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Martingale concentration inequalities constitute a powerful mathematical toolin the analysis of problems in a wide variety of fields ranging fromprobability and statistics to information theory and machine learning. Here weapply techniques borrowed from this field to quantum hypothesis testing, whichis the problem of discriminating quantum states belonging to two differentsequences $\{\rho_n\}_{n}$ and $\{\sigma_n\}_n$. We obtain upper bounds on thefinite blocklength type II Stein- and Hoeffding errors, which, for i.i.d.states, are in general tighter than the corresponding bounds obtained byAudenaert, Mosonyi and Verstraete [Journal of Mathematical Physics, 53(12),2012]. We also derive finite blocklength bounds and moderate deviation resultsfor pairs of sequences of correlated states satisfying a (non-homogeneous)factorization property. Examples of such sequences include Gibbs states of spinchains with translation-invariant finite range interaction, as well as finitelycorrelated quantum states. We apply our results to find bounds on the capacityof a certain class of classical-quantum channels with memory, which satisfy aso-called channel factorization property- both in the finite blocklength andmoderate deviation regimes.

Finite blocklength and moderate deviation analysis of hypothesis testing  of correlated quantum states and application to classical-quantum channels  with memory