Holographic purification complexity

Elena Cáceres, Josiah Couch, Stefan Eccles, Willy Fischler

Abstract:

We study holographic subregion complexity, and its possible connection to purification complexity
suggested recently by Agón et al. In particular, we study the conjecture that subregion complexity is the
purification complexity by considering holographic purifications of a holographic mixed state. We argue
that these include states with any amount of coarse-graining consistent with being a purification of the
mixed state in question, corresponding holographically to different choices of the cutoff surface. We find
that within the complexity ¼ volume and complexity ¼ spacetime volume conjectures, the subregion
complexity is equal to the holographic purification complexity. For complexity ¼ action (CA), the
subregion complexity seems to provide an upper bound on the holographic purification complexity, though
we show cases where this bound is not saturated. One such example is provided by black holes with a large
genus behind the horizon, which were studied by Fu et al. As such, one must conclude that these offending
geometries are not holographic, that CA must be modified, or else that holographic subr