Analysis of finite sample size quantum hypothesis testing via martingale concentration inequalities

Martingale concentration inequalities constitute a powerful mathematical tool in the analysis of problems in a wide variety of fields ranging from probability and statistics to information theory and machine learning. Here we apply such inequalities to finite sample size quantum hypothesis testing, which is the problem of discriminating quantum states belonging to two different sequences {ρn}n and {σn}n, for a finite n. We obtain upper bounds on the type II Steinand Hoeffding errors, which, for i.i.d. states, are in general tighter than the corresponding bounds obtained by Audenaert, Mosonyi and Verstrate in [4]. Moreover, our bounds are also valid for sequences of non-i.i.d. states which satisfy certain conditions. We also use a martingale concentration inequality to obtain an upper bound on the second order asymptotics of the type II error exponent in the setting of quantum Stein’s lemma, for certain classes of states.