  Quantum Stein's Lemma is a cornerstone of quantum statistics and concerns the
problem of correctly identifying a quantum state, given the knowledge that it
is one of two specific states ($\rho$ or $\sigma$). It was originally derived
in the asymptotic i.i.d. setting, in which arbitrarily many (say, $n$)
identical copies of the state ($\rho^{\otimes n}$ or $\sigma^{\otimes n}$) are
considered to be available. In this setting, the lemma states that, for any
given upper bound on the probability $\alpha_n$ of erroneously inferring the
state to be $\sigma$, the probability $\beta_n$ of erroneously inferring the
state to be $\rho$ decays exponentially in $n$, with the rate of decay
converging to the relative entropy of the two states. The second order
asymptotics for quantum hypothesis testing, which establishes the speed of
convergence of this rate of decay to its limiting value, was derived in the
i.i.d. setting independently by Tomamichel and Hayashi, and Li. We extend this
result to settings beyond i.i.d.. Examples of these include Gibbs states of
quantum spin systems (with finite-range, translation-invariant interactions) at
high temperatures.
