  One of the central problems in quantum theory is to characterize, detect, and
quantify quantumness in terms of classical strategies. Dephasing processes,
caused by non-dissipative information exchange between quantum systems and
environments, provides a natural platform for this purpose, as they control the
quantum-to-classical transition. Recently, it has been shown that dephasing
dynamics itself can exhibit (non)classical traits, depending on the nature of
the system-environment correlations and the related (im)possibility to simulate
these dynamics with Hamiltonian ensembles---the classical strategy. Here we
establish the framework of detecting and quantifying the nonclassicality for
pure dephasing dynamics. The uniqueness of the canonical representation of
Hamiltonian ensembles is shown, and a constructive method to determine the
latter is presented. We illustrate our method for qubit, qutrit, and qubit-pair
pure dephasing and describe how to implement our approach with quantum process
tomography experiments. Our work is readily applicable to present-day quantum
experiments.
