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 general properties of this measure of non-classicality, and use it to identify for a given dimension of Hilbert space what are the "Queen of Quantum" states, i.e. the most non-classical quantum states. In three dimensions we obtain the Queen of Quantum state analytically and show that it is unique up to rotations. In up to 11-dimensional Hilbert spaces, we find the Queen of Quantum 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.
Comment: 23 pages, 4 figures