A simple approximation of minimum error probability using trace distance

A simple lower bound on the average error probability using the trace distance is considered. We compute the bound for M-ary PSK and ASK signals and show that the simple bound is better than conventional lower bounds L₂ and L₄, which are defined using trace distances, in almost region of the average number of photons. Then we propose an approximation of the minimum error probability, which is defined as the product of the average number of neighboring signals and the simple lower bound. It is clarified that the approximation is useful for the case of large number of signals and small error probability or large number of photons.