  We study subadditivity of the minimum output entropy ($H_{min}$) of quantum
multiple access channels (MACs). We provide an example of violation of the
additivity theorem for $H_{min}$ known in classical information theory. Our
result is based on a fundamental property of $MACs$ i.e. independence of each
sender. The channels used in the example can be constructed explicitly. On the
basis of subadditivity of $H_{min}$ we also provide an example of extremal
superadditivity (super activation) of the classical capacity region of MACs.
