The density matrix is the quantum analogue of a classical probability density.
If you write quantum mechanics from the start in terms of an algebra of observables and density matrices (rather than wave functions and operators), it looks very much like a direct generalization of classical mechanics, in the sense that almost everything is the same except that commutativity of multiplication is lost.
(Indeed, in mathematics, the subject studying the probabilistic aspects of the quantum mechanical formalism is usually called ''noncommutative probability''.)
You can convince yourself of this by looking at my book ''Classical and Quantum Mechanics via Lie algebras'' (http://lanl.arxiv.org/abs/0810.1019), where this approach is followed systematically.
For example (directly relating to your question), in classical probability theory a pure state (with a degenerate distribution in your terminology) is characterized by zero entropy, and in quantum mechanics, a pure state (i.e., one in which the density matrix can be written as $\psi\psi^*$ with a wave function $\psi$ is also characterized by zero entropy. The formula for the entropy is also the same; $S=\langle -\log\rho\rangle$, with expectation taken in the state $\rho$.
Expressed directly in terms of the density matrix, quantum mechanics is governed by the following six axioms and their explanation. (This is taken from the Section ''Postulates for the formal core of quantum mechanics'' of Chapter A1: Fundamental concepts in quantum mechanics of my theoretical physics FAQ.)
Note that the only difference between classical and quantum mechanics in this axiomatic setting is that
the classical case only works with diagonal operators, where all operations happen pointwise on the diagonal elements. Thus multiplication is commutative, and one can identify operators and functions. In particular, the density mattrix degenerates into a probability density.
the quantum case allows for noncommutative operators, hence both observable quantities and the density are (usually infinite-dimensional) matrices.
A1. A generic system (e.g., a 'hydrogen molecule') is defined by
specifying a Hilbert space $K$ and a (densely defined, self-adjoint)
Hermitian linear operator $H$ called the Hamiltonian or the energy.
A2. A particular system (e.g., 'the ion in the ion trap on this
particular desk') is characterized by its state $\rho(t)$
at every time $t \in R$ (the set of real numbers).
Here $\rho(t)$ is a Hermitian, positive semidefinite, linear trace class
operator on $K$ satisfying at all times the condition
$Tr\ \rho(t) = 1$, (normalization)
where $Tr$ denotes the trace.
A3. A system is called closed in a time interval $[t_1,t_2]$
if it satisfies the evolution equation
$d/dt\ \rho(t) = i/\hbar [\rho(t),H] \mbox{ for } t \in [t_1,t_2]$,
and open otherwise. ($\hbar$ is Planck's constant, and is often set
to 1.)
If nothing else is apparent from the context, a system is assumed to
be closed.
A4. Besides the energy $H$, certain other (densely defined, self-adjoint)
Hermitian operators (or vectors of such operators) are distinguished
as observables.
(E.g., the observables for a system of $N$ distinguishable particles
conventionally include for each particle several 3-dimensional vectors:
the position $x^a$, momentum $p^a$, _orbital_angular_momentum_ $L^a$
and the _spin_vector_ (or Bloch vector) $\sigma^a$ of the particle with
label $a$. If $u$ is a 3-vector of unit length then $u \cdot p^a$,
$u \cdot L^a$ and $u \cdot \sigma^a$ define the momentum, orbital angular
momentum, and spin of particle $a$ in direction $u$.)
A5. For any particular system, and for every vector $X$ of observables
with commuting components, one associates a time-dependent monotone
linear functional $\langle \cdot\rangle_t$ defining the expectation
$\langle f(X)\rangle_t:=Tr\ \rho(t) f(X)$
of bounded continuous functions $f(X)$ at time $t$.
(This is equivalent to a multivariate probability measure $d\mu_t(X)$
on a suitable sigma algebra over the spectrum $spec(X)$ of $X$) defined by
$\int d\mu_t(X) f(X) := Tr\ \rho(t) f(X) =\langle f(X)\rangle _t$.
This sigma algebra is uniquely determined.)
A6. Quantum mechanical predictions consist of predicting properties
(typically expectations or conditional probabilities) of the measures
defined in Axiom A5, given reasonable assumptions about the states
(e.g., ground state, equilibrium state, etc.)
Axiom A6 specifies that the formal content of quantum mechanics is
covered exactly by what can be deduced from Axioms A1-A5 without
anything else added (except for restrictions defining the specific
nature of the states and observables), and hence says that
Axioms A1-A5 are complete.
The description of a particular closed system is therefore given by
the specification of a particular Hilbert space in A1, the
specification of the observable quantities in A4, and the
specification of conditions singling out a particular class of
states (in A6). Given this, everything else is determined by the theory,
and hence is (in principle) predicted by the theory.
The description of an open system involves, in addition, the
specification of the details of the dynamical law.
