Problems with wormholes which involve arbitrarily small amounts of exotic matter

A quantum inequality bound on the expectation value of the null-contracted stress tensor in an arbitrary Hadamard state is used to obtain constraints on the geometries of traversable wormholes. Particular attention is given to the wormhole models of Visser, Kar, and Dadhich (VKD) and to those of Kuhfittig. These are models which use arbitrarily small amounts of exotic matter for wormhole maintenance. It is shown that macroscopic VKD models are either ruled out or severely constrained, while a recent model of Kuhfittig is shown to be, in fact, non-traversable.