  Doubly-parametric quantum transducers, such as electro-opto-mechanical
devices, are quickly approaching quantum operation as decoherence mechanisms
such as thermal noise, loss, and limited cooperativities are improved. These
devices show potential as the critical link between quantum information
contained at frequencies as disparate as those in the optical and microwave
domains, thus enabling applications such as long distance networking of
superconducting quantum computers. However, the requirements on the operating
parameters of the transducers necessary to achieve quantum operation have yet
to be characterized. In this work we find simple, explicit expressions for the
necessary and sufficient conditions under which doubly-parametric transducers
in the resolved-sideband, steady-state limit are capable of entangling optical
and microwave modes. Our analysis treats the transducer as a two-mode bosonic
Gaussian channel capable of both beamsplitter-type and two-mode squeezing-type
interactions between optical and microwave modes. For the beamsplitter-type
interaction, we find parameter thresholds which distinguish regions of the
channel's separability, capacity for bound entanglement, and capacity for
distillable entanglement. By contrast, the two-mode squeezing-type interaction
always produces distillable entanglement with no restrictions on temperature,
cooperativities, or losses. Finally, we find the entanglement breaking
conditions on the reduced one-mode upconversion and downconversion channels
obtained by initializing the unused input with vacuum while ignoring the unused
output. These differences between the entanglement thresholds of the
beamsplitter-type and two-mode squeezing-type interactions are then important
considerations in the construction of larger quantum networks that integrate
multiple transducers.
