization
procedure introduced by F. Verstraete et al. in 2005 and characterize the
tensors corresponding to the fixed points. We relate them to the states
possessing zero correlation length, saturation of the area law, as well as to
those which generate ground states of local and commuting Hamiltonians. For
mixed states, we introduce the concept of renormalization fixed points and
characterize the corresponding tensors. We also relate them to concepts like
finite correlation length, saturation of the area law, as well as to those
which generate Gibbs states of local and commuting Hamiltonians. One of the
main result of this work is that the resulting fixed points can be associated
to the boundary theories of two-dimensional topological states, through the
bulk-boundary correspondence introduced by Cirac et al. in 2011.Comment: 63 pages, Annals of Physics (2016). Accepted versio
Nonlocal resources in the presence of Superselection Rules
Superselection rules severely alter the possible operations that can be
implemented on a distributed quantum system. Whereas the restriction to local
operations imposed by a bipartite setting gives rise to the notion of
entanglement as a nonlocal resource, the superselection rule associated with
particle number conservation leads to a new resource, the \emph{superselection
induced variance} of local particle number. We show that, in the case of pure
quantum states, one can quantify the nonlocal properties by only two additive
measures, and that all states with the same measures can be asymptotically
interconverted into each other by local operations and classical communication.
Furthermore we discuss how superselection rules affect the concepts of
majorization, teleportation and mixed state entanglement.Comment: 4 page
Edge theories in Projected Entangled Pair State models
We study the edge physics of gapped quantum systems in the framework of
Projected Entangled Pair State (PEPS) models. We show that the effective
low-energy model for any region acts on the entanglement degrees of freedom at
the boundary, corresponding to physical excitations located at the edge. This
allows us to determine the edge Hamiltonian in the vicinity of PEPS models, and
we demonstrate that by choosing the appropriate bulk perturbation, the edge
Hamiltonian can exhibit a rich phase diagram and phase transitions. While for
models in the trivial phase any Hamiltonian can be realized at the edge, we
show that for topological models, the edge Hamiltonian is constrained by the
topological order in the bulk which can e.g. protect a ferromagnetic Ising
chain at the edge against spontaneous symmetry breaking.Comment: 5 pages, 4 figure
One-shot entanglement generation over long distances in noisy quantum networks
We consider the problem of creating a long-distance entangled state between
two stations of a network, where neighboring nodes are connected by noisy
quantum channels. We show that any two stations can share an entangled pair if
the effective probability for the quantum errors is below a certain threshold,
which is achieved by using local redundant encoding to preserve the global
phase and network-based correction for the bit-flip errors. In contrast to the
convensional quantum repeater schemes we are not limited by the memory
coherence time, because all quantum operations only use one-way classical
communication and can be done in one shot. Meanwhile, the overhead of local
resources only increases logarithmically with the size of the network, making
our proposal favorable to practical applications of long-distance quantum
communication.Comment: revtex4, 6 pages, 5 figures (.eps
Entanglement renormalization and boundary critical phenomena
The multiscale entanglement renormalization ansatz is applied to the study of
boundary critical phenomena. We compute averages of local operators as a
function of the distance from the boundary and the surface contribution to the
ground state energy. Furthermore, assuming a uniform tensor structure, we show
that the multiscale entanglement renormalization ansatz implies an exact
relation between bulk and boundary critical exponents known to exist for
boundary critical systems.Comment: 6 pages, 4 figures; for a related work see arXiv:0912.164
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
Exact symmetry breaking ground states for quantum spin chains
We introduce a family of spin-1/2 quantum chains, and show that their exact
ground states break the rotational and translational symmetries of the original
Hamiltonian. We also show how one can use projection to construct a spin-3/2
quantum chain with nearest neighbor interaction, whose exact ground states
break the rotational symmetry of the Hamiltonian. Correlation functions of both
models are determined in closed form. Although we confine ourselves to
examples, the method can easily be adapted to encompass more general models.Comment: 4 pages, RevTex. 4 figures, minor changes, new reference
From density-matrix renormalization group to matrix product states
In this paper we give an introduction to the numerical density matrix
renormalization group (DMRG) algorithm, from the perspective of the more
general matrix product state (MPS) formulation. We cover in detail the
differences between the original DMRG formulation and the MPS approach,
demonstrating the additional flexibility that arises from constructing both the
wavefunction and the Hamiltonian in MPS form. We also show how to make use of
global symmetries, for both the Abelian and non-Abelian cases.Comment: Numerous small changes and clarifications, added a figur
On entropy growth and the hardness of simulating time evolution
The simulation of quantum systems is a task for which quantum computers are
believed to give an exponential speedup as compared to classical ones. While
ground states of one-dimensional systems can be efficiently approximated using
Matrix Product States (MPS), their time evolution can encode quantum
computations, so that simulating the latter should be hard classically.
However, one might believe that for systems with high enough symmetry, and thus
insufficient parameters to encode a quantum computation, efficient classical
simulation is possible. We discuss supporting evidence to the contrary: We
provide a rigorous proof of the observation that a time independent local
Hamiltonian can yield a linear increase of the entropy when acting on a product
state in a translational invariant framework. This criterion has to be met by
any classical simulation method, which in particular implies that every global
approximation of the evolution requires exponential resources for any MPS based
method.Comment: 15 pages. v2: Published version, Journal-Ref. adde
Entanglement and correlation functions following a local quench: a conformal field theory approach
We show that the dynamics resulting from preparing a one-dimensional quantum
system in the ground state of two decoupled parts, then joined together and
left to evolve unitarily with a translational invariant Hamiltonian (a local
quench), can be described by means of quantum field theory. In the case when
the corresponding theory is conformal, we study the evolution of the
entanglement entropy for different bi-partitions of the line. We also consider
the behavior of one- and two-point correlation functions. All our findings may
be explained in terms of a picture, that we believe to be valid more generally,
whereby quasiparticles emitted from the joining point at the initial time
propagate semiclassically through the system.Comment: 19 pages, 4 figures, v2 typos corrected and refs adde