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arXiv:quant-ph/0605113

Geometric approach to the discrete Wigner function

A. B. Klimov, Munoz, C., José Luis Romero

Published May 13, 2006• 2 citations
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Abstract

We analyze the Wigner function constructed on the basis of the discrete rotation and displacement operators labeled with elements of the underlying finite field. We separately discuss the case of odd and even characteristics and analyze the algebraic origin of the non uniqueness of the representation of the Wigner function. Explicit expressions for the Wigner kernel are given in both cases.

DOI
10.48550/arxiv.quant-ph/0605113

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We analyze the Wigner function constructed on the basis of the discrete rotation and displacement operators labeled with elements of the underlying finite field. We separately discuss the case of odd and even characteristics and analyze the algebraic origin of the non uniqueness of the representation of the Wigner function. Explicit expressions for the Wigner kernel are given in both cases.
We study the Wigner Transform (WT) in the case of an infinite-dimensional continuous Hilbert space and a discrete Hilbert space of finite prime-number dimensionality NNN. In the discrete case we define a family of WT's as a function of a phase parameter. It is only for a specific value of the parameter that all the properties we have examined, including the matrix elements of the phase-space operator PPP and the relation between Wigner's function and Kirkwood's distribution, have a parallel in both types of Hilbert space. This close parallel was found constructing the two WT's in terms of a complete set of operators having similar properties, in the sense that the position and momentum operators appear disentangled in the two situations. We give a geometric interpretation of the properties involving the phase-space operator in the discrete case.
A phase-space approach to finite-dimensional systems is developed from basic principles. For a system describable by a Hilbert space of dimension d we define a one-to-one correspondence between operators and functions on a discrete and finite phase space with points valid for any dimension d . The properties fulfilled by this correspondence and its uniqueness are examined. This formalism is applied to the number difference and phase difference of a two-mode field. This case is compared with the marginal distribution for these variables arising from a two-mode Wigner function for number and phase.
The general Weyl–Wigner formalism in finite dimensional phase spaces is investigated. Then this formalism is specified to the case of symmetric ordering of operators in an odd-dimensional Hilbert space. A respective Wigner function on the discrete phase space is found and the limit, when the dimension of Hilbert space tends to infinity, is considered. It is shown that this limit gives the number–phase Wigner function in quantum optics. Analogous results for the “almost” symmetric ordering in an even-dimensional Hilbert space are obtained. Relations between the discrete Wigner functions introduced in our paper and some other discrete Wigner functions appearing in literature are studied.
The Wigner function provides a useful quasiprobability representation of quantum mechanics, with applications in various branches of physics. Many nice properties of the Wigner function are intimately connected with the high symmetry of the underlying operator basis composed of phase point operators: any pair of phase point operators can be transformed to any other pair by a unitary symmetry transformation. We prove that, in the discrete scenario, this permutation symmetry is equivalent to the symmetry group being a unitary 2 design. Such a highly symmetric representation can only appear in odd prime power dimensions besides dimensions 2 and 8. It suffices to single out a unique discrete Wigner function among all possible quasiprobability representations. In the course of our study, we show that this discrete Wigner function is uniquely determined by Clifford covariance, while no Wigner function is Clifford covariant in any even prime power dimension.
A comprehensive study is presented on the Weyl-Wigner formalism for rotation-angle and angular-momentum variables: the elements of kinematics are extended, the elements of dynamics are established, and the implications of rotational perodicity and angular-momentum quantization are investigated. Particular attention is paid to discreteness, and two of its consequences are emphasized: the importance of evenness and oddness, and the need to use two difference operators in a discrete domain, whereas one differential operator suffices in a continuous domain. These consequences are shown to strongly distinguish the Weyl-Wigner formalism for rotation-angle and angular-momentum variables from the well-known Weyl-Wigner formalism for Cartesian-position and linear-momentum variables. The point is made clear that the first of these formalisms cannot be regarded as a trivial and straightforward extension of the second. The rotational Wigner function is derived as the only bilinear form of the state vector that is real, has the natural invariances for rotational motion, and yields the correct distributions for the rotation-angle and angular-momentum variables as well as the appropriate expression for the transition probability between states. The conditions for its uniqueness are thus established. Its properties are described in detail and, in particular, its uniform boundedness is demonstrated. The rotational Wigner function and the associated correspondence between quantum operators and classical-like functions, as well as the relations they obey, are explored and are written so as to clearly exhibit the distinct contributions of evenness and oddness in the discrete domain of the angular-momentum eigenvalues, thus providing a most natural way to account for periodicity.Using the derivative, which acts on the continuous rotation-angle variable, and the forward and backward differences, which act on the discrete angular-momentum variable, the equation of motion for the rotational Wigner function is established. This equation is detailed for the following Hamiltonian forms: those that depend only on the angular-momentum variable, including, in particular, the free rotator, and those that are in the cosine of the rotation-angle variable. It is verified that the Weyl-Wigner formalism for rotation-angle and angular-momentum variables has the correct nonperiodic limit and that it properly reduces to the Weyl-Wigner formalism for Cartesian-position and linear-momentum variables. In order to illustrate the formalism, a careful analysis is carried out for the rotational Wigner function representing the energy eigenstates of a hindered rotator whose Hamiltonian is the sum of a term in the absolute value of the angular-momentum variable with a term in the cosine of the rotation-angle variable. For this hindered rotator, and within the approximation of a large absolute value of the angular-momentum variable, the equation of motion for the rotational Wigner function is solved for its stationary solutions, and the time-independent Schr\"odinger equation is also solved.