Quantum Multiple Hypothesis Testing Based on a Sequential Discarding Scheme

We consider the quantum multiple hypothesis testing problem, focusing on the case of hypothesis represented by pure states. A sequential adaptive algorithm is derived and analyzed first. This strategy exhibits a decay rate in the error probability with respect to the expected value of measurements greater than the optimal decay rate of the fixed-length methods. A more elaborated scheme is developed next, by serially concatenating multiple implementations of the first scheme. In this case each stage considers as
a priori
hypothesis probability the
a posteriori
probability of the previous stage. We show that, by means of a fixed number of concatenations, the expected value of measurements to be performed decreases considerably. We also analyze one strategy based on an asymptotically large concatenation of the initial scheme, demonstrating that the expected number of measurements in this case is upper bounded by a constant, even in the case of zero average error probability. A lower bound for the expected number of measurements in the zero error probability setting is also derived.