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Chinese Physics B

Radon transforms of the Wigner operator on hyperplanes

Junhua Chen, Fan Hong‐Yi

Published Sep 1, 2009• 6 citations
Publisher

Abstract

The generalization of tomographic maps to hyperplanes is considered. We find that the Radon transform of the Wigner operator in multi-dimensional phase space leads to a normally ordered operator in binomial distribution—a mixed-state density operator. Reconstruction of the Wigner operator is also feasible. The normally ordered form and the Weyl ordered form of the Wigner operator are used in our derivation. The operator quantum tomography theory is expressed in terms of some operator identities, with the merit of revealing the essence of the theory in a simple and concise way.

DOI
10.1088/1674-1056/18/9/018

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The generalization of tomographic maps to hyperplanes is considered. We find that the Radon transform of the Wigner operator in multi-dimensional phase space leads to a normally ordered operator in binomial distribution—a mixed-state density operator. Reconstruction of the Wigner operator is also feasible. The normally ordered form and the Weyl ordered form of the Wigner operator are used in our derivation. The operator quantum tomography theory is expressed in terms of some operator identities, with the merit of revealing the essence of the theory in a simple and concise way.
The two-dimensional Radon transform of the Wigner quasiprobability is introduced in canonical form and the functions playing a role in its inversion are discussed. The transformation properties of this Radon transform with respect to displacement and squeezing of states are studied and it is shown that the latter is equivalent to a symplectic transformation of the variables of the Radon transform with the contragredient matrix to the transformation of the variables in the Wigner quasiprobability. The reconstruction of the density operator from the Radon transform and the direct reconstruction of its Fock-state matrix elements and of its normally ordered moments are discussed. It is found that for finite-order moments the integration over the angle can be reduced to a finite sum over a discrete set of angles. The reconstruction of the Fock-state matrix elements from the normally ordered moments leads to a new representation of the pattern functions by convergent series over even or odd Hermite polynomials which is appropriate for practical calculations. The structure of the pattern functions as first derivatives of the products of normalizable and non-normalizable eigenfunctions to the number operator is considered from the point of view of this new representation.
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.
Using the normally ordered Gaussian form of the Wigner operator we recapitulate the quantum phase space representation, we derive a new formula for searching for the classical correspondence of quantum mechanical operators; we also show that if there exists the eigenvector |q⟩ λ,ν of linear combination of the coordinate and momentum operator, (λQ + νP), where λ, ν are real numbers, and |q⟩ λν is complete, then the projector |q⟩ λν ⟨q| must be the Radon transform of Wigner operator. This approach seems concise and physical appealing.