  Recall the classical hypothesis testing setting with two convex sets of
probability distributions P and Q. One receives either n i.i.d. samples from a
distribution p in P or from a distribution q in Q and wants to decide from
which set the points were sampled. It is known that the optimal exponential
rate at which errors decrease can be achieved by a simple maximum-likelihood
ratio test which does not depend on p or q, but only on the sets P and Q.
  We consider an adaptive generalization of this model where the choice of p in
P and q in Q can change in each sample in some way that depends arbitrarily on
the previous samples. In other words, in the k'th round, an adversary, having
observed all the previous samples in rounds 1,...,k-1, chooses p_k in P and q_k
in Q, with the goal of confusing the hypothesis test. We prove that even in
this case, the optimal exponential error rate can be achieved by a simple
maximum-likelihood test that depends only on P and Q.
  We then show that the adversarial model has applications in hypothesis
testing for quantum states using restricted measurements. For example, it can
be used to study the problem of distinguishing entangled states from the set of
all separable states using only measurements that can be implemented with local
operations and classical communication (LOCC). The basic idea is that in our
setup, the deleterious effects of entanglement can be simulated by an adaptive
classical adversary.
  We prove a quantum Stein's Lemma in this setting: In many circumstances, the
optimal hypothesis testing rate is equal to an appropriate notion of quantum
relative entropy between two states. In particular, our arguments yield an
alternate proof of Li and Winter's recent strengthening of strong subadditivity
for quantum relative entropy.
