Phase-space distributions in information theory

We use phase-space distributions – specifically, the Wigner distribution (WD) and Husimi distribution (HD) – to investigate certain information-theoretic measures as descriptors for a given system. We extensively investigate and analyse Shannon, Wehrl and Rényi entropies, their divergences, mutual information and other correlation measures within the context of these phase-space distributions. The analysis is illustrated with an anharmonic oscillator and is studied with respect to the perturbation parameter (\(\lambda \)) and states (n). The entropies associated with WD are observed to be lower than those of HD, which aligns with the findings regarding the marginals. Moreover, the real components of the entropies associated with WD tend to approach the entropic uncertainty bound more closely than those of the corresponding HD. Moreover, we quantify the precise amount of information lost when opting for HD over WD for characterising the specified system. Since it is not always positive definite, the entropies cannot always be defined.