174 research outputs found
Duality symmetry for star products
A duality property for star products is exhibited. In view of it, known
star-product schemes, like the Weyl-Wigner-Moyal formalism, the Husimi and the
Glauber-Sudarshan maps are revisited and their dual partners elucidated. The
tomographic map, which has been recently described as yet another star product
scheme, is considered. It yields a noncommutative algebra of operator symbols
which are positive definite probability distributions. Through the duality
symmetry a new noncommutative algebra of operator symbols is found, equipped
with a new star product. The kernel of the new star product is established in
explicit form and examples are considered.Comment: 14 pages, no figure
Remarks on the star product of functions on finite and compact groups
Using the formalism of quantizers and dequantizers, we show that the
characters of irreducible unitary representations of finite and compact groups
provide kernels for star products of complex-valued functions of the group
elements. Examples of permutation groups of two and three elements, as well as
the SU(2) group, are considered. The k-deformed star products of functions on
finite and compact groups are presented. The explicit form of the quantizers
and dequantizers, and the duality symmetry of the considered star products are
discussed.Comment: 17 pages, minor changes with respect to the published version of the
pape
Robustness of raw quantum tomography
We scrutinize the effects of non-ideal data acquisition on the homodyne
tomograms of photon quantum states. The presence of a weight function,
schematizing the effects of the finite thickness of the probing beam or
equivalently noise, only affects the state reconstruction procedure by a
normalization constant. The results are extended to a discrete mesh and show
that quantum tomography is robust under incomplete and approximate knowledge of
tomograms.Comment: 7 pages, 1 figure, published versio
Alternative commutation relations, star products and tomography
Invertible maps from operators of quantum obvservables onto functions of
c-number arguments and their associative products are first assessed. Different
types of maps like Weyl-Wigner-Stratonovich map and s-ordered quasidistribution
are discussed. The recently introduced symplectic tomography map of observables
(tomograms) related to the Heisenberg-Weyl group is shown to belong to the
standard framework of the maps from quantum observables onto the c-number
functions. The star-product for symbols of the quantum-observable for each one
of the maps (including the tomographic map) and explicit relations among
different star-products are obtained. Deformations of the Moyal star-product
and alternative commutation relations are also considered.Comment: LATEX plus two style files, to appear in J. Phys.
Density Matrix From Photon Number Tomography
We provide a simple analytic relation which connects the density operator of
the radiation field with the number probabilities. The problem of
experimentally "sampling" a general matrix elements is studied, and the
deleterious effects of nonunit quantum efficiency in the detection process are
analyzed showing how they can be reduced by using the squeezing technique. The
obtained result is particulary useful for intracavity field reconstruction
states.Comment: LATEX,6 pages,accepted by Europhysics Letter
On the nonlinearity interpretation of q- and f-deformation and some applications
q-oscillators are associated to the simplest non-commutative example of Hopf
algebra and may be considered to be the basic building blocks for the symmetry
algebras of completely integrable theories. They may also be interpreted as a
special type of spectral nonlinearity, which may be generalized to a wider
class of f-oscillator algebras. In the framework of this nonlinear
interpretation, we discuss the structure of the stochastic process associated
to q-deformation, the role of the q-oscillator as a spectrum-generating algebra
for fast growing point spectrum, the deformation of fermion operators in
solid-state models and the charge-dependent mass of excitations in f-deformed
relativistic quantum fields.Comment: 11 pages Late
Radon transform on the cylinder and tomography of a particle on the circle
The tomographic probability distribution on the phase space (cylinder)
related to a circle or an interval is introduced. The explicit relations of the
tomographic probability densities and the probability densities on the phase
space for the particle motion on a torus are obtained and the relation of the
suggested map to the Radon transform on the plane is elucidated. The
generalization to the case of a multidimensional torus is elaborated and the
geometrical meaning of the tomographic probability densities as marginal
distributions on the helix discussed.Comment: 9 pages, 3 figure
Beyond the Standard "Marginalizations" of Wigner Function
We discuss the problem of finding "marginal" distributions within different
tomographic approaches to quantum state measurement, and we establish
analytical connections among them.Comment: 12 pages, LaTex, no figures, to appear in Quantum and Semiclass. Op
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