  To unequivocally distinguish the genuine quantumness from classicality, a
widely adopted approach appeals to the negativity within a join
quasi-distribution representation as a compelling evidence for the nonclassical
essence. However, to construct a joint quasi-distribution with negativity from
experimental data typically proves to be highly cumbersome. Here we propose a
computational approach utilizing a deep generative model integrated with color
mapping to construct the bivariate joint quasi-distribution functions by
processing three marginals. We first apply our model to predict the Wigner
functions subject to thermal noises. Our model successfully predicts the Wigner
functions with a prominent accuracy by processing three marginals of
probability distributions. We also tackle a challenging problem of the
canonical Hamiltonian ensemble representation (CHER), which is developed for
characterizing the dynamical process nonclassicality. Furthermore, we also
design optimal synthetic datasets to train the model for overcoming the
ground-truth deficiency of the CHER problem. While trained with synthetic data,
the physics-informed optimization enables our model to capture the detrimental
effect of the thermal fluctuations on nonclassicality. Our approach also
provides a significant reduction of the experimental efforts of constructing
the Wigner functions of quantum states.
