Non-negative discord strengthens the subadditivity of quantum entropy functions

The definition of quantum discord is generalized to allow for any concave entropy function, and its non-negativity strengthens the subadditivity condition for that entropy function. In a sense, this condition is intermediate in strength between subadditivity and strong subadditivity, hence called firm subadditivity, allowing one to further classify entropy functions based on whether they satisfy this intermediate condition. It is proven that the quadratic entropy $1-\Tr(\rho^2)$ satisfies the firm subadditivity condition, whereas some other subadditive Tsallis entropies are not firmly subadditive.