  In circuit and cavity quantum electrodynamics devices where control qubits
are dispersively coupled to high-quality-factor cavities, characteristic
functions of cavity states can be directly probed with conditional displacement
(CD) gates. In this Letter, I propose a method to certify non-Gaussian
entanglement between cavities using only CD gates and qubit readouts. The CD
witness arises from an application of Bochner's theorem to a surprising
connection between two negativities: that of the reduced Wigner function, and
that of the partial transpose. Non-Gaussian entanglement of some common states,
like entangled cats and photon-subtracted two-mode squeezed vacua, can be
detected by measuring as few as four points of the characteristic function.
Furthermore, the expectation value of the witness is a simultaneous lower bound
to the Wigner negativity volume and a geometric measure of entanglement
conjectured to be the partial transpose negativity. Both negativities are
strong monotones of non-Gaussianity and entanglement, respectively, so the CD
witness provides experimentally accessible lower bounds to quantities related
to these monotones without the need for tomography on the cavity states.
