We study a generalization of the Wigner function to arbitrary tuples of hermitian operators, which is a distribution uniquely characterized by the property that the marginals for all linear combinations of the given operators agree with the quantum mechanical distributions. Its role as a joint quasi-probability distribution is underlined by the property that its support always lies in the set of expectation value tuples of the operators. We characterize the set of singularities and positivity, and provide some basic examples.
Wigner phase space quasi-probability distribution function is a Fourier transform related to a given quantum mechanical wave function. It is shown that for the wave functions of type
ψ(q)=e−aq2ϕ(q), the Wigner function can be defined in terms of differential operators acting on a given function, independently from the integral formula which appears in the standard definition. Gaussian wave packet, harmonic and singular oscillators are given as the examples.
A classical simulation scheme of quantum computation given a restricted set of states and measurements may be---occasionally, but only occasionally---interpreted naturally as a statistical simulation of positive quasi-probability distributions on a phase space. In this dissertation, we explore phase space representations for finite-dimensional quantum systems and their negativities beyond the usual analogues of the Wigner function. The first line of study focuses on a characterization tool for valid quasi-probability distributions of (possibly mixed) quantum states. A quantum generalization of Bochner's theorem from classical probability theory simultaneously characterizes both the set of valid Wigner functions and the subset of positive ones. We extend this theorem to discrete phase spaces based on projective representations of abelian groups, including the discrete Wigner functions of Gross (J. Math. Phys. 47, 122107 (2006)) and Gibbons et al. (Phys. Rev. A 70, 062101 (2004)). The rest of the dissertation is then dedicated to phase space representations covariant with respect to a general Lie group. This means, in particular, that operators that lie in the same orbit of the group have the same amount of negativity in their quasi-probabilities---a natural requirement if one would like to interpret negativity as non-classicality, and the group action as a classical dynamics. We show that the construction due to Brif and Mann (Phys. Rev. A 59, 971 (1999)) of group covariant phase space representations satisfying the so-called Stratonovich-Weyl correspondence is essentially unique, given the group and an input state in the relevant representation. As an application, we construct quasi-probability representations on a compact phase space of fermionic Gaussian states tailored to a classically simulatable problem of fermionic linear optics and find that, in the first non-trivial case of four fermionic modes, the quasi-probabilities of the Gaussian states exhibit a considerable amount of negativity.
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].