Dynamically equivalent states and entanglement entropy

The entanglment entropy of spatial regions has been a very fruitful quantity to study in quantum field theories. In this work, I study an equivalence relation between states supported in the exterior of a given region, based on how they affect the dynamics of the interior for a given amount of time. I show how this equivalence can be formulated in a way similar to quantum field theories possessing a gauge symmetry. The resulting modified entanglement entropy can be interpreted as a measure of local complexity. Explicit results are given for some example systems and possible applications indicated.