A number of entropy inequalities have proved useful for quantifying the degree of entanglement of bipartite (or multipartite) quantum states. The problem of determining all of the cases of equality in these inequalities is largely complete but less is known about lower bounds on the deficit in these inequalities in terms of the distance to the set of states for which equality obtains. We prove several theorems giving remainder terms that quantify the degree of inequality in various entropy inequalities in terms of the distance to the states for which equality obtains. We apply them to the problem of quantifying the degree of entanglement in fermionic states, which can never be separable.; i.e., unentangled, in the usual sense. For fermionic states, one may expect reduced density matrices of Slater determinant states to have minimal entanglement, and in certain cases, our results bear out this expectation. This is joint work with Elliott Lieb.