36 research outputs found
Multiplicativity of the maximal output 2-norm for depolarized Werner-Holevo channels
We study the multiplicativity of the output 2-norm for depolarized
Werner-Holevo channels and show that multiplicativity holds for a product of
two identical channels in this class. Moreover, it shown that the depolarized
Werner-Holevo channels do not satisfy the entrywise positivity condition
introduced by C. King and M.B. Ruskai, which suggests that the main result is
non-trivial.Comment: 3 page
Quantization of Hall Conductance For Interacting Electrons on a Torus
We consider interacting, charged spins on a torus described by a gapped
Hamiltonian with a unique groundstate and conserved local charge. Using
quasi-adiabatic evolution of the groundstate around a flux-torus, we prove,
without any averaging assumption, that the Hall conductance of the groundstate
is quantized in integer multiples of e^2/h, up to exponentially small
corrections in the linear size of the system. In addition, we discuss
extensions to the fractional quantization case under an additional topological
order assumption on the degenerate groundstate subspace.Comment: 28 pages, 4 figures, This paper significantly simplifies the proof
and tightens the bounds previously shown in arXiv:0911.4706 by the same
authors. Updated to reflect published versio
Space from Hilbert Space: Recovering Geometry from Bulk Entanglement
We examine how to construct a spatial manifold and its geometry from the
entanglement structure of an abstract quantum state in Hilbert space. Given a
decomposition of Hilbert space into a tensor product of factors,
we consider a class of "redundancy-constrained states" in that
generalize the area-law behavior for entanglement entropy usually found in
condensed-matter systems with gapped local Hamiltonians. Using mutual
information to define a distance measure on the graph, we employ classical
multidimensional scaling to extract the best-fit spatial dimensionality of the
emergent geometry. We then show that entanglement perturbations on such
emergent geometries naturally give rise to local modifications of spatial
curvature which obey a (spatial) analog of Einstein's equation. The Hilbert
space corresponding to a region of flat space is finite-dimensional and scales
as the volume, though the entropy (and the maximum change thereof) scales like
the area of the boundary. A version of the ER=EPR conjecture is recovered, in
that perturbations that entangle distant parts of the emergent geometry
generate a configuration that may be considered as a highly quantum wormhole.Comment: 37 pages, 5 figures. Updated notation, references, and
acknowledgemen
Persistence of locality in systems with power-law interactions
Motivated by recent experiments with ultra-cold matter, we derive a new bound
on the propagation of information in -dimensional lattice models exhibiting
interactions with . The bound contains two terms: One
accounts for the short-ranged part of the interactions, giving rise to a
bounded velocity and reflecting the persistence of locality out to intermediate
distances, while the other contributes a power-law decay at longer distances.
We demonstrate that these two contributions not only bound but, except at long
times, \emph{qualitatively reproduce} the short- and long-distance dynamical
behavior following a local quench in an chain and a transverse-field Ising
chain. In addition to describing dynamics in numerous intractable long-range
interacting lattice models, our results can be experimentally verified in a
variety of ultracold-atomic and solid-state systems.Comment: 5 pages, 4 figures, version accepted by PR
Approximating the ground state of gapped quantum spin systems
We consider quantum spin systems defined on finite sets equipped with a
metric. In typical examples, is a large, but finite subset of Z^d. For
finite range Hamiltonians with uniformly bounded interaction terms and a
unique, gapped ground state, we demonstrate a locality property of the
corresponding ground state projector. In such systems, this ground state
projector can be approximated by the product of observables with quantifiable
supports. In fact, given any subset, X, of V the ground state projector can be
approximated by the product of two projections, one supported on X and one
supported on X^c, and a bounded observable supported on a boundary region in
such a way that as the boundary region increases, the approximation becomes
better. Such an approximation was useful in proving an area law in one
dimension, and this result corresponds to a multi-dimensional analogue
Stability of Frustration-Free Hamiltonians
We prove stability of the spectral gap for gapped, frustration-free Hamiltonians under general, quasi-local perturbations. We present a necessary and sufficient condition for stability, which we call Local Topological Quantum Order and show that this condition implies an area law for the entanglement entropy of the groundstate subspace. This result extends previous work by Bravyi et al. on the stability of topological quantum order for Hamiltonians composed of commuting projections with a common zero-energy subspace. We conclude with a list of open problems relevant to spectral gaps and topological quantum order
Stability of Local Quantum Dissipative Systems
This is the author accepted manuscript. The final version is available from Springer at http://link.springer.com/article/10.1007%2Fs00220-015-2355-3.Open quantum systems weakly coupled to the environment are modeled
by completely positive, trace preserving semigroups of linear maps. The
generators of such evolutions are called Lindbladians. In the setting of quantum
many-body systems on a lattice it is natural to consider Lindbladians that decompose
into a sum of local interactions with decreasing strength with respect to
the size of their support. For both practical and theoretical reasons, it is crucial
to estimate the impact that perturbations in the generating Lindbladian, arising
as noise or errors, can have on the evolution. These local perturbations are potentially
unbounded, but constrained to respect the underlying lattice structure.
We show that even for polynomially decaying errors in the Lindbladian, local
observables and correlation functions are stable if the unperturbed Lindbladian
has a unique fixed point and a mixing time which scales logarithmically with
the system size. The proof relies on Lieb-Robinson bounds, which describe a
finite group velocity for propagation of information in local systems. As a main
example, we prove that classical Glauber dynamics is stable under local perturbations,
including perturbations in the transition rates which may not preserve
detailed balance
Robustness in projected entangled pair states
We analyze a criterion which guarantees that the ground states of certain many-body systems are stable under perturbations. Specifically, we consider PEPS, which are believed to provide an efficient description, based on local tensors, for the low energy physics arising from local interactions. In order to assess stability in the framework of PEPS, one thus needs to understand how physically allowed perturbations of the local tensor affect the properties of the global state. In this paper, we show that a restricted version of the local topological quantum order (LTQO) condition [Michalakis and Pytel, Commun. Math. Phys. 322, 277 (2013)] provides a checkable criterion which allows us to assess the stability of local properties of PEPS under physical perturbations. We moreover show that LTQO itself is stable under perturbations which preserve the spectral gap, leading to nontrivial examples of PEPS which possess LTQO and are thus stable under arbitrary perturbations