# 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.