Tom Weinstein wrote:

We used this formulation of entropy in Statistical Mechanics. It's especially useful in Quantum Thermo where you can actually enumerate all of the states instead of relying on probabilistic arguments.

Sure, this formulation can be used. As a pedagogic tool for explaining what a theory is all about, many formulations are discussed as if they have application in real world situations. Of course (for pedagogic reasons), these discussions focus on systems in which there is a definition, and typically a well-behaved mathematical model, for all of the significant states. Some instructors believe this will assist students in appreciating the concepts of statistical mechanics and quantum thermodynamics. To build a working apparatus (or software systems, as we are discussing here), the designer is typically faced with the breakdown of well-behaved mathematical models. Everything from degenerate states to the "baked in" uncertainty of certain states tends to undermine the mathematical foundations of a theorist's constructions. Of course the theoretical models are absolutely critical, but the designer must always caution themselves against drawing inferences without measurements and clearly stated rationales that speak to these physical realities that lead to mathematical weaknesses. Ultimately, the probabilistic nature of such systems may be "moved around," but not removed from the model! Since the real world of actual measurements interferes with essentially everything we claim to "know" about quantities such as entropy, the real danger is assigning an independent "meaning" to these constructs. Why? Because these quantities do not exist independently, they only exist with respect to our predictive models of a system's behavior. So these models do not really "enumerate" anything about states, but rather restate the probability assumptions of the model in the form of a "working equation." In addition, drawing inferences as to the behavior of systems based on common mathematical form is simply inviting trouble, even at the theoretical level. Mathematical models are not the real world, and the superficial mathematical consistency between say, the functional form of a resonance in a quantum well and a marble in a bowl, does not mean that the marble gives any special insight into the nature of the quantum well. In fact, beyond the curiosity of similar equations, the most important information is in the distinctions and clarifications (emanating from theory) between the systems from a practical, apparatus building, real world perspective (as contrasted with the "everything is just a special case of X" perspective). This danger is also present in designs for sources of entropy to seed RNGs for random data or to create uniformly distributed keys. Well designed models will avoid rephrasing assumptions as conclusions, and will explicitly address the mathematical weaknesses upon which the theoretical arguments in support of the model are ultimately based. dvw