  In this work, we consider the phase-space picture of quantum mechanics. We
then introduce non-negative Wigner-like (operational) distributions
\widetilde{\mathcal W}_{rho;alpha}(x,p) corresponding to the density operator
\hat{rho} and being proportional to {W_{rho^(alpha/2)}(x,p)}^2, where
W_{rho}(x,p) denotes the usual Wigner function. In doing so, we utilize the
formal symmetry between the purity measure Tr(rho^2) and its Wigner
representation (2 pi hbar) \int dx dp {W_{rho}(x,p)}^2 and then consider, as a
generalization, such symmetry between the fractional moment
Tr(\hat{rho}^{alpha}) and its Wigner representation (2 pi hbar) \int dx dp
{W_{rho^{alpha/2}}(x,p)}^2. Next, we create a framework that enables explicit
evaluation of the Renyi-Wigner entropies for the classical-like distributions
\widetilde{\mathcal W}_{rho;alpha}(x,p). Consequently, a better understanding
of some non-Gaussian features of a given state rho will be given, by comparison
with the Gaussian state rho_G defined in terms of its Wigner function
W_{rho_G}(x,p) and essentially determined by its purity measure T(rho_G)^2
alone. To illustrate the validity of our framework, we evaluate the
distributions \widetilde{\mathcal W}_{beta;alpha}(x,p) corresponding to the
(non-Gaussian) thermal state rho_{\beta} of a single particle confined by a
one-dimensional infinite potential well with either the Dirichlet or Neumann
boundary condition and then analyze the resulting Renyi entropies. Our
phase-space approach will also contribute to a deeper understanding of
non-Gaussian states and their properties either in the semiclassical limit
(hbar \to 0) or in the high-temperature limit (beta \to 0), as well as enabling
us to systematically discuss the quantal-classical Second Law of Thermodynamics
on the single footing.
