  For any pair of quantum states (the hypotheses), the task of binary quantum
hypotheses testing is to derive the tradeoff relation between the probability
$p_{01}$ of rejecting the null hypothesis and $p_{10}$ of accepting the
alternative hypothesis. The case when both hypotheses are explicitly given was
solved in the pioneering work by Helstrom. Here, instead, for any given null
hypothesis as a pure state, we consider the worst-case alternative hypothesis
that maximizes $p_{10}$ under a constraint on the distinguishability of such
hypotheses. Additionally, we restrict the optimization to separable
measurements, in order to describe tests that are performed locally. The case
$p_{01}=0$ has been recently studied under the name of "quantum state
verification". We show that the problem can be cast as a semi-definite program
(SDP). Then we study in detail the two-qubit case. A comprehensive study in
parameter space is done by solving the SDP numerically. We also obtain
analytical solutions in the case of commuting hypotheses, and in the case where
the two hypotheses can be orthogonal (in the latter case, we prove that the
restriction to separable measurements generically prevents perfect
distinguishability). In regards to quantum state verification, our work shows
the existence of more efficient strategies for noisy measurement scenarios.
