128,983 research outputs found
Generalized Performance of Concatenated Quantum Codes -- A Dynamical Systems Approach
We apply a dynamical systems approach to concatenation of quantum error
correcting codes, extending and generalizing the results of Rahn et al. [1] to
both diagonal and nondiagonal channels. Our point of view is global: instead of
focusing on particular types of noise channels, we study the geometry of the
coding map as a discrete-time dynamical system on the entire space of noise
channels. In the case of diagonal channels, we show that any code with distance
at least three corrects (in the infinite concatenation limit) an open set of
errors. For Calderbank-Shor-Steane (CSS) codes, we give a more precise
characterization of that set. We show how to incorporate noise in the gates,
thus completing the framework. We derive some general bounds for noise
channels, which allows us to analyze several codes in detail.Comment: 12 pages two-column format, no figures, slightly revised versio
Spectral noncommutative geometry and quantization: a simple example
We explore the relation between noncommutative geometry, in the spectral
triple formulation, and quantum mechanics. To this aim, we consider a dynamical
theory of a noncommutative geometry defined by a spectral triple, and study its
quantization. In particular, we consider a simple model based on a finite
dimensional spectral triple (A, H, D), which mimics certain aspects of the
spectral formulation of general relativity. We find the physical phase space,
which is the space of the onshell Dirac operators compatible with A and H. We
define a natural symplectic structure over this phase space and construct the
corresponding quantum theory using a covariant canonical quantization approach.
We show that the Connes distance between certain two states over the algebra A
(two ``spacetime points''), which is an arbitrary positive number in the
classical noncommutative geometry, turns out to be discrete in the quantum
theory, and we compute its spectrum. The quantum states of the noncommutative
geometry form a Hilbert space K. D is promoted to an operator *D on the direct
product *H of H and K. The triple (A, *H, *D) can be viewed as the quantization
of the family of the triples (A, H, D).Comment: 7 pages, no figure
Nonlinear mobility continuity equations and generalized displacement convexity
We consider the geometry of the space of Borel measures endowed with a
distance that is defined by generalizing the dynamical formulation of the
Wasserstein distance to concave, nonlinear mobilities. We investigate the
energy landscape of internal, potential, and interaction energies. For the
internal energy, we give an explicit sufficient condition for geodesic
convexity which generalizes the condition of McCann. We take an eulerian
approach that does not require global information on the geodesics. As
by-product, we obtain existence, stability, and contraction results for the
semigroup obtained by solving the homogeneous Neumann boundary value problem
for a nonlinear diffusion equation in a convex bounded domain. For the
potential energy and the interaction energy, we present a non-rigorous argument
indicating that they are not displacement semiconvex.Comment: 33 pages, 1 figur
Strong Brane Gravity and the Radion at Low Energies
For the 2-brane Randall-Sundrum model, we calculate the bulk geometry for
strong gravity, in the low matter density regime, for slowly varying matter
sources. This is relevant for astrophysical or cosmological applications. The
warped compactification means the radion can not be written as a homogeneous
mode in the orbifold coordinate, and we introduce it by extending the
coordinate patch approach of the linear theory to the non-linear case. The
negative tension brane is taken to be in vacuum. For conformally invariant
matter on the positive tension brane, we solve the bulk geometry as a
derivative expansion, formally summing the `Kaluza-Klein' contributions to all
orders. For general matter we compute the Einstein equations to leading order,
finding a scalar-tensor theory with ,
and geometrically interpret the radion. We comment that this radion scalar may
become large in the context of strong gravity with low density matter.
Equations of state allowing to be negative, can exhibit behavior
where the matter decreases the distance between the 2 branes, which we
illustrate numerically for static star solutions using an incompressible fluid.
For increasing stellar density, the branes become close before the upper mass
limit, but after violation of the dominant energy condition. This raises the
interesting question of whether astrophysically reasonable matter, and initial
data, could cause branes to collide at low energy, such as in dynamical
collapse.Comment: 24 pages, 3 figure
(Quantum) Space-Time as a Statistical Geometry of Fuzzy Lumps and the Connection with Random Metric Spaces
We develop a kind of pregeometry consisting of a web of overlapping fuzzy
lumps which interact with each other. The individual lumps are understood as
certain closely entangled subgraphs (cliques) in a dynamically evolving network
which, in a certain approximation, can be visualized as a time-dependent random
graph. This strand of ideas is merged with another one, deriving from ideas,
developed some time ago by Menger et al, that is, the concept of probabilistic-
or random metric spaces, representing a natural extension of the metrical
continuum into a more microscopic regime. It is our general goal to find a
better adapted geometric environment for the description of microphysics. In
this sense one may it also view as a dynamical randomisation of the causal-set
framework developed by e.g. Sorkin et al. In doing this we incorporate, as a
perhaps new aspect, various concepts from fuzzy set theory.Comment: 25 pages, Latex, no figures, some references added, some minor
changes added relating to previous wor
Hamilton-Jacobi Theory and Information Geometry
Recently, a method to dynamically define a divergence function for a
given statistical manifold by means of the
Hamilton-Jacobi theory associated with a suitable Lagrangian function
on has been proposed. Here we will review this
construction and lay the basis for an inverse problem where we assume the
divergence function to be known and we look for a Lagrangian function
for which is a complete solution of the associated
Hamilton-Jacobi theory. To apply these ideas to quantum systems, we have to
replace probability distributions with probability amplitudes.Comment: 8 page
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