# Roughness as Classicality Indicator of a Quantum State. (arXiv:1802.02880v1 [quant-ph])

We define a new quantifier of classicality for a quantum state, the
Roughness, which is given by the $\mathcal{L}^2 (\R^2)$ distance between Wigner
and Husimi functions. We show that the Roughness is bounded and therefore it is
a useful tool for comparison between different quantum states for single
bosonic systems. The state classification via the Roughness is not binary, but
rather it is continuous in the interval [0,1], being the state more classic as
the Roughness approaches to zero, and more quantum when it is closer to the
unity. The Roughness is maximum for Fock states when its number of photons is
arbitrarily large, and also for squeezed states at the maximum compression
limit. On the other hand, the Roughness reaches its minimum value for thermal
states at infinite temperature and, more generally, for infinite entropy
states. The Roughness of a coherent state is slightly below one half, so we may
say that it is more a classical state than a quantum one. Another important
result is that the Roughness performs well for discriminating both pure and
mixed states. Since the Roughness measures the inherent quantumness of a state,
we propose another function, the Dynamic Distance Measure (DDM), which is
suitable for measure how much quantum is a dynamics. Using DDM, we studied the
quartic oscillator, and we observed that there is a certain complementarity
between dynamics and state, i.e. when dynamics becomes more quantum, the
Roughness of the state decreases, while the Roughness grows as the dynamics
becomes less quantum.