Quantifying Quantumness and the Quest for Queens of Quantum

TL;DR

This paper introduces a measure of 'quantumness' for quantum states based on their distance from classical states, identifies the most non-classical states in various dimensions, and reveals their geometric symmetries.

Contribution

It defines a new quantumness measure, analytically finds the most non-classical states in 3D, and numerically characterizes them in higher dimensions with geometric insights.

Findings

The 'Queen of Quantum' states are highly symmetric and often correspond to Platonic bodies.

The measure effectively distinguishes classical from non-classical states.

Most non-classical states exhibit high symmetry in their Majorana representation.

Abstract

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.