  We consider a distributed quantum hypothesis testing problem with
communication constraints, in which the two hypotheses correspond to two
different states of a bipartite quantum system, multiple identical copies of
which are shared between Alice and Bob. They are allowed to perform local
operations on their respective systems and send quantum information to Charlie
at limited rates. By doing measurements on the systems that he receives,
Charlie needs to infer which of the two different states the original bipartite
state was in, that is, which of the two hypotheses is true. We prove that the
Stein exponent for this problem is given by a regularized quantum relative
entropy. The latter reduces to a single letter formula when the alternative
hypothesis consists of the products of the marginals of the null hypothesis,
and there is no rate constraint imposed on Bob. Our proof relies on certain
properties of the so-called quantum information bottleneck function.
  The second part of this paper concerns the general problem of finding finite
blocklength strong converse bounds in quantum network information theory. In
the classical case, the analogue of this problem has been reformulated in terms
of the so-called image size characterization problem. Here, we extend this
problem to the classical-quantum setting and prove a second order strong
converse bound for it. As a by-product, we obtain a similar bound for the Stein
exponent for distributed hypothesis testing in the special case in which the
bipartite system is a classical-quantum system, as well as for the task of
quantum source coding with compressed classical side information. Our proofs
use a recently developed tool from quantum functional inequalities, namely, the
tensorization property of reverse hypercontractivity for the quantum
depolarizing semigroup.
