Reference-frame-independent approach dimensionality and classicality in quantum theory

What is the dimension of a quantum ensemble? The standard answer is to count how many classical states are superposed, but that depends entirely on the reference frame of the observer. I discuss a different approach to the dimensionality of quantum systems which is independent of the frame in which we view the quantum state. I then take these ideas further and address the question of when a set of quantum states can be considered classical. The standard answer is if they commute. I propose a stronger (frame-independent) notion of classicality and discuss its consequences for the quantumness of noisy ensembles. If time permits, I will also discuss these ideas applied to the classicality of quantum channels.