2,671 research outputs found
Postquantum Br\`{e}gman relative entropies and nonlinear resource theories
We introduce the family of postquantum Br\`{e}gman relative entropies, based
on nonlinear embeddings into reflexive Banach spaces (with examples given by
reflexive noncommutative Orlicz spaces over semi-finite W*-algebras,
nonassociative L spaces over semi-finite JBW-algebras, and noncommutative
L spaces over arbitrary W*-algebras). This allows us to define a class of
geometric categories for nonlinear postquantum inference theory (providing an
extension of Chencov's approach to foundations of statistical inference), with
constrained maximisations of Br\`{e}gman relative entropies as morphisms and
nonlinear images of closed convex sets as objects. Further generalisation to a
framework for nonlinear convex operational theories is developed using a larger
class of morphisms, determined by Br\`{e}gman nonexpansive operations (which
provide a well-behaved family of Mielnik's nonlinear transmitters). As an
application, we derive a range of nonlinear postquantum resource theories
determined in terms of this class of operations.Comment: v2: several corrections and improvements, including an extension to
the postquantum (generally) and JBW-algebraic (specifically) cases, a section
on nonlinear resource theories, and more informative paper's titl
Nonextensive statistical mechanics and central limit theorems II - Convolution of q-independent random variables
In this article we review recent generalisations of the central limit theorem
for the sum of specially correlated (or q-independent) variables, focusing on q
greater or equal than 1. Specifically, this kind of correlation turns the
probability density function known as q-Gaussian, which emerges upon
maximisation of the entropy Sq, into an attractor in probability space.
Moreover, we also discuss a q-generalisation of a-stable Levy distributions
which can as well be stable for this special kind of correlation.Within this
context, we verify the emergence of a triplet of entropic indices which relate
the form of the attractor, the correlation, and the scaling rate, similar to
the q-triplet that connects the entropic indices characterising the sensitivity
to initial conditions, the stationary state, and relaxation to the stationary
state in anomalous systems.Comment: 14 pages, 4 figures, and 1 table. To appear in the Proceedings of the
conference CTNEXT07, Complexity, Metastability and Nonextensivity, Catania,
Italy, 1-5 July 2007, Eds. S. Abe, H.J. Herrmann, P. Quarati, A. Rapisarda
and C. Tsallis (American Institute of Physics, 2008) in pres
Tensorial dynamics on the space of quantum states
A geometric description of the space of states of a finite-dimensional
quantum system and of the Markovian evolution associated with the
Kossakowski-Lindblad operator is presented. This geometric setting is based on
two composition laws on the space of observables defined by a pair of
contravariant tensor fields. The first one is a Poisson tensor field that
encodes the commutator product and allows us to develop a Hamiltonian
mechanics. The other tensor field is symmetric, encodes the Jordan product and
provides the variances and covariances of measures associated with the
observables. This tensorial formulation of quantum systems is able to describe,
in a natural way, the Markovian dynamical evolution as a vector field on the
space of states. Therefore, it is possible to consider dynamical effects on
non-linear physical quantities, such as entropies, purity and concurrence. In
particular, in this work the tensorial formulation is used to consider the
dynamical evolution of the symmetric and skew-symmetric tensors and to read off
the corresponding limits as giving rise to a contraction of the initial Jordan
and Lie products.Comment: 31 pages, 2 figures. Minor correction
Correlating matched-filter model for analysis and optimisation of neural networks
A new formalism is described for modelling neural networks by means of which a clear physical understanding of the network behaviour can be gained. In essence, the neural net is represented by an equivalent network of matched filters which is then analysed by standard correlation techniques. The procedure is demonstrated on the synchronous Little-Hopfield network. It is shown how the ability of this network to discriminate between stored binary, bipolar codes is optimised if the stored codes are chosen to be orthogonal. However, such a choice will not often be possible and so a new neural network architecture is proposed which enables the same discrimination to be obtained for arbitrary stored codes. The most efficient convergence of the synchronous Little-Hopfield net is obtained when the neurons are connected to themselves with a weight equal to the number of stored codes. The processing gain is presented for this case. The paper goes on to show how this modelling technique can be extended to analyse the behaviour of both hard and soft neural threshold responses and a novel time-dependent threshold response is described
Correlating matched-filter model for analysis and optimisation of neural networks
A new formalism is described for modelling neural networks by means of which a clear physical understanding of the network behaviour can be gained. In essence, the neural net is represented by an equivalent network of matched filters which is then analysed by standard correlation techniques. The procedure is demonstrated on the synchronous Little-Hopfield network. It is shown how the ability of this network to discriminate between stored binary, bipolar codes is optimised if the stored codes are chosen to be orthogonal. However, such a choice will not often be possible and so a new neural network architecture is proposed which enables the same discrimination to be obtained for arbitrary stored codes. The most efficient convergence of the synchronous Little-Hopfield net is obtained when the neurons are connected to themselves with a weight equal to the number of stored codes. The processing gain is presented for this case. The paper goes on to show how this modelling technique can be extended to analyse the behaviour of both hard and soft neural threshold responses and a novel time-dependent threshold response is described
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