Super-exponential distinguishability of correlated quantum states

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