8,353 research outputs found
Computation of the entropy of polynomials orthogonal on an interval.
We give an effective method to compute the entropy for polynomials orthogonal on a segment of the real axis that uses as input data only the coefficients of the recurrence relation satisfied by these polynomials. This algorithm is based on a series expression for the mutual energy of two probability measures naturally connected with the polynomials. The particular case of Gegenbauer polynomials is analyzed in detail. These results are applied also to the computation of the entropy of spherical harmonics, important for the study of the entropic uncertainty relations as well as the spatial complexity of physical systems in central potentials
Spreading lengths of Hermite polynomials
The Renyi, Shannon and Fisher spreading lengths of the classical or
hypergeometric orthogonal polynomials, which are quantifiers of their
distribution all over the orthogonality interval, are defined and investigated.
These information-theoretic measures of the associated Rakhmanov probability
density, which are direct measures of the polynomial spreading in the sense of
having the same units as the variable, share interesting properties: invariance
under translations and reflections, linear scaling and vanishing in the limit
that the variable tends towards a given definite value. The expressions of the
Renyi and Fisher lengths for the Hermite polynomials are computed in terms of
the polynomial degree. The combinatorial multivariable Bell polynomials, which
are shown to characterize the finite power of an arbitrary polynomial, play a
relevant role for the computation of these information-theoretic lengths.
Indeed these polynomials allow us to design an error-free computing approach
for the entropic moments (weighted L^q-norms) of Hermite polynomials and
subsequently for the Renyi and Tsallis entropies, as well as for the Renyi
spreading lengths. Sharp bounds for the Shannon length of these polynomials are
also given by means of an information-theoretic-based optimization procedure.
Moreover, it is computationally proved the existence of a linear correlation
between the Shannon length (as well as the second-order Renyi length) and the
standard deviation. Finally, the application to the most popular
quantum-mechanical prototype system, the harmonic oscillator, is discussed and
some relevant asymptotical open issues related to the entropic moments
mentioned previously are posed.Comment: 16 pages, 4 figures. Journal of Computational and Applied Mathematics
(2009), doi:10.1016/j.cam.2009.09.04
Computing the Entropy of a Large Matrix
Given a large real symmetric, positive semidefinite m-by-m matrix, the goal
of this paper is to show how a numerical approximation of the entropy, given by
the sum of the entropies of the individual eigenvalues, can be computed in an
efficient way. An application from quantum-optics illustrates the new
algorithm
Random Matrix Theory and Entanglement in Quantum Spin Chains
We compute the entropy of entanglement in the ground states of a general
class of quantum spin-chain Hamiltonians - those that are related to quadratic
forms of Fermi operators - between the first N spins and the rest of the system
in the limit of infinite total chain length. We show that the entropy can be
expressed in terms of averages over the classical compact groups and establish
an explicit correspondence between the symmetries of a given Hamiltonian and
those characterizing the Haar measure of the associated group. These averages
are either Toeplitz determinants or determinants of combinations of Toeplitz
and Hankel matrices. Recent generalizations of the Fisher-Hartwig conjecture
are used to compute the leading order asymptotics of the entropy as N -->
infinity . This is shown to grow logarithmically with N. The constant of
proportionality is determined explicitly, as is the next (constant) term in the
asymptotic expansion. The logarithmic growth of the entropy was previously
predicted on the basis of numerical computations and conformal-field-theoretic
calculations. In these calculations the constant of proportionality was
determined in terms of the central charge of the Virasoro algebra. Our results
therefore lead to an explicit formula for this charge. We also show that the
entropy is related to solutions of ordinary differential equations of
Painlev\'e type. In some cases these solutions can be evaluated to all orders
using recurrence relations.Comment: 39 pages, 1 table, no figures. Revised version: minor correction
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