Detecting continuous variable entanglement in phase space with the $Q$-distribution

We prove a general class of continuous variable entanglement criteria based on the Husimi $Q$-distribution, which represents a quantum state in canonical phase space, by employing a theorem by Lieb and Solovej. We discuss their generality, which roots in the possibility to optimize over the set of concave functions, from the perspective of continuous majorization theory and show that with this approach families of entropic as well as second moment criteria follow as special cases. All derived criteria are compared to corresponding marginal based criteria and the strength of the phase space approach is demonstrated for a family of prototypical example states where only our criteria flag entanglement. Further, we explore their optimization prospects in two experimentally relevant scenarios characterized by sparse data: finite detector resolution and finite statistics. In both scenarios optimization leads to clear improvements enlarging the class of detected states and the signal-to-noise ratio of the detection, respectively.