  It is well known that, due to the uncertainty principle, the Planck constant
sets a resolution boundary in phase space and the resulting trade-off in
resolution between incompatible measurements has been thoroughly investigated.
It is also known that, in the classical regime, sufficiently coarse
measurements of position and momentum can simultaneously be determined.
However, the picture of how the uncertainty principle gradually disappears as
we transition from the quantum to the classical regime is not so vivid. In the
present work we clarify this picture by studying the associated probabilities
that quantify the effects of the uncertainty principle in the framework of
finite-dimensional quantum mechanics on a lattice. We also study how these
probabilities are perturbed by the granularity of the lattice and show that
they can signal the discontinuity of the underlying space.
