Information-theoretic significance of the Wigner distribution

Abstract

A coarse-grained Wigner distribution ${p}_{W}(x,\ensuremath{\mu})$ obeying positivity derives out of information-theoretic considerations. Let $p(x,\ensuremath{\mu})$ be the unknown joint probability density function (PDF) on position and momentum fluctuations $x$, $\ensuremath{\mu}$ for a particle in a pure state $\ensuremath{\psi}(x)$. Suppose that the phase part $\ensuremath{\Psi}(x,z)$ of its Fourier transform ${T}_{F}[p(x,\ensuremath{\mu})]\ensuremath{\equiv}\ensuremath{\mid}G(x,z)\ensuremath{\mid}\mathrm{exp}[i\ensuremath{\Psi}(x,z)]$ is constructed as a hologram. (Such a hologram is often used in heterodyne interferometry.) Consider a particle randomly illuminating this phase hologram. Let its two position coordinates be measured. Require that the measurements contain an extreme amount of Fisher information about true position, through variation of the phase function $\ensuremath{\Psi}(x,z)$. The extremum solution gives an output PDF $p(x,\ensuremath{\mu})$ that is the convolution of the Wigner ${p}_{W}(x,\ensuremath{\mu})$ with an instrument function defining uncertainty in either position $x$ or momentum $\ensuremath{\mu}$. The convolution arises naturally out of the approach, and is one dimensional, in comparison with the ad hoc two-dimensional convolutions usually proposed for coarse graining purposes. The output obeys positivity, as required of a PDF, if the one-dimensional instrument function is sufficiently wide. The result holds for a large class of systems: those whose amplitudes $\ensuremath{\psi}(x)$ are the same at their boundaries [examples: states $\ensuremath{\psi}(x)$ with positive parity; with periodic boundary conditions; free particle trapped in a box].