Is entropy in quantum mechanics emergent or fundamental? Since a quantum mechanical system, even an isolated system containing one particle, can be described by a density matrix, with entropy for the system given by $\langle S\rangle=-k \rho\ln(\rho)$, is not entropy therefore a property of the system like mass or energy?
 A: First of all, unlike mass/energy, the entropy is not an observable. The word "observable" may be understood both as an adjective and as a noun ("an observable"). Entropy isn't an observable because it is not given by a linear operator acting on the Hilbert space (or on the space of density matrices), 
$$ L: |\psi\rangle \to L|\psi\rangle $$
Instead, the entropy (nonlinearly) depends on $\rho$ itself i.e. on the observer's knowledge of the system. Note that $\rho$ generalizes the probability distribution on the phase space in classical physics, and the latter is clearly related to subjective ignorance. It is not an operationally measurable quantity.
The entropy, like the related notion of information, is such a universal concept applicable to any theory (classical or quantum) that it makes no sense to ask whether it is emergent. It is as ill-defined as asking whether Thursdays are green. The adjective "emergent" and its approximately opposite adjective "fundamental" may only be applied to dynamical laws of physics (laws dictating how things evolve in time) and concepts governed by these laws. The entropy also evolves with time but its evolution is just a small aspect of the evolution of more general, system-specific observables such as particles' coordinates or fields.
OK, perhaps I just de facto argued that the entropy is always "emergent" because it is calculated from some more detailed degrees of freedom. But entropy is so conceptually different from other things that may be said to be "emergent" that this label attached to entropy is useless and meaningless. Entropy is "fundamental" in the sense that it is important for the understanding of information and thermodynamics in any system; it is "emergent" because its value always depend on the state of some more detailed, typically microscopic, degrees of freedom, and the "forgetting" of the microscopic details when discussing a notion may be thought to define "emergence".
A: You have provided the von Neumann entropy definition which is derived from its density matrix. I would consider it an intrinsic rather than fundamental property, but this is just semantics.
Some recent work by John Baez has investigated the dynamics of quantum entropy called quantropy.
