Quantifying quantumness and the quest for Queens of Quantum

We introduce a measure of 'quantumness' for any quantum state in a finite-dimensional Hilbert space, based on the distance between the state and the convex set of classical states. The latter are defined as states that can be written as a convex sum of projectors onto coherent states. We derive the general properties of this measure of non-classicality and use it to identify, for a given dimension of Hilbert space, the 'Queen of Quantum' (QQ) states, i.e. the most non-classical quantum states. In three dimensions, we obtain the QQ state analytically and show that it is unique up to rotations. In up to 11-dimensional Hilbert spaces, we find the QQ states numerically, and show that in terms of their Majorana representation they are highly symmetric bodies, which for dimensions 5 and 7 correspond to Platonic bodies.