  In this paper we discuss local averages of the energy density for the
non-minimally coupled scalar quantum field, extending a previous investigation
of the classical field. By an explicit example, we show that such averages are
unbounded from below on the class of Hadamard states. This contrasts with the
minimally coupled field, which obeys a state-independent lower bound known as a
Quantum Energy Inequality (QEI). Nonetheless, we derive a generalised QEI for
the non-minimally coupled scalar field, in which the lower bound is permitted
to be state-dependent. This result applies to general globally hyperbolic
curved spacetimes for coupling constants in the range $0<\xi\leq 1/4$. We
analyse the state-dependence of our QEI in four-dimensional Minkowski space and
show that it is a nontrivial restriction on the averaged energy density in the
sense that the lower bound is of lower order, in energetic terms, than the
averaged energy density itself.
