Adversarial Hypothesis Testing and a Quantum Stein's Lemma for Restricted Measurements

Recall the classical hypothesis testing setting with two 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,\ldots,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