  In this work, we analyze the local certification of unitary quantum channels
and von Neumann measurements, which is a natural extension of quantum
hypothesis testing. A particular case of a quantum channel and von Neumann
measurement, operating on two systems corresponding to product states at the
input, is considered. The goal is to minimize the probability of the type II
error, given a specified maximum probability of the type I error, considering
assistance through entanglement. We introduce a new mathematical structure
q-product numerical range, which is a natural generalization of the q-numerical
range, used to obtain result, when dealing with one system. In our findings, we
employ the q-product numerical range as a pivotal tool, leveraging its
properties to derive our results and minimize the probability of type II error
under the constraint of type I error probability. We show a fundamental
dependency: for local certification, the tensor product structure inherently
manifests, necessitating the transition from q-numerical range to q-product
numerical range.
