  The task of binary quantum hypothesis testing is to determine the state of a
quantum system via measurements on it, given the side information that it is in
one of two possible states, say $\rho$ and $\sigma$. This task is generally
studied in either the symmetric setting, in which the two possible errors
incurred in the task (the so-called type I and type II errors) are treated on
an equal footing, or the asymmetric setting in which one minimizes the type II
error probability under the constraint that the corresponding type I error
probability is below a given threshold. Here we define a one-parameter family
of binary quantum hypothesis testing tasks, which we call $s$-hypothesis
testing, and in which the relative significance of the two errors are weighted
by a parameter $s$. In particular, $s$-hypothesis testing interpolates
continuously between the regimes of symmetric and asymmetric hypothesis
testing. Moreover, if arbitrarily many identical copies of the system are
assumed to be available, then the minimal error probability of $s$-hypothesis
testing is shown to decay exponentially in the number of copies, with a decay
rate given by a quantum divergence which we denote as $\xi_s(\rho\|\sigma)$,
and which satisfies a host of interesting properties. Moreover, this
one-parameter family of divergences interpolates continuously between the
corresponding decay rates for symmetric hypothesis testing (the quantum
Chernoff divergence) for $s = 1$, and asymmetric hypothesis testing (the
Umegaki relative entropy) for $s = 0$.
