  Quantum networks play a key role in many scenarios of quantum information
theory. Here we consider the quantum causal networks in the manner of entropy.
First we present a revised smooth max-relative entropy of quantum combs, then
we present a lower and upper bound of a type II error of the hypothesis
testing. Next we present a lower bound of the smooth max-relative entropy for
the quantum combs with asymptotic equipartition. At last, we consider the score
to quantify the performance of an operator. We present a quantity equaling to
the smooth asymptotic version of the performance of a quantum positive
operator.
