  Open-air microwave quantum communication and metrology protocols must be able
to transfer quantum resources from a cryostat, where they are created, to an
environment dominated by thermal noise. Indeed, the states carrying such
quantum resources are generated in a cryostat characterized by a temperature
$T_\text{in} \simeq 50 $~mK and an intrinsic impedance $Z_\text{in} = 50 \,
\Omega$. Then, an antenna-like device is required to transfer them with minimal
losses into open air, characterized by an intrinsic impedance of $Z_\text{out}
= 377 \, \Omega$ and a temperature $T_\text{out} \simeq 300$~K. This device
accomplishes a smooth impedance matching between the cryostat and the open air.
Here, we study the transmission of two-mode squeezed thermal states, developing
a technique to design the optimal shape of a coplanar antenna to preserve the
entanglement. Based on a numerical optimization procedure, we find the optimal
shape of the impedance, and we propose a functional ansatz to qualitatively
describe this shape. Additionally, this study reveals that the reflectivity of
the antenna is very sensitive to this shape, so that small changes dramatically
affect the outcoming entanglement, which could have been a limitation in
previous experiments employing commercial antennae. This work is relevant in
the fields of microwave quantum sensing and quantum metrology with special
application to the development of the quantum radar, as well as any open-air
microwave quantum communication protocol.
