  Entropy is the distinguishing and most important concept of our efforts to
understand and regularize our observations of a very large class of natural
phenomena, and yet, it is one of the most contentious concepts of physics. In
this article, we review two expositions of thermodynamics, one without
reference to quantum theory, and the other quantum mechanical without
probabilities of statistical mechanics. In the first, we show that entropy is
an inherent property of any system in any state, and that its analytical
expression must conform to eight criteria. In the second, we recognize that
quantum thermodynamics: (i) admits quantum probabilities described either by
wave functions or by nonstatistical density operators; and (ii) requires a
nonlinear equation of motion that is delimited by but more general than the
Schroedinger equation, and that accounts for both reversible and irreversible
evolutions of the state of the system in time. Both the more general quantum
probabilities, and the equation of motion have been defined, and the three laws
of thermodynamics are shown to be theorems of this equation.
