  A full-rank lattice in the Euclidean space is a discrete set formed by all
integer linear combinations of a basis. Given a probability distribution on
$\mathbb{R}^n$, two operations can be induced by considering the quotient of
the space by such a lattice: wrapping and quantization. For a lattice
$\Lambda$, and a fundamental domain $D$ which tiles $\mathbb{R}^n$ through
$\Lambda$, the wrapped distribution over the quotient is obtained by summing
the density over each coset, while the quantized distribution over the lattice
is defined by integrating over each fundamental domain translation. These
operations define wrapped and quantized random variables over $D$ and
$\Lambda$, respectively, which sum up to the original random variable. We
investigate information-theoretic properties of this decomposition, such as
entropy, mutual information and the Fisher information matrix, and show that it
naturally generalizes to the more abstract context of locally compact
topological groups.
