  The concepts of independence and totalness of subspaces are introduced in the
context of quasi-probability distributions in phase space, for quantum systems
with finite-dimensional Hilbert space. It is shown that due to the
non-distributivity of the lattice of subspaces, there are various levels of
independence, from pairwise independence up to (full) independence. Pairwise
totalness, totalness and other intermediate concepts are also introduced, which
roughly express that the subspaces overlap strongly among themselves, and they
cover the full Hilbert space. A duality between independence and totalness,
that involves orthocomplementation (logical NOT operation), is discussed.
Another approach to independence is also studied, using Rota's formalism on
independent partitions of the Hilbert space. This is used to define
informational independence, which is proved to be equivalent to independence.
As an application, the pentagram (used in discussions on contextuality) is
analyzed using these concepts.
