Quantum phase space functions and relations of entropic localisation measures

The concept of quantum phase space offers a view on quantum mechanics, which is different from the standard Hilbert space approach, but which more closely resembles the classical phase space. Due to the properties of quantum mechanics there are several equivalent quantum phase space descriptions, and one cannot always prefer one or another as they all have certain merits and drawbacks. For example, the Husimi-Kano Q function is a probability distribution and thus gives rise to entropic quantities, namely the Renyi-Wehrl entropies, of which several properties are known. The Wigner function, on the other hand, has an easier physical explanation, but may take negative values. In this article, we investigate entropic measures of localisation for a state in quantum phase space by using the Beckner-Brascamp-Lieb inequality to relate different phase-space functions.