  In the problem of asymptotic binary i.i.d. state discrimination, the optimal
asymptotics of the type I and the type II error probabilities is in general an
exponential decrease to zero as a function of the number of samples; the set of
achievable exponent pairs is characterized by the quantum Hoeffding bound
theorem. A super-exponential decrease for both types of error probabilities is
only possible in the trivial case when the two states are orthogonal, and hence
can be perfectly distinguished using only a single copy of the system.
  In this paper we show that a qualitatively different behaviour can occur when
there is correlation between the samples. Namely, we use gauge-invariant and
translation-invariant quasi-free states on the algebra of the canonical
anti-commutation relations to exhibit pairs of states on an infinite spin chain
with the properties that a) all finite-size restrictions of the states have
invertible density operators, and b) the type I and the type II error
probabilities both decrease to zero at least with the speed $e^{-nc\log n}$
with some positive constant $c$, i.e., with a super-exponential speed in the
sample size $n$. Particular examples of such states include the ground states
of the $XX$ model corresponding to different transverse magnetic fields. In
fact, we prove our result in the setting of binary composite hypothesis
testing, and hence it can be applied to prove super-exponential
distinguishability of the hypotheses that the transverse magnetic field is
above a certain threshold vs. that it is below a strictly lower value.
