  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.
