We review the framework of operator Hilbert space and introduce the one- and two-particle super reduced density matrices (1-SRDMs and 2-SRDMs), as well as the super mutual information (SMI). The eigenvectors of the 1-SRDMs define what we term natural single particle operator bases, which provides a way to compress vibrational and vibronic Hamiltonians with controlled error. The SMI is defined from the operator entanglement entropy of the 1-SRDMs and 2-SRDMs, and captures the correlation between operators acting on different one-mode subspaces, and may be used to reveal and quantify both direct and indirect couplings that might otherwise be difficult to extract. These analysis tools are applied to a set of prototypical vibrational and vibronic Hamiltonians, as well as approximations to the corresponding time-evolution operators. Through this, we demonstrate that: (i) commonly used vibronic Hamiltonians are amenable to extremely high levels of compression without compromising accuracy, and; (ii) SMI analysis can be used to systematically and quantitatively reveal both direct and indirect couplings that might otherwise be difficult to extract, including indirect couplings of vibrational modes via intermediary electronic-vibrational interactions.