758 research outputs found
Symbolic calculus on the time-frequency half-plane
The study concerns a special symbolic calculus of interest for signal
analysis. This calculus associates functions on the time-frequency half-plane
f>0 with linear operators defined on the positive-frequency signals. Full
attention is given to its construction which is entirely based on the study of
the affine group in a simple and direct way. The correspondence rule is
detailed and the associated Wigner function is given. Formulas expressing the
basic operation (star-bracket) of the Lie algebra of symbols, which is
isomorphic to that of the operators, are obtained. In addition, it is shown
that the resulting calculus is covariant under a three-parameter group which
contains the affine group as subgroup. This observation is the starting point
of an investigation leading to a whole class of symbolic calculi which can be
considered as modifications of the original one.Comment: 25 pages, Latex, minor changes and more references; to be published
in the "Journal of Mathematical Physics" (special issue on "Wavelet and
Time-Frequency Analysis"
Uncertainty principles for magnetic structures on certain coadjoint orbits
By building on our earlier work, we establish uncertainty principles in terms
of Heisenberg inequalities and of the ambiguity functions associated with
magnetic structures on certain coadjoint orbits of infinite-dimensional Lie
groups. These infinite-dimensional Lie groups are semidirect products of
nilpotent Lie groups and invariant function spaces thereon. The recently
developed magnetic Weyl calculus is recovered in the special case of function
spaces on abelian Lie groups.Comment: 19 page
Covariant Affine Integral Quantization(s)
Covariant affine integral quantization of the half-plane is studied and
applied to the motion of a particle on the half-line. We examine the
consequences of different quantizer operators built from weight functions on
the half-plane. To illustrate the procedure, we examine two particular choices
of the weight function, yielding thermal density operators and affine inversion
respectively. The former gives rise to a temperature-dependent probability
distribution on the half-plane whereas the later yields the usual canonical
quantization and a quasi-probability distribution (affine Wigner function)
which is real, marginal in both momentum p and position q.Comment: 36 pages, 10 figure
Smoothed Affine Wigner Transform
We study a generalization of Husimi function in the context of wavelets. This
leads to a nonnegative density on phase-space for which we compute the
evolution equation corresponding to a Schr\"Aodinger equation
Hudson's Theorem for finite-dimensional quantum systems
We show that, on a Hilbert space of odd dimension, the only pure states to
possess a non-negative Wigner function are stabilizer states. The Clifford
group is identified as the set of unitary operations which preserve positivity.
The result can be seen as a discrete version of Hudson's Theorem. Hudson
established that for continuous variable systems, the Wigner function of a pure
state has no negative values if and only if the state is Gaussian. Turning to
mixed states, it might be surmised that only convex combinations of stabilizer
states give rise to non-negative Wigner distributions. We refute this
conjecture by means of a counter-example. Further, we give an axiomatic
characterization which completely fixes the definition of the Wigner function
and compare two approaches to stabilizer states for Hilbert spaces of
prime-power dimensions. In the course of the discussion, we derive explicit
formulas for the number of stabilizer codes defined on such systems.Comment: 17 pages, 3 figures; References updated. Title changed to match
published version. See also quant-ph/070200
Tomograms and other transforms. A unified view
A general framework is presented which unifies the treatment of wavelet-like,
quasidistribution, and tomographic transforms. Explicit formulas relating the
three types of transforms are obtained. The case of transforms associated to
the symplectic and affine groups is treated in some detail. Special emphasis is
given to the properties of the scale-time and scale-frequency tomograms.
Tomograms are interpreted as a tool to sample the signal space by a family of
curves or as the matrix element of a projector.Comment: 19 pages latex, submitted to J. Phys. A: Math and Ge
The Wigner caustic on shell and singularities of odd functions
We study the Wigner caustic on shell of a Lagrangian submanifold L of affine
symplectic space. We present the physical motivation for studying singularities
of the Wigner caustic on shell and present its mathematical definition in terms
of a generating family. Because such a generating family is an odd deformation
of an odd function, we study simple singularities in the category of odd
functions and their odd versal deformations, applying these results to classify
the singularities of the Wigner caustic on shell, interpreting these
singularities in terms of the local geometry of L.Comment: 24 page
Geometrical Structures for Classical and Quantum Probability Spaces
On the affine space containing the space of quantum states of
finite-dimensional systems there are contravariant tensor fields by means of
which it is possible to define Hamiltonian and gradient vector fields encoding
relevant geometrical properties of . Guided by Dirac's analogy
principle, we will use them as inspiration to define contravariant tensor
fields, Hamiltonian and gradient vector fields on the affine space containing
the space of fair probability distributions on a finite sample space and
analyse their geometrical properties.
Most of our considerations will be dealt with for the simple example of a
three-level system.Comment: 16 page
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