  Since it began \cite{CocconiMorrison}, the search for extraterrestrial
intelligence (SETI) has focused on interstellar \emph{classical} communication.
Recently, Berera \cite{Berera:2020rpl} pointed out that, at certain
frequencies, photon qubits can retain their quantum coherence over interstellar
(and even intergalactic) distances, raising the prospect of interstellar
\emph{quantum} communication. This is an intriguing possibility, since quantum
communication permits certain tasks that would be impossible with classical
communication, and allow exponential speed-ups for others. (We suggest some
motivations in the interstellar context.) But quantum coherence alone is not
sufficient for quantum communication: here, for the first time, we analyze the
\emph{quantum capacity} $Q$ of an interstellar channel. We point out that, to
have non-zero quantum capacity $Q>0$, interstellar communication over a
distance $L$ must use wavelengths $\lambda < 26.5\,cm$ (to avoid depolarization
by the cosmic microwave background), and \emph{enormous} telescopes of
effective diameter $D>0.78\sqrt{\lambda L}$ (to satisfy quantum erasure
constraints). For example, for two telescopes of diameter $D$ on Earth and
Proxima Centauri, this implies $D>100\,km$! This is a technological threshold
that remains to be crossed in order for reliable one-way quantum communication
to become possible, and suggests a fundamental new resolution of the Fermi
paradox.
