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Journal of Physics Conference Series

Phase space regions operators and ISp(2) maps

Demosthenes Ellinas, Anthony J. Bracken

Published Apr 1, 2019• 1 citation
Publisher

Abstract

Region operators (generalized observable) (RO), defined as operator valued probability measures over classical phase space ensuing from quantization of phase space characteristic functions are investigated. The problem of constructing of RO is treated by means of positive trace increasing maps (pti maps). Exemplary domains of RO include: sets of points, line segments and rotated line segments, circles and disks, straight and rotated lines, canonical polygons, squeezed line segments, and lines related to Radon transform in phase space. Phase plane RO constructing pti maps, in their operator sum representation, are generated by elements of the quantum symplectic in-homogeneous group ISp(2). The group ISp(2) and its double cover metaplectic group Mp(2) are the kinematical groups for RO construction on the plane.

DOI
10.1088/1742-6596/1194/1/012033

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Region operators (generalized observable) (RO), defined as operator valued probability measures over classical phase space ensuing from quantization of phase space characteristic functions are investigated. The problem of constructing of RO is treated by means of positive trace increasing maps (pti maps). Exemplary domains of RO include: sets of points, line segments and rotated line segments, circles and disks, straight and rotated lines, canonical polygons, squeezed line segments, and lines related to Radon transform in phase space. Phase plane RO constructing pti maps, in their operator sum representation, are generated by elements of the quantum symplectic in-homogeneous group ISp(2). The group ISp(2) and its double cover metaplectic group Mp(2) are the kinematical groups for RO construction on the plane.
Quasiprobability measures on a canonical phase space give rise through the action of Weyl's quantization map to operator-valued measures and, in particular, to region operators. Spectral properties, transformations, and general construction methods of such operators are investigated. Geometric trace-increasing maps of density operators are introduced for the construction of region operators associated with one-dimensional domains, as well as with two-dimensional shapes (segments, canonical polygons, lattices, etc.). Operational methods are developed that implement such maps in terms of unitary operations by introducing extensions of the original quantum system with ancillary spaces (qubits). Tomographic methods of reconstruction of the Wigner function based on the radon transform technique are derived by the construction methods for region operators. A Hamiltonian realization of the region operator associated with the radon transform is provided, together with physical interpretations.
An integral of the Wigner function of a wavefunction |psi >, over some region S in classical phase space is identified as a (quasi) probability measure (QPM) of S, and it can be expressed by the |psi > average of an operator referred to as the region operator (RO). Transformation theory is developed which provides the RO for various phase space regions such as point, line, segment, disk and rectangle, and where all those ROs are shown to be interconnected by completely positive trace increasing maps. The latter are realized by means of unitary operators in Fock space extended by 2D vector spaces, physically identified with finite dimensional systems. Bounds on QPMs for regions obtained by tiling with discs and rectangles are obtained by means of majorization theory.
The symplectic group Sp(n) acts on phase space while the unitary representation of its double cover, Mp(n), the metaplectic group, acts on functions defined on configuration space. We will construct an extension Mp(n) of Mp(n) acting on square integrable functions on phase space. This is performed using previous results of ours involving explicit expressions of the twisted Weyl symbols of metaplectic operators and Bopp pseudodifferential operators, which are phase space extensions of the usual Weyl operators.
The integral of the Wigner function of a quantum-mechanical system over a region or its boundary in the classical phase plane, is called a quasiprobability integral. Unlike a true probability integral, its value may lie outside the interval [0, 1]. It is characterized by a corresponding selfadjoint operator, to be called a region or contour operator as appropriate, which is determined by the characteristic function of that region or contour. The spectral problem is studied for commuting families of region and contour operators associated with concentric discs and circles of given radius a . Their respective eigenvalues are determined as functions of a , in terms of the Gauss–Laguerre polynomials. These polynomials provide a basis of vectors in a Hilbert space carrying the positive discrete series representation of the algebra su (1, 1) ≈ so (2, 1). The explicit relation between the spectra of operators associated with discs and circles with proportional radii, is given in terms of the discrete variable Meixner polynomials.
Some non-linear generalizations of classical Radon tomography were recently introduced by M. Asorey et al [Phys. Rev. A 77, 042115 (2008), where the straight lines of the standard Radon map are replaced by quadratic curves (ellipses, hyperbolas, circles) or quadratic surfaces (ellipsoids, hyperboloids, spheres). We consider here the quantum version of this novel non-linear approach and obtain, by systematic use of the Weyl map, a tomographic encoding approach to quantum states. Non-linear quantum tomograms admit a simple formulation within the framework of the star-product quantization scheme and the reconstruction formulae of the density operators are explicitly given in a closed form, with an explicit construction of quantizers and dequantizers. The role of symmetry groups behind the generalized tomographic maps is analyzed in some detail. We also introduce new generalizations of the standard singular dequantizers of the symplectic tomographic schemes, where the Dirac delta-distributions of operator-valued arguments are replaced by smooth window functions, giving rise to the new concept of "thick" quantum tomography. Applications for quantum state measurements of photons and matter waves are discussed.