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Journal of Physics A Mathematical and Theoretical

Doubling feature of the Wigner function: finite phase space

J. Žák

Published Aug 8, 2011• 19 citations
Publisher

Abstract

The Wigner function has a doubling feature which is expressed in the fact that an x, p-point in the Wigner function leads to a (2x, 2p)-point in the wavefunctions. This feature is of no special consequence in the continuous phase plane. But in the finite phase plane, it is of crucial importance. This doubling feature enables one to define the finite phase plane Wigner function directly from the Wigner function in the continuous phase plane. Bearing in mind the doubling feature of the Wigner function, we define the inversion operator not only around the points of the discrete phase plane lattice but also around their mid-points. Our approach makes it clear why there is a difference in constructing the Wigner function in odd- and even-dimensional phase spaces.

DOI
10.1088/1751-8113/44/34/345305

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The Wigner function has a doubling feature which is expressed in the fact that an x, p-point in the Wigner function leads to a (2x, 2p)-point in the wavefunctions. This feature is of no special consequence in the continuous phase plane. But in the finite phase plane, it is of crucial importance. This doubling feature enables one to define the finite phase plane Wigner function directly from the Wigner function in the continuous phase plane. Bearing in mind the doubling feature of the Wigner function, we define the inversion operator not only around the points of the discrete phase plane lattice but also around their mid-points. Our approach makes it clear why there is a difference in constructing the Wigner function in odd- and even-dimensional phase spaces.
A discussion of discrete Wigner functions in phase space related to mutually unbiased bases is presented. This approach requires mathematical assumptions which limits it to systems with density matrices defined on complex Hilbert spaces of dimension pn where p is a prime number. With this limitation it is possible to define a phase space and Wigner functions in close analogy to the continuous case. That is, we use a phase space that is a direct sum of n two-dimensional vector spaces each containing p2 points. This is in contrast to the more usual choice of a two-dimensional phase space containing p2n points. A useful aspect of this approach is that we can relate complete separability of density matrices and their Wigner functions in a natural way. We discuss this in detail for bipartite systems and present the generalization to arbitrary numbers of subsystems when p is odd. Special attention is required for two qubits (p=2) and our technique fails to establish the separability property for more than two qubits.
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.
A discussion of discrete Wigner functions in phase space related to mutually unbiased bases is presented. This approach requires mathematical assumptions, which limits it to systems with density matrices defined on complex Hilbert spaces of dimension p n where p is a prime number. With this limitation, it is possible to define a phase space and Wigner functions in close analogy to the continuous case. That is, we use a phase space that is a direct sum of n two-dimensional vector spaces each containing p 2 points. This is in contrast to the more usual choice of a two-dimensional phase space containing p 2 n points. A useful aspect of this approach is that we can relate complete separability of density matrices and their Wigner functions in a natural way. We discuss this in detail for bipartite systems and present the generalization to arbitrary numbers of subsystems when p is odd. Special attention is required for two qubits ( p = 2) and our technique fails to establish the separability property for more than two qubits. Finally, we give a brief discussion of Hamiltonian dynamics in the language developed in the paper.
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.
We define a Wigner distribution function for a one-dimensional finite quantum system, in which the position and momentum operators have a finite (multiplicity-free) spectrum. The distribution function is thus defined on discrete phase-space, i.e. on a finite discrete square grid. These discrete Wigner functions possess a number of properties similar to the Wigner function for a continuous quantum system such as the quantum harmonic oscillator. As an example, we consider the so-called su(2) oscillator model in dimension 2j + 1, which is known to tend to the canonical oscillator when j tends to infinity. In particular, we compare plots of our discrete Wigner functions for the su(2) oscillator with the well known plots of Wigner functions for the canonical quantum oscillator. This comparison supports our approach to discrete Wigner functions. 1