  Logical entropy gives a measure, in the sense of measure theory, of the
distinctions of a given partition of a set, an idea that can be naturally
generalized to classical probability distributions. Here, we analyze how
fundamental concepts of this entropy and other related definitions can be
applied to the study of quantum systems, leading to the introduction of the
quantum logical entropy. Moreover, we prove several properties of this entropy
for generic density matrices that may be relevant to various areas of quantum
mechanics and quantum information. Furthermore, we extend the notion of quantum
logical entropy to post-selected systems.
