  We study circuit complexity for spatial regions in holographic field
theories. We study analogues based on the entanglement wedge of the bulk
quantities appearing in the "complexity = volume" and "complexity = action"
conjectures. We calculate these quantities for one exterior region of an
eternal static neutral or charged black hole in general dimensions, dual to a
thermal state on one boundary with or without chemical potential respectively,
as well as for a shock wave geometry. We then define several analogues of
circuit complexity for mixed states, and use tensor networks to gain intuition
about them. We find a promising qualitative match between the holographic
action and what we call the purification complexity, the minimum number of
gates required to prepare an arbitrary purification of the given mixed state.
On the other hand, the holographic volume does not appear to match any of our
definitions of mixed-state complexity.
