66,320 research outputs found
Tomographic map within the framework of star-product quantization
Tomograms introduced for the description of quantum states in terms of
probability distributions are shown to be related to a standard star-product
quantization with appropriate kernels. Examples of symplectic tomograms and
spin tomograms are presented.Comment: LATEX plus sprocl.sty, to appear in the Proceedings of the conference
``Quantum Theory and Symmetries'' (Krakow, July 2001), World Scietifi
Characteristic Kernels and Infinitely Divisible Distributions
We connect shift-invariant characteristic kernels to infinitely divisible
distributions on . Characteristic kernels play an important
role in machine learning applications with their kernel means to distinguish
any two probability measures. The contribution of this paper is two-fold.
First, we show, using the L\'evy-Khintchine formula, that any shift-invariant
kernel given by a bounded, continuous and symmetric probability density
function (pdf) of an infinitely divisible distribution on is
characteristic. We also present some closure property of such characteristic
kernels under addition, pointwise product, and convolution. Second, in
developing various kernel mean algorithms, it is fundamental to compute the
following values: (i) kernel mean values , , and
(ii) kernel mean RKHS inner products , for probability measures . If , and
kernel are Gaussians, then computation (i) and (ii) results in Gaussian
pdfs that is tractable. We generalize this Gaussian combination to more general
cases in the class of infinitely divisible distributions. We then introduce a
{\it conjugate} kernel and {\it convolution trick}, so that the above (i) and
(ii) have the same pdf form, expecting tractable computation at least in some
cases. As specific instances, we explore -stable distributions and a
rich class of generalized hyperbolic distributions, where the Laplace, Cauchy
and Student-t distributions are included
Mutually unbiased bases: tomography of spin states and star-product scheme
Mutually unbiased bases (MUBs) are considered within the framework of a
generic star-product scheme. We rederive that a full set of MUBs is adequate
for a spin tomography, i.e. knowledge of all probabilities to find a system in
each MUB-state is enough for a state reconstruction. Extending the ideas of the
tomographic-probability representation and the star-product scheme to
MUB-tomography, dequantizer and quantizer operators for MUB-symbols of spin
states and operators are introduced, ordinary and dual star-product kernels are
found. Since MUB-projectors are to obey specific rules of the star-product
scheme, we reveal the Lie algebraic structure of MUB-projectors and derive new
relations on triple- and four-products of MUB-projectors. Example of qubits is
considered in detail. MUB-tomography by means of Stern-Gerlach apparatus is
discussed.Comment: 11 pages, 1 table, partially presented at the 17th Central European
Workshop on Quantum Optics (CEWQO'2010), June 6-11, 2010, St. Andrews,
Scotland, U
Orthogonal polynomial kernels and canonical correlations for Dirichlet measures
We consider a multivariate version of the so-called Lancaster problem of
characterizing canonical correlation coefficients of symmetric bivariate
distributions with identical marginals and orthogonal polynomial expansions.
The marginal distributions examined in this paper are the Dirichlet and the
Dirichlet multinomial distribution, respectively, on the continuous and the
N-discrete d-dimensional simplex. Their infinite-dimensional limit
distributions, respectively, the Poisson-Dirichlet distribution and Ewens's
sampling formula, are considered as well. We study, in particular, the
possibility of mapping canonical correlations on the d-dimensional continuous
simplex (i) to canonical correlation sequences on the d+1-dimensional simplex
and/or (ii) to canonical correlations on the discrete simplex, and vice versa.
Driven by this motivation, the first half of the paper is devoted to providing
a full characterization and probabilistic interpretation of n-orthogonal
polynomial kernels (i.e., sums of products of orthogonal polynomials of the
same degree n) with respect to the mentioned marginal distributions. We
establish several identities and some integral representations which are
multivariate extensions of important results known for the case d=2 since the
1970s. These results, along with a common interpretation of the mentioned
kernels in terms of dependent Polya urns, are shown to be key features leading
to several non-trivial solutions to Lancaster's problem, many of which can be
extended naturally to the limit as .Comment: Published in at http://dx.doi.org/10.3150/11-BEJ403 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
- …