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We prove a general class of continuous variable entanglement criteria basedon the Husimi $Q$-distribution, which represents a quantum state in canonicalphase space, by employing a theorem by Lieb and Solovej. We discuss theirgenerality, which roots in the possibility to optimize over the set of concavefunctions, from the perspective of continuous majorization theory and show thatwith this approach families of entropic as well as second moment criteriafollow as special cases. All derived criteria are compared to correspondingmarginal based criteria and the strength of the phase space approach isdemonstrated for a family of prototypical example states where only ourcriteria flag entanglement. Further, we explore their optimization prospects intwo experimentally relevant scenarios characterized by sparse data: finitedetector resolution and finite statistics. In both scenarios optimization leadsto clear improvements enlarging the class of detected states and thesignal-to-noise ratio of the detection, respectively.

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