640 research outputs found
A note on the factorization conjecture
We give partial results on the factorization conjecture on codes proposed by
Schutzenberger. We consider finite maximal codes C over the alphabet A = {a, b}
with C \cap a^* = a^p, for a prime number p. Let P, S in Z , with S = S_0 +
S_1, supp(S_0) \subset a^* and supp(S_1) \subset a^*b supp(S_0). We prove that
if (P,S) is a factorization for C then (P,S) is positive, that is P,S have
coefficients 0,1, and we characterize the structure of these codes. As a
consequence, we prove that if C is a finite maximal code such that each word in
C has at most 4 occurrences of b's and a^p is in C, then each factorization for
C is a positive factorization. We also discuss the structure of these codes.
The obtained results show once again relations between (positive)
factorizations and factorizations of cyclic groups
Global-to-local incompatibility, monogamy of entanglement, and ground-state dimerization: Theory and observability of quantum frustration in systems with competing interactions
Frustration in quantum many body systems is quantified by the degree of
incompatibility between the local and global orders associated, respectively,
to the ground states of the local interaction terms and the global ground state
of the total many-body Hamiltonian. This universal measure is bounded from
below by the ground-state bipartite block entanglement. For many-body
Hamiltonians that are sums of two-body interaction terms, a further inequality
relates quantum frustration to the pairwise entanglement between the
constituents of the local interaction terms. This additional bound is a
consequence of the limits imposed by monogamy on entanglement shareability. We
investigate the behavior of local pair frustration in quantum spin models with
competing interactions on different length scales and show that valence bond
solids associated to exact ground-state dimerization correspond to a transition
from generic frustration, i.e. geometric, common to classical and quantum
systems alike, to genuine quantum frustration, i.e. solely due to the
non-commutativity of the different local interaction terms. We discuss how such
frustration transitions separating genuinely quantum orders from classical-like
ones are detected by observable quantities such as the static structure factor
and the interferometric visibility.Comment: 11 pages, 7 figures. Matches published versio
Code properties from holographic geometries
Almheiri, Dong, and Harlow [arXiv:1411.7041] proposed a highly illuminating
connection between the AdS/CFT holographic correspondence and operator algebra
quantum error correction (OAQEC). Here we explore this connection further. We
derive some general results about OAQEC, as well as results that apply
specifically to quantum codes which admit a holographic interpretation. We
introduce a new quantity called `price', which characterizes the support of a
protected logical system, and find constraints on the price and the distance
for logical subalgebras of quantum codes. We show that holographic codes
defined on bulk manifolds with asymptotically negative curvature exhibit
`uberholography', meaning that a bulk logical algebra can be supported on a
boundary region with a fractal structure. We argue that, for holographic codes
defined on bulk manifolds with asymptotically flat or positive curvature, the
boundary physics must be highly nonlocal, an observation with potential
implications for black holes and for quantum gravity in AdS space at distance
scales small compared to the AdS curvature radius.Comment: 17 pages, 5 figure
Holographic Renyi Entropy from Quantum Error Correction
We study Renyi entropies in quantum error correcting codes and compare
the answer to the cosmic brane prescription for computing . We find that general operator
algebra codes have a similar, more general prescription. Notably, for the
AdS/CFT code to match the specific cosmic brane prescription, the code must
have maximal entanglement within eigenspaces of the area operator. This gives
us an improved definition of the area operator, and establishes a stronger
connection between the Ryu-Takayanagi area term and the edge modes in lattice
gauge theory. We also propose a new interpretation of existing holographic
tensor networks as area eigenstates instead of smooth geometries. This
interpretation would explain why tensor networks have historically had trouble
modeling the Renyi entropy spectrum of holographic CFTs, and it suggests a
method to construct holographic networks with the correct spectrum.Comment: 24 pages, 1 figure, V2: Fixed typos and revised explanation
Task-based adaptive multiresolution for time-space multi-scale reaction-diffusion systems on multi-core architectures
A new solver featuring time-space adaptation and error control has been
recently introduced to tackle the numerical solution of stiff
reaction-diffusion systems. Based on operator splitting, finite volume adaptive
multiresolution and high order time integrators with specific stability
properties for each operator, this strategy yields high computational
efficiency for large multidimensional computations on standard architectures
such as powerful workstations. However, the data structure of the original
implementation, based on trees of pointers, provides limited opportunities for
efficiency enhancements, while posing serious challenges in terms of parallel
programming and load balancing. The present contribution proposes a new
implementation of the whole set of numerical methods including Radau5 and
ROCK4, relying on a fully different data structure together with the use of a
specific library, TBB, for shared-memory, task-based parallelism with
work-stealing. The performance of our implementation is assessed in a series of
test-cases of increasing difficulty in two and three dimensions on multi-core
and many-core architectures, demonstrating high scalability
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